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Volume 7 No. V, September 2013

ISSN 0973-9424

INTERNATIONAL JOURNAL OF MATHEMATICAL SCIENCES AND ENGINEERING APPLICATIONS

(IJMSEA)

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ASCENT N PU

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www.ascent-journals.com

INTERNATIONAL JOURNAL OF MATHEMATICAL SCIENCES AND ENGINEERING APPLICATIONS (IJMSEA) ISSN 0973-9424

EDITORIAL COMMITTEE

Dr. S. M. Khairnar (India) Dr. H. M. Srivastava (Canada) Dr. S. K. Lee (Korea) Dr. Khalida Inayat Noor (Pakistan) Dr. Maslina Darus (Malaysia) • Dr. S. R. Kulkarni (India) • Dr. G. K. Srinivasan (India) • Dr. H. Silverman (USA) • Dr. Qaiser Mushtaq (Pakistan) • Dr. T. W. Ma (Australia) • Dr. Kanhaiya Jha (Nepal) • Dr. Suthep Suantai (Thailand) • Dr. David A. Herron (USA) • Dr. R. Michael Porter K. (Mexico) • Dr. Fathia Mohammed Ali Al Samman (Yeman) • Dr. Moustafa El-Shahed (Saudi Arabia) • Dr. Xiang Li (China) • Dr. Muhammad Aslam Noor (Islamabad) • Dr. Mahmoud Abdel-Aty ( Bahrain) • Dr. Haydar Akca (UAE) • Dr. U. R. Karnos (Qatar) • Dr. S. Kouachi (Algeria)

International Journal of Mathematical Sciences and Engineering Applications (IJMSEA) Volume 7 No. V (September 2013)

ISSN 0973 9424

CONTENTS 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20.

E-Dynamic Sir Epidemic Model and Stability Analysis Under... Prasant Kumar Nayak & Bimal Kumar Mishra Image Enhancement Using Laplace and Lehmann-Type Laplace... R. Poornima & V. Saavithri Subclasses of Meromorphically Univalent Functions... R. A. Sukne Subset Cordial Graphs D. K. Nathan & K. Nagarajan On Nonlinear Mixed Functional Integrodifferential Equation With... Machindra B.Dhakne & Poonam S. Bora On the Range RA When A is K ∗ -Paranormal Operator Nuha H. Hamada The Relationship Between Two Different Integral... A. K. Thakur, Ashish Tiwari & Saheb Ram Chandel Reliability Evaluation of Safety-Critical Component... Nikolay Petrov & Angel Tanev Semirings Satisfying the Identity a + b + ab = a K. Mrudula Devi & G. Shobha Latha Inversion of Integral Transform Involving Legendre’s... A. K. Thakur, Kalyani Thakur & Shashi Sukla A Study of Incomplete Generalized Mittag-Lefflerfunctions Rashmi Jain & Padama Kumawat Composition of p- (α, β) - Normal Operators D. Senthilkumar & R. Santhi MHD Peristaltic Motion of a Williamson Fluid Through... S. V. H. N. Krishna Kumari P., V. V. Ravi Kumar, M. V. Ramana Murthy & S. Sreenadh The Disjoint Vertex Covering Number of a Graph V. R. Kulli Integral Representation of Generalized M -Series Kuldeep Singh Gehlot Two-Dimensional Theory of a Flapping Wing Ahmad Ahmadihaghighat & Habib Molaei On Anti-Homomorphism in Fuzzy Ideals of Near-Rings Venkatesh Kulkarni & Shabbir Ahmed New and Old Theories of Flight Ahmad Ahmadihaghighat & Aisan Aribi Trimodal Weakly Blending Maps with Positive... Iftichar Mudhar Al-Sharaa Mathematical Model of Wing Habib Molaei & Ahmad Ahmadihaghighat Continued inside

1-13 15-26 27-42 43-56 57-76 77-81 83-88 89-96 97-104 105-108 109-115 117-122

123-133 135-141 143-150 151-158 159-168 169-175 177-184 185-191

21. 22. 23. 24. 25.

26. 27.

28. 29.

30. 31.

32. 33.

34. 35.

36. 37. 38. 39. 40.

41. 42.

Identifying an Unknown Source in the Poisson Equation... Ai-Lin Qian Fixed Points for Ciric’s Multi-Valued Quasi-... Luljeta Kikina, Kristaq Kikina & Ilir Vardhami On Intuitionistic Fuzzy Magnified Translation in Γ-... N. Meenakumari & S. Sujitha Solving Fully Fuzzy Linear Systems With Trapezoidal... N. Jayanth Karthik & E. Chandrasekaran Degree of Approximation of Continuous Functions of LIP... Ripendra Kumar, B. K. Singh & Aditya Kumar Raghuvanshi Fuzzy β-Subalgebras f β-Algebras M. Abu Ayub Ansari & M. Chandramouleeswaran Degree of Approximation of Function Belonging to... Aditya Kumar Raghuvanshi, B. K. Singh & Ripendra Kumar Intuitionistic Fuzzy Strongly Preopen Sets and... B. Krsteska & H. Snopce Free Convective MHD Jeffrey Fluid Flow Between Two Coaxial... S. Sreenadh, B. Govindarajulu, E. Sudhakara & A. Parandhama The Effects of Mass/Heat, Rotation on the Linearized... A. R. Vijayalkshmi & P. M. Balagondar MHD Flow of an Ionized Gas in a Parallel Plate Channel With... A. Rama Devi, S. Sreenadh, V. Ramesh Babu & E. Sudhakara On Binary Quadratic Equation 2x2 − y 2 + 2(2x − 2y − 1) = 1 P. Thirunavukarasu & S. Sriram Modified Sigmoid Function in Univalent Function Theory O. A. Fadipe-Joseph, A. T. Oladipo & A. Uzoamaka Ezeafulukwe On Binary Quadratic Equation 3x2 − 2y 2 + 6(2x − 2y − 1) = 1 P. Thirunavukarasu & S. Sriram Determination and Stabilization of the Attitude of an... B. B. Salmer´ on-Quiroz, G. Villegas-Medina, S. A. Rodriguez-Paredes, J. R. Aguilar S´ anchez & P. Ni˜ no-Suarez Certain Family of Analytic and Multivalent Function With... Pravin Ganapat Jadhav A Note on Secondary k-Normal Matrices S. Krishnamoorthy & G. Bhuvaneswari Starting One Step, Five Off Steps Stomer Cowell Method By... Adesanya A. Olaide, Abubakar B. Binta & Alkali M. Adamu Model of the Transmission Dynamics of the East African... Inyama, Simeon Chioma Neural, Fuzzy and Econometric Techniques for the Calibration... Kingsley Adjenughwure, George N. Botzoris & Basil K. Papadopoulos Elasticity Based vs Conventional Stress Analysis:... Sunil Bhat A Study of a Rayleigh System A. M. Marin, R. D. Ortiz & J. A. Rodriguez

193-201 203-209 211-224 225-232

233-238 239-249

251-257 259-270

271-281 283-294

295-308 309-312

313-317 319-322

323-341 343-362 363-366 367-375 377-384

385-403 405-414 415-418

International J. of Math. Sci. & Engg. Appls. (IJMSEA) ISSN 0973-9424, Vol. 7 No. V (September, 2013), pp. 1-13

E-DYNAMIC SIR EPIDEMIC MODEL AND STABILITY ANALYSIS UNDER VACCINATION IN COMPUTER NETWORK PRASANT KUMAR NAYAK1 AND BIMAL KUMAR MISHRA2 1 Department of Mathematics, C.V.R.C.E., Bhubaneswar, Orissa-752054, India E-mail: [email protected] 2 Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi - 835 215, India E-mail: [email protected]

Abstract We develop a three dimensional e-epidemic model to control spread of malicious objects in computer network under vaccination. Stability analysis show that the diseases free equilibrium is globally asymptotic if R0V < 1 on the other hand if R0V > 1 then endemic equilibrium appears. Numerical methods are employed to solve and simulate the system of equations. The simulated results may help us to understand the spread and control of malicious objects.

1. Introduction Developments of communication networks have made computers more and more important in our daily life. Different type of communication devices increased human dependence on computers. Thus the indefinitely number of existing malicious codes and their extreme destructivity appear as an important risk factor for large sectors and −−−−−−−−−−−−−−−−−−−−−−−−−−−−

Key Words : Epidemic model, Malicious object, Stability Analysis, Threshold. c http: //www.ascent-journals.com

1

2

PRASANT KUMAR NAYAK & BIMAL KUMAR MISHRA

individuals. Since Transmission of malicious codes in computer network is epidemic in nature, the action of malicious objects throughout a network can be studied by using epidemiological models for disease propagation [1, 2, 3, 16, 26, 27, 28, 30, 33]. Based on the Kermack and McKendrick SIR model [17, 18, 19], dynamical models for malicious objects propagation were proposed, providing estimations for temporal evolutions of infected nodes depending on network parameters considering topological aspects of the network [ 6, 7, 8, 9, 12, 13, 14, 15, 22, 23, 24, 25, 28, 29, 32, 35, 38]. Recently, more research attention has been paid to the combination of virus propagation models and antivirus countermeasures to study the prevalence of virus, for example, virus immunization [1,] quarantine [7], vaccination [10, 11, 14, 15, 20, 21, 34, 36, 37]. Since the malicious objects differ in their attacking behaviour, a non-linear incidence rates can give a reasonable qualitative description of the disease dynamics. Many authors have developed mathematical models with non-linear incidence rate [4, 8, 22, 23, 35]. In a certain sense, the propagation of virtual malicious objects in a system of interacting computers could be compared with a disease transmitted by vectors when dealing with public health. Concerning diseases transmitted by vectors, one has to take into account that the parasites spend part of its lifetime inhabiting the vector, so that the infection switches back and forth between host and vector. Anderson and May [32, 33] discussed the spreading nature of biological viruses, parasites etc. leading to infectious diseases in human population through several epidemic models. 1.1 Key Word Malicious object: is a code that infects computer systems. There are different kinds of malicious objects such as: Worm, Virus, Trojan etc., which differ according to the way they attack computer systems and the malicious actions they perform.

2. Mathematical Model and the Basic Reproduction Number A population size N (t) is partitioned into subclasses of nodes which are susceptible, vaccinated, infected; recover with sizes denoted by S(t), V (t), I(t), R(t) respectively. Our assumptions on the dynamical transfer of the population are depicted in the Figure 1. In this model, the flow of viruses is from class S to class V , class S to class I, class V to class I, class I to class R, class R to class S. The vaccinated nodes again enter into

E-DYNAMIC SIR EPIDEMIC MODEL AND STABILITY...

3

the susceptible class due to the lack of updated anti-virus. The transmission between model classes can be expressed by the following system of differential equations: ds dt

= ∧ + θV − βSI − (γ + µ)S + ωR

dV dt

= γS − (θ + µ)V − σβV I

dI dt

= βSI + σβV I − (µ + α)I − ηI

dR dt

= ηI − (µ + ω)R

dN dt

= ∧ − µN − αI

(1)

2.1 Equilibrium Point The system (1) is defined on the closed, positive invariant set E = {(S, V, I, R); S, V, I, R ≥ 0 : S + V + I + R = N } which has two possible equilibriums, first, the virus free equilibrium, E0 = (S, V, 0, 0) and second, the endemic equilibrium E ∗ = (S ∗ , V ∗ , I ∗ , R∗ ) which is the interior of D and can be obtained by taking all the equations of system (1) equal to zero. Disease free equilibrium E0 = (S, V, 0, 0) =

∧(θ + µ) ∧γ , , 0, 0 . (θ + µ)(γ + µ) − γθ (θ + µ)(γ + µ) − γθ

(2)

2.2 Endemic Equilibrium 0 = ∧ − µN − αI 0 = ∧ + θV + ωR − βSI − (γ + µ)S 0 = γS − (θ + µ)V − σβV I

(3)

0 = βSI + σβV I − (µ + α)I − ηI 0 = ηI − (µ + ω)R µN

= ∧ − αI (4)

ηI

= (µ + ω)R

4


((θ + µ) + σβI)V

= γ(N − I − R)

(µ + α + η)I

= βSI + σβV I = Iβ(N − V (1 − σ) − I − R)

N∗ =

∧−αI ∗ µ

R∗

=

ηI ∗ (µ+ω)

V∗

=

γ(N −I−R) ((θ+µ)+σβI)

= =

γ

““

(5)

” ” ηI ∧−αI −I− (µ+ω) µ ((θ+µ)+σβI)

γ((∧−αI)(µ+ω)−Iµ(µ+ω)−ηIµ) . µ(µ+ω)((θ+µ)+σβI)

Put N, R, V in last equation of (4) we obtain a quadratic equation for the equilibrium values of I of the form AI 2 + βI + C = 0 where A − [σβ 2 αµ + σβ 2 ωα + σβ 2 µ2 + σβ 2 µω + σβ 2 ηω] B = [β + θαµ + βθωα + βθµ2 + βθµω + βαµ2 + βµωα + βµ3 + βµ2 ω + βηωµ −σβ 2 ∧ ω − βγαµ − βγαω − βγµ2 − βµωγ − βηγµ + βσγαµ + βσγαω +βσµ2 γ + βσµωγ + βηγµ − µ3 σβ − µ2 ωσβ − αµ2 σβ − αµωσβ − ηµ2 σβ −ηµωσβ] C = [−βθ ∧ µ − βθω ∧ +βθ2 ηω − β ∧ µ2 − βµω ∧ +β ∧ γµ + βγµω − βσγ ∧ µ −βσγµω − µ3 θ − µ4 − µ2 ωθ − µ3 ω − αµ2 θ − αµ3 − αµωθ − αµ2 ω − ηµ2 θ −ηµ3 − ηµωθ − ηµ2 ω]. Let us first consider the basic reproductive number in the absence of vaccination. The Basic reproductive number of the model without vaccination is R0 =

∧β . µ(µ + α + η)

(6)

The reproductive number of the model with vaccination is R0V =

∧(θ + µ)β ∧γσβ + ((θ + µ)(γ + µ) − γθ)(µ + α + η) ((θ + µ)(γ + µ) − γθ)(µ + α + η)

(7)

5


R0V = R0

(θ + µ) γσ + (θ + γ + µ) (θ + γ + µ)

3. Stability Analysis 3.1 Locally Stability at Disease Free Equilibrium ds dt

= ∧ + θV − βSI − (γ + µ)S + ωR

dV dt

= γS − (θ + µ)V − σβV I

dI dt

= βSI + σβV I − (µ + α)I − ηI

dR dt

= ηI − (µ + ω)R.

(8)

Theorem 1 (local stability) : The disease free equilibrium point of system (8) is locally asymptotically stable, If R0V < 1 and unstable R0V > 1. Linearizing system (8) at disease free equilibrium points we will get  −(γ + µ) θ −βS ω    γ −(θ + µ) −σβV 0     0 0 βS + σβV − (µ + α + η) 0   0

0

η

         

−ω

The Eigen value of above matrix is λ1 = −(µ + ω), λ2 = βS + σβV − (µ + α + η) λ2 = βS + σβV − (µ + α + η) = (µ + α +

η)(R0V

if R0V < 1, λ2 is negative − 1)

And other two Eigen value having negative real part can be finding from the quadratic equation λ2 + aλ + b = 0 where a = (γ + θ + 2µ), b = (γ + µ)(θ + µ) − γθ and a, b > 0 by Hurwitz condition it is stable. Therefore all the Eigen value are negative if R0V < 1. Therefore the system is locally asymptotic stable at disease free equilibrium.

6


3.2 Local Stability at Endemic Equilibrium  −βI+ − (γ + µ) θ −βS+    γ −(θ + µ) − σβI+ −σβV+     βI+ σβI+ βS+ + σβV+ − (µ + α + η)   0

0 

−βS+

0 0

         

−(µ + ω)

η

−βI+ − (γ + µ)

θ

γ

γS+ V

−σβV+

0

βI+

σβI+

0

0

0

0

η

−(µ + ω)

        

ω

ω

         

The characteristic equation of above model is λ4 + (βI+ + γ + θ + σβI+ + 3µ + ω)λ3 +[(βI+ + γ + θ + σβI+ + 2µ)(µ + ω) + σ 2 β 2 V+ I+ + β 2 S+ I+ +θγ(µ + ω) + (βI+ + γ + µ)(θ + µ + σβI+ )]λ2 +[(βI+ + γ + µ)(θ + µ + σβI+ )(µ + ω) + (σ 2 β 2 V+ I+ )(βI+ + γ + 2µ + ω) +θσβ 2 V+ I+ + β 2 γσS+ I+ (θ + 2µ + ω + βσI+ )β 2 S+ I+ + ωβηI+ + θγ(µ + ω)]λ +[(βI+ + γ + µ)(µ + ω)σ 2 β 2 V+ I+ + θσβ 2 V+ (µ + ω)I+ + β 2 S+ γσI+ (µ + ω) +(µ + ω)(θ + µ + σβI+ )β 2 S+ I+ + ωβηI+ (−γσ + θ + µ + σβI+ )] In short we can write the characteristic equation of above model is λ4 + Aλ3 + Bλ2 + Cλ + D = 0

(9)

where A, B, C, D positive constant. A > 0, B > 0, C > 0, D > 0 and by straight forward calculation we will get ABC − C 2 − B D > 0 therefore by Routh -Hurwitz condition the system is locally stable at endemic equilibrium. Another way of proof. Now, (9) can be expressed as a product of two quadratic equations i.e.: f (λ) = (λ2 + bλ + c)(λ2 + dλ + e)

(10)


7

b+D =A (11) c + bd + e = B be + cd = C

ce = D.

Since, the coefficients C and D are generally much smaller than ‘A’ and ‘B’, the quantities d and e in Eq. (11) are much smaller than ‘b’ and ‘c’. Hence, the first approximations of b, c, e and d, denoted by : b1 , c1 , e1 , d1 and are written as b1 ≈ A > 0 c1 ≈ B > 0 D D e1 ≈ ≈ >0 c B BC − AD >0 d1 ≈ B2 Hence

if R0v > 1.

BC − AD D f (λ) ≈ (λ2 + Aλ + B) λ2 + λ = B2 B

are stable. Therefore their product is stable. Hence f (λ) is stable. Proposition 1 : If A and B are two polynomials with real coefficients, then A and B are stable if and only if A · B is stable. [31] 3.3 Global Stability Let us consider the Lyapunov function at the disease free equilibrium is L = S − S0 ln

S V + V − V0 − V0 ln + I. S0 V0

(12)

L0 is negative definite if R0V < 1. For global stability let us consider the Lyapunov function at the endemic equilibrium is L = S − S+ − S+ ln L0 is negative definite if R0V > 1.

S V I + V − V+ − V+ ln + I − I+ ln S+ V+ I+

(13)

8


4. Effect of σ and Endemic Equilibrium Case 1 : When σ = 1. dS dt dV dt dI dt dR dt

= ∧ + θV − βSI − (γ + µ)S + ωR = γS − (θ + µ)V − σβV I = βSI + σβV I − (µ + α)I − ηI = ηI − (µ + ω)R.

Put σ = 1 in last equation of (4). We will get I=

∧β(µ + ω) − (µ + α + η)µ(µ + ω) R0 − 1 = µβ(µ + α + ω + η) + αωβ µβ(µ + α + ω + η) + αωβ

which exists only when R0 > 1. Case 2 : when σ = 0 < σ < 1. Last Equation (4) is (µ + α + η) = βSI + σβV I = Iβ(N − V (1 − σ) − I − R). Put N, R, V in last equation of (4) we will get a quadratic equation having I is the variable AI 2 + BI + C = 0

(14)

where A − [σβ 2 αµ + σβ 2 ωα + σβ 2 µ2 + σβ 2 µω + σβ 2 ηω] B = [β + θαµ + βθωα + βθµ2 + βθµω + βαµ2 + βµωα + βµ3 + βµ2 ω + βηωµ −σβ 2 ∧ ω − βγαµ − βγαω − βγµ2 − βµωγ − βηγµ + βσγαµ + βσγαω +βσµ2 γ + βσµωγ + βηγµ − µ3 σβ − µ2 ωσβ − αµ2 σβ − αµωσβ − ηµ2 σβ −ηµωσβ] C = [−βθ ∧ µ − βθω ∧ +βθ2 ηω − β ∧ µ2 − βµω ∧ +β ∧ γµ + βγµω − βσγ ∧ µ −βσγµω − µ3 θ − µ4 − µ2 ωθ − µ3 ω − αµ2 θ − αµ3 − αµωθ − αµ2 ω − ηµ2 θ −ηµ3 − ηµωθ − ηµ2 ω].


9

Note that B 2 −4AC > 0 when C < 0 and A > 0 one endemic equilibrium when R0V > 1, since there are two real roots and the product of those two roots is negative. On the other hand we can see that C > 0 if R0V < 1. Also Note that there are exactly two changes in the sign of coefficients of equation (11) if coefficient B < 0 and none when B > 0. By Descartes’ rule of signs one can conclude that the maximum number of endemic equilibrium is two when R0V < 1 and B < 0 and that there is no endemic equilibrium when R0V < 1 and B > 0.

5. Numerical Methods and Discussion SVIRS model have been developed for the transmission of malicious objects in computer network. The model has a constant recruitment of the nodes and exponential natural and infection-related death (crashing) of the nodes. Global stability of the unique endemic equilibrium for the epidemic model has been established. Numerical methods are employed to solve the system (1) and the behaviour of the susceptible, vaccinated, infected, And recovered nodes with respect to time are observed which is depicted in Fig. 2, and Fig 3. From Fig. 2, Fig 3 we observes that the system is asymptotically stable. The effect of V on I is also observed and is depicted in figure 4.

Figure 1 : Flow of virus in computer network

10


Figure 2 : SVIR epidemic models with low vaccination

Figure 3 : Dynamical behaviour of the system (1) with the real parameters ∧ = 0.01; β = 0.15; µ = 0.05; γ = 0.85; θ = 0.05; α = 0.035; η = 0.35; σ = 0.10; ω = 0.01.


Figure 4 : Vaccination vs infection

5. Model Parameters Notation N S V I R β η γ θ µ α σ ω R0 R0V

Explanation Total number of node under consideration Number of susceptible at time t Number of vaccinated individuals at time t Number of infective at time t Number of recovered people with immunity at time t Number of new born Contact rate Recovery rate Vaccination rate Rate at which the vaccine wears off Natural death rate that is not related to the disease Disease-related death rate Factor by which the vaccine has the effect of reducing the infection Rate susceptible from recovery class Basic reproductive number Vaccination reproductive number

11

12


References [1] Mishra Bimal Kumar, Saini Dinesh, Mathematical models on computer viruses, Applied Mathematics and Computation, 187 (2007), 929-936. [2] Zou C. C., Gong W. B., Towsley D., Gao L. X., The monitoring and early detection of internet worms, IEEE/ACM Transactions on Networking, 13(5) (2005), 961-974. [3] Moore D., Shannon C., Voelker G. M., Savage S., Internet quarantine: requirements for containing self-propagating code, in: Proceedings of IEEE INFOCOM 2003, IEEE, (April, 2003). [4] Xiao D., Ruan S., Global Analysis of an Epidemic Model with Nonmonotone Incidence Rate, Math. Biosci., 208 (2007), 419-429. [5] Eunha Shim, An Epidemic Model with Immigration of Infectives and Vaccination, The University of British Columbia, (2002). [6] Guihua Li*, Zhen Jin Global stability of a SEIR epidemic model with infectiousforce in latent, infected and immune period Chaos, Solitons and Fractals, 25 (2005), 1177-1184. [7] Hethcote H., Zhein M., Shengbing L., Effects of quarantine in six endemic models for infectious diseases, Math. Biosc., 180 (2002), 141-160. [8] Hethcote H. W., P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Bio., 29 (1991), 271-287. [9] Hethcote H. W., The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. [10] Hai-Feng Huo and Li-Xiang Feng, Global Stability of an Epidemic Model with Incomplete Treatment and Vaccination Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2012, Article ID 530267, 14 pagesdoi:10.1155/2012/530267. [11] Yu Jiang, Huiming Wei, Xinyu Song, Liquan Mei, Guanghui Su, Suizheng Qiu, Global attractivity and permanence of a delayed SVEIR epidemic model with pulse vaccination and saturation incidence, 213 (2009), 312-321. [12] Hale J. K., Ordinary Differential Equations, 2nd ed, Krieger, Basel, (1980). [13] Kephart J. O., A Biologically Inspired Immune System for Computers, Proceedings of International Joint Conference on Artificial Intelligence, (1995). [14] Jianwen Jia Ping Li, Global Analysis of an SVEIR Epidemic Model with Partial Immunity Mathematica Aeterna, 1 (2011), 547-561. [15] Cook K. L., P. van den Driessche, Analysis of SEIRS epidemic model with two delays, J. Math. Bio., 35 (1996), 240-260. [16] Kephart J. O., White SR, Chess DM, Computers and epidemiology, IEEE Spectrum, (1993). [17] Kermack WO, McKendrick AG, Contributions of mathematical theory to epidemics, Proceedings of the Royal Society of London - Series A, 141 (1933), 94-122. [18] Kermack WO, McKendrick AG, Contributions of mathematical theory to epidemics, Proceedings of the Royal Society of London - Series A, 115 (1927), 700-21.


[19] Kermack WO, McKendrick AG, Contributions of mathematical theory to epidemics. Proceedings of the Royal Society of London - Series A, 138(1932), 55-83. [20] Li-Ming Cai, Xue Zhi Li, Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl. Math. Modelling (2008), doi: 10.1016/j.apm.2008.01.005. [21] van der Wal M. F., Diepenmaat A. C. M., Pel J. M., Hirasing R. A., Vaccination rates in a multicultural population Downloaded from adc.bmj.com on December 17, 2012 - Published by group.bmj.com. [22] Alexander M. E., Moghadas S. M., Bifurcation analysis of an SIRS Epidemic model with generalized incidence, SIAM J. Appl. Math., 65 (2005), 1794-1816. [23] Alexander M. E., Moghadas S. M., Periodicity in an epidemic model with a generalized nonlinear incidence, Math. Biosci., 189 (2004), 75-96. [24] Li M. Y., Graff J. R., Wang L. C., Karsai J., Global dynamics of a SEIR model with a varying total population size, Math. Biosc., 160 (1999) 191-213. [25] Li M. Y., Muldowney J. S., Global stability for the SEIR model in epidemiology, Math. Biosc., 125 (1995), 155-164. [26] Keeling M. J., Eames K. T. D., Networks and epidemic models, Journal of the Royal Society Interface, 2(4) (2005), 295-307. [27] Ma. M. Williamson, Leill J., An epidemiological model of virus spread and cleanup, http://www.hpl.hp.com/techreports/, (2003). [28] Newman MEJ, Forrest S, Balthrop J, Email networks and the spread of computer viruses, Physical Review E, 66:035101-1-035101-4, (2002). [29] Ping Yan, Shengqiang Liu, SEIR epidemic model with delay, Journal of Australian Mathematical Society, Series B- Applied Mathematics, 48 (2006), 119134. [30] Piqueira JRC, Navarro BF, Monteiro LHA, Epidemiological models applied to viruses in computer networks, Journal of Computer Science, 1(1) (2005), 31-4. [31] Raymond Seroul Programming for Mathematicians Springer, (2000). [32] Anderson R. M., May R. M., Population Biology of infectious disease I, Nature, 180 (1999), 361-367. [33] May R. M., Lloyd A. L., Infection dynamics on scale-free networks, Physical Review, (2001), 1-3. [34] Datta S., Wang H., The effectiveness of vaccinations on the spread of emailborne computer viruses, in: IEEE CCECE/CCGEI, IEEE, (2005), 219-223. [35] Ruan S., Wang W., Dynamical behavior of an epidemic model with a non linear incidence rate, Journal of Differential Equations, 188 (2003), 135-163. [36] Sylvain Gandon and Troy Day, The evolutionary epidemiology of vaccination, J. R. Soc. Interface, 4 (2007), 803-817, doi: 10.1098/rsif.2006.0207. [37] Xianning Liu, Yasuhiro Takeuchi, Shingo Iwami SVIR epidemic models with vaccination strategies. [38] Michael Y., Smith H., Wang L., Global dynamics of SIER epidemic model with vertical transmission, SIAM Journal of Applied Mathematics, 62(1) (2001), 5869.

13


IMAGE ENHANCEMENT USING LAPLACE AND LEHMANN-TYPE LAPLACE DISTRIBUTIONS R. POORNIMA1 AND V. SAAVITHRI2 1,2 Nehru Memorial College, Puthanampatti, Tiruchirappalli, Tamilnadu, India E-mail: 1 [email protected], 2 [email protected]

Abstract In this work, Laplace distribution and a new form of distribution called the Lehmanntype Laplace Distribution (LLD) are used to process and enhance SAR and medical images. The processed images have better clarity than the original images. The images are also processed using Weibull and Gamma distributions and it is observed that the Laplace family competes with other family of distributions in the process of image enhancement.

1. Introduction Statistical distributions are used by many authors for digital image processing. Among the various statistical distributions, K distribution, Rayleigh, Weibull and Gamma distributions are used in image segmentation of SAR images. K distribution is used by Yanasse et al (1994) for modelling SAR image. In the case of SAR images of oceanic areas, K distribution is used for ship detection by Vachon et al (1997). According to −−−−−−−−−−−−−−−−−−−−−−−−−−−−

Key Words : Image enhancement, Image processing, Laplace distribution, Lehmann-type Laplace distribution, SAR. c http: //www.ascent-journals.com

15

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R. POORNIMA & V. SAAVITHRI

Oliver and Quegan (1998) the Rayleigh, Weibull and Gamma distributions can also be used for the SAR image processing depending on the area of study. The Gamma distribution is used for the segmentation of SAR images by Zaart et al (1999). Ferreira et al (2000), Fernandes (1998) and Macedo et al (2001) used Weibull distribution in the segmentation of SAR images. Eldhuset (1996) generated a new image after using the statistical parameters such as mean and variance. Rocha et al (2006), developed a software for ship detection where some routines were implemented for the use of several statistical distributions for the segmentation of images. The possible targets in SAR images are detected by means of the great difference of values of grey level showed by ships in contrast with the water. In SAR images, the high value grey level points represent ships and low grey level points represent the water. In this paper, the observed images are processed using two types of Laplace family of distributions namely Laplace and Lehmann-type Laplace distribution. For the pixel values of the images, various distributions are fitted by estimating the statistical parameters of the corresponding distributions. In most of the cases, it is observed that the Laplace family processed images are enhanced images and the probable targets are identified clearly. The remaining sections are organized as follows. In Section 2 a new family of Laplace distribution namely Lehmann-type Laplace distribution is defined. Its parameters are estimated using profile likelihood estimation method. In section 3, various digital images are processed by using Weibull, Gamma, Laplace and Lehmann-type Laplace distributions. SAR images and medical images are processed using these distributions and it is observed that in most of the cases Laplace family performs well to give a very good result.

2. Lehmann-type Laplace Distribution (LLD) The density function of the three parameter family of distribution, namely, Lehmanntype Laplace distribution (LLD), (Poornima and Saavithri, 2013), is

fX (x) =

    

α (α+1) 2φ e

   

α φ

1−

“

x−θ φ

”

,

“ ” “ ” θ−x α θ−x 1 φ φ e , 2e

x≤θ (2.1) x≥θ

17

IMAGE ENHANCEMENT USING LAPALCE AND...

where θ ∈ (−∞, ∞) is a location parameter, α > 0 is a weight parameter, φ > 0 is a scale parameter. When α = 1, this form coincides with the distribution obtained by Balakrishnan and Ambagaspitiya (1994). The parameter ‘α’ controls the skewness and kurtosis of the distribution. The cumulative distribution function corresponding to density (2.1) is

FX (x) =

      

(α+1) α 2(α+1) e

“

x−θ φ

”

x≤θ

,

(2.2) α

1−e

“

θ−x φ

”

+

(α+1) α 2(α+1) e

“

θ−x φ

”

, x≥θ

2.1 Profile Likelihood Estimation We estimate the parameters using the Profile likelihood estimation. Based on the observations (x1 , y1 ), (x2 , y2 ), · · · , (xn , yn ), from LLD, the unknown parameters can be obtained by maximizing the log-likelihood function. Let I1 = {i/xi ≤ θ} and I2 = {i/xi > θ}, |I1 | = n1 , |I2 | = n2 and n = n1 + n2 . The likelihood function of the observed data can be Y α (α+1)“ xi −θ ” Y α φ e 1− L= 2φ φ i∈I1

i∈I2

written as “ θ−x ” i i 1 θ−x α φ e φ . e 2

(2.3)

The log-likelihood function of the observed data can be written as θ−xi P 1 φ log L = l = n log α − n1 log 2φ − n2 log φ + log 1 − 2 e i∈I2

+ α+1 φ

P

(xi − θ) +

i∈I1

α φ

P

(2.4)

(θ − xi )

i∈I2

Differentiating equation (2.4) partially with respect to α, φ and equating to zero, we get n n1 X 1 n1 θ n2 θ n2 X 2 ∂l = + − + − =0 ∂α α φ φ φ φ ∂l ∂φ

=

−n1 φ

−

n2 φ

− nφ2 αθ + 2

+

1 2φ2

n2 P

(θ−xi )e

i=1

(α+1) n1 θ φ2

θ−xi φ

θ−xi 1− 12 e φ

+

αn2 X 2 φ2

−

=0

(2.5)

(α+1) n1 X 1 φ2

(2.6)

18


ˆ θˆ Then max log L(α, φ, θ|x) = max[max log L(α, φ|x)] is calculated. The estimators α, ˆ φ, α,θ,φ

θ

α,φ

can be estimated using numerical techniques and MATLAB tools.

3. Image Enhancement The observed image is read by its pixel value matrix. The distribution is fitted to this pixel values in the pixel matrix by estimating the parameters by profile likelihood method. Then the image is enhanced in its spatial domain by the point operation. SAR Images Fig. 1 is the RADARSAT SAR image acquired on December 17, 2003, from the busy harbor of Brazil situated in the area of Santos(SP). (R. F. Rocha, 2006).


19

The image generated by Lehmann-type Laplace distribution is shown in Fig. 2. The black spots are the ships which are the white speckles (ships) in the original image. In this Fig. the target (ships) are in high grey level, almost black and the sea has low grey level. In Fig.3, the inverse of the Lehmann-type Laplace distribution processed image, the ships are clearly visible as white spots. Fig. 4 shows the image processed using Laplace distribution and the inverse of Fig. 4 is Fig. 5. Fig. 6 shows the image processed using Weibull distribution and the inverse of Fig. 6 is Fig. 7. Fig. 8 shows the image processed using Gamma distribution and the inverse of Fig. 8 is Fig. 9. Comparing all the figures we can observe that the image in Fig. 2 is free from noise and is clearer than all the other images. Fig. 10 is the Malaspina Glacier in Alaska. This image was captured by the ERS-1 SAR on July 18, 1992. The glacier has a dark core surrounded by radiating bright lines. The open ocean is at the top of the image. A ship (bright dot) and its dark wake can also be seen.

20


The image generated by Lehmann-type Laplace distribution is shown in Fig. 11. The inverse of the Lehmann-type Laplace distribution processed image is shown in Fig. 12.


21

The image of the ship and its wake is clear in Fig.12. Similarly the ship and its wake are also clear in the images processed by Weibull and Gamma distribution. Segmented SAR Images

Fig. 19 is a part of the image taken from Fig. 1. The selected part is processed using Lehmann-type Laplace distribution, Laplace, Weibull and Gamma distributions.

22


In Figures 20 and 21, which were processed using Lehmann-type Laplace distribution and its inverse image, the ships are clearly visible as black and white spots respectively. Comparing with the other figures from 20 to 27, figures 21 and 25 show good result. Fig 21 is the inverse of the image generated by Lehmann-type Laplace distribution and Fig. 25 is the inverse of the image generated by Weibull distribution. In Fig 28 we see the composition among the images generated by the use of distributions Lehmann-type Laplace and Weibull. We can observe the ships as dark black spots which are clearly visible. In Fig 29 we see the composition among the images generated by the use of distributions Lehmann-type Laplace, Weibull and Gamma. Fig. 29 is clearer than Fig 28. Fig 30 is the composition among the images generated by the use of distributions Lehmann-type Laplace and Gamma.


23

Fig. 31 is a part of the image taken from Fig. 10. Here the image of the ship and its wake are taken. The image is processed using Lehmann-type Laplace, Laplace, Weibull and Gamma distributions. In Figures 34 and 35, which were processed using Laplace distribution and its inverse image, the ship and its wake are clearly visible. Comparing with the other figures from 32 to 39, figures 34, 35 and 32, 33 shows good result. Medical Image : Fig. 40 is the image of an aorta

24

Flower Images



25

In all the above medical image and flower images the inverse of the image processed by LLD is of better quality.

26


In general, we can infer that the images enhanced by Laplace distribution and Lehmanntype Laplace distribution are better than the ones processed by Weibull and Gamma distributions.

References [1] Eldhuset K., An automatic ship and ship wake detection system for spaceborn SAR images in coastal regions. IEEE Transcations on Geoscience and Remote sensing, 34(4) (Jul. 1996), 1010-1018. [2] Fernandes D., Segmentation of SAR Images with Weibull Distribution. Proceedings of the International Geoscience and Remote Sensing Symposium (IGARSS), Seatle, USA, (1998), 1456-1458. [3] Ferreira A. F. C., Fernandes D., Speckle filter for Weibull-distributed SAR images. Proceedings of the International Geoscience and Remote Sensing Symposium (IGARSS), Honolulu, USA, (2000), 12-16. [4] Macedo K. A. C., Fernandes D., Schneider R. Z., Separacao entre alvos naturais e construidos em imagens SAR. Anais do X Simposio Brasileiro de Sensoriamento Remoto, Foz do Iguacu, Brasil, (2001). [5] Oliver C., Quegan S., Understanding Synthetic Aperture Radar Images. Artec House Inc., Norwood, (1998), 479. [6] Poornima R. and Saavithri V., Lehmann-Type Weighted Laplace Distribution. International Journal of Mathematical Sciences & Engineering Applications (IJMSEA) ISSN 0973-9424, 7(1) (January, 2013), 413-423. [7] Rocha R. F., Use of statistical distribution for segmentation of SAR images of oceanic areas, Remote Sensing Division, Hydrographic Navy Center, Av. Barao de Jaceguay s/n, 24048-900, Niteroi, RJ, [email protected] [8] Vachon P. W., Campbell J. W. M., Bjerkelund C. A., Dobson F. W., Rey M. T., Ship detection by the RADARSAT SAR: Validation of Detection Model Predictions. Canadian Journal of Remote Sensing, 23(1) (1997), 48-59. [9] Yanasse C. C. F., Frery A. C., Sant’anna S. J. S., Dutra L. V., On the use of Multilook Amplitude K distribution for SAR image analysis. Proceedings of the International Geoscience and Remote Sensing Symposium (IGARSS), Pasadena, California, USA, (1994). [10] Zaart A. E., Ziou D., Wang S., Jiang Q., Benie G. B., SAR images segmentation using mixture using Gamma Distribution. Vision Interface ’99, Trois-Rivieres, Canada, (1999).


SUBCLASSES OF MEROMORPHICALLY UNIVALENT FUNCTIONS ASSOCIATED WITH A DIFFERENTIAL OPERATOR RAJKUMAR ANANDRAO SUKNE RESEARCH SCHOLAR SHRI JAGDISH PRASAD JHABARMAL TIBREWALA UNIVERSITY,JHUNHUNU,RAJASTHAN. E-mail: [email protected]

Abstract In this paper authors introduced two new subclasses Σ%τ ζξmp (α, β, η) and Σ+ %τ ζξmp (α, β, η) of meromorphically univalent functions which are defined by means of a new differential operator. By making use of the principle of differential subordination, authors investigated several inclusion relationships and properties of certain subclasses which are defined here by means of a differential operator. Some results connected to subordination properties, coefficient estimates, convolution properties, integral representation, distortion theorems are obtained. We also extend the familiar concept of (n, δ) neighborhoods of analytic functions to these subclasses of meromorphically univalent functions.

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Key Words and Phrases : Analytic functions, Meromorphic functions, Univalent functions, Differential operator, Subordination, Neighborhoods and Convolution. c http: //www.ascent-journals.com

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R. A. SUKNE

1. Introduction Let $ be the class of analytic functions in the unit disk U = {z ∈ C : |z| < 1}. ∴ Ω = {w ∈ $ : w(0) = 0 and |w(z)| < 1, z ∈ U }

(1.1.1)

Ω the class of Schwarz functions. For 0 ≤ α < 1 let    p ∈ $ : p(0) = 1  P (α) =   and Rep(z) > α, z ∈ U.

(1.1.2)

Note that P = P (0) is the well known Caratheodory class of functions. It is easy to see that p ∈ P (α) if and only if 9(z) − α ∈ P. 1−α

(1.1.3)

Lemma 1.1.1 : Let p ∈ $. Then p ∈ P (α) if and only if there exists w ∈ Ω such that p(z) =

1 − (4(1 − η)α − 1)w(z) . 1 − w(z)

(1.1.4)

Lemma 1.1.2 (Herglotz formula) : A function p ∈ $ belongs to the class P (α) if and only if there exists a probability measure m(x) on ∂U such that Z p(z) = |x|

1 − (4(1 − η)α − 1)x(z) dµ(x)‘ (z ∈ U ). 1 − x(z)

(1.1.5)

In particular, if g is univalent in U we have the equivalence: f (z) ≺ g(z) if and only if f (0) = g(0) and f (U ) ⊂ g(U ). Let Σ1 denote the class of all meromorphic functions f of the form f (z) = z −1 +

∞ X

ak z k

(ak ≥ 0).

(1.1.6)

k=0

Subclass of Σ1 +consisting of functions of the form f (z) = z −1 +

∞ X

ak z k ≥ 0,

(z ∈ U ∗ ).

(1.1.7)

k=0

A function f ∈ Σ1 is meromorphically univalent starlike of order α (0 ≤ α < 1). If n 0 o (z) −Re zff (z) > α (z ∈ U ). The class of all such functions is denoted by Σ+ 1 (α), ∞ P if f ∈ Σ1 is given by (1.1.6) and g ∈ Σ1 is given by f (z) = z −1 + bk z k . Then k=0

29

SUBCLASSES OF MEROMORPHICALLY UNIVALENT FUNCTIONS...

the Hadamard product (or convolution) of f and g is defined by (f ∗ g)(z) = z −1 + ∞ P ak bk z k ≥ 0 (z ∈ U ∗ ). For a function f ∈ Σ1 , we define the differential operator k=0 m D%τ ζξ1

in the following way 0 1 D%τ ζξ1 f (z) = f (z)D%τ ζξ1 f (z) = D%τ ζξ1 f (z)

= (τ − ξ)(τ − ζ)[z 2 f (z)]00 + +

h

%−(τ −ξ)−%2 (τ −ζ) %

i

(τ −ξ)−%2 (τ −ζ) [z 2 f (z)]0 % z

(1.1.8)

f (z).

and in general i h m−1 m D%τ ζξ1 = D%τ ζξ1 D%τ ζξ1 f (z)

(1.1.9)

m m For our convenience throughout the paper we are using D%τ ζξ1 = Dµ and h4(η − 1)α +

1) = λ, where, 

0 < % ≤ 12 , 0 ≤ ξ < 1, τ ≥ 1, 0 ≤ ζ < 1, 0 ≤ η < 1

 .

 0 ≤ α < 1, 0 < β ≤ 1 and m ∈ N

If the functionf ∈ Σ1 is given by (1.1.6) then, from (1.1.8) and (1.1.9), we obtain Dµm f (z) = z −1 +

∞ X

Φk (%, τ, ζ, ξ, m, 1)ak z k

(m ∈ N, z ∈ U ∗ ),

(1.1.10)

k=0

where, Φkn (%, τ, ζ, ξ, m, h1) = 1 + (k + p)

(τ −ξ)−%2 (τ −ζ) %

iom + (k + p + 1)(τ 2 − τ (ξ + ζ) + ξζ) .

(1.1.11)

From (1.1.10) it follows that Dµm f (z) can be written in terms of convolution as Dµm f (z) = (f ∗ h)(z) where h(z) = z

−1

+

∞ X

Φk (%, τ, ζ, ξ, m, 1)z k .

(1.1.12)

(1.1.13)

k=0

Note that, the case ξ =

1 2

m and τ = ζ of the differential operator D%τ ζξ1 was introduced

by Srivastava and Patel [18]. For differential operator Dµm f (z), we define a new subclass of the function class Σ1 as follows.

30

R. A. SUKNE

Definition 1.1.1 : A function f ∈ Σ1 is said to be in the class Σ%τ ζξm1 (α, βη) if it satisfies the condition m f (z)]0 z[Dµ + 2(1 − η) m f (z) Dµ 0) and decreases with the decreasing values of β (for β < 0). We note that the there is no effect of the Jeffrey parameter on the temperature θ0 . This is because, the temperature θ0 is independent of λ1 . The effect of non-Newtonian behaviour of the Jeffrey parameter λ1 on the temperature may be seen in the first order solution θ1 . We observe from Fig. 9 that for given free convection parameter K, Nusselt number decreases with increasing heat source parameter β at the outer wall when the temperature currents flow from inner wall to outer wall. In view of Fig. 10. we infer that the behavior of N u2 is otherwise owing to an increase in β at the inner wall. The solutions for higher order may be studied using numerical techniques like finite Difference method as exact solutions are complex in such cases.

FREE CONVECTIVE MHD JEFFREY FLUID FLOW BETWEEN...

279

280

S. SREENADH, B. GOVINDARAJULU, E. SUDHAKARA & A. PARANDHAMA

Acknowledgement The author Prof. S. Sreenadh gratefully acknowledges the financial support from UGC, New Delhi through Major Research Project.

References [1] Soundalgekar V. M., Free convection effects on steady MHD flow past a vertical porous plate. I. Fluid Mech., 66 (1974), 541-55. [2] Gupta M., Dubey G. K. and Sharma H. S., Laminar free convection flow with and without heat sources through coaxial circular pipes. Indian Jour. Pure appl. Maths., 10(7) (1979), 792. [3] Goren S. L., On free convective in water at 40 c. chem. Engg. Sci., 20 (1966), 515. [4] Bal Krishan, Gupta G. D. and Sharma G. C., Free convective MHD flow between two coaxial circular pipes. Bull. Cal. Math. Sec., 76 (1984), 315-320. [5] Vajravelu K., Sreenadh S. and Arunachalam P. V., Combined free and forced convection in an inclined channel with permeable boundaries. Journal of Mathematical analysis and Applications, 166(2) (1992), 393-403. [6] Angel M., Pop I. and Hossain M. A., Combined heat and mass transfer on mixed convection flow over a horizontal plate. Applied Mechanics and Engineering, 4 (1999), 773-788. [7] Das U. N., Aziz A. and Ahmed S., Free convective steady flow and heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall with equal transpiration. placeFar East J Appl. Maths., 3(3.3) (1999), 263-286. [8] Deka R. K. Das, Soundalgekar U. N., Sarmah V. M. A. and Takhar H. S., Flow of viscous incompressible fluid past a semi-infinite plate with variable viscosity. Intl. J. of Heat and Technology, 17(3.1) (1999), 71-73.

FREE CONVECTIVE MHD JEFFREY FLUID FLOW BETWEEN...

[9] Chen Chien-Hsin, Combined heat and mass transfer in MHD free convection from a vertical surface with Ohmic heating and viscous dissipation. International Journal of Engineering Science, 42(3.7) (2004), 699-713. [10] Chamkha Ali J., Unsteady MHD convective heat and mass transfer past a semiinfinite vertical permeable moving plate with heat absorption. International Journal of Engineering Science, 42(3.2) (2004), 217-230. [11] Aydin Orhan and Avci, Mete. Laminar forced convection with viscous dissipation in a Couette- Poiseuille flow between parallel plates. Applied Energy, 83(3.8), (2006) 856-867. [12] Krishna Gopal Singha, International Journal of Applied Mathematics and Computation, 1(4) (2009), 183-193. [13] Sreenadh S., Prakash J. and Sudhakara E., MHD free convective flow of Jeffrey fluid between two coaxial impermeable and permeable cylinders. International journal of engineering and interdisplinary mathematics, 4(2) (2012), 13-22.

281


THE EFFETS OF MASS/HEAT, ROTATION ON THE LINEARIZED PERTURBATIONS IN A SHEAR FLOW A. R. VIJAYALAKSHMI1 AND P. M. BALAGONDAR2 1 Department of Mathematics, Maharani’s Science College for women, Bangalore - 560 001, India E-mail: [email protected] 2 Department of Mathematics, Bangalore University, Central College, Bangalore - 560 001, India E-mail: drp [email protected]

Abstract In this paper, we have considered the effects of heat/mass and rotation on the linearized perturbations in a shear flow. We have derived a differential equation in time for the wave amplitudes that is solved for real time, providing the asymptotic behaviour directly. It is found that the asymptotic time behaviour in the inviscid problem is only valid immediately after the initial moment. Diffusion of heat can be destabilizing, where as mass diffusion is stabilizing and viscosity no matter however small, dominates the final period of decay ultimately becomes exponential. The numerical solution for the governing equation is obtained and is compared with the analytical solution for different values of Richardson number, viscosity and rotation parameter.

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Key Words : Mass / Heat, Rotation, Linearized perturbations, Shear flow. 2000 AMS Subject Classification : 76E19, 76E25. c http: //www.ascent-journals.com

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A. R. VIJAYALAKSHMI & P. M. BALAGONDAR

1. Introduction Stratified shear flows are the flows in which the density varies with the height together with velocity and these flows are of interest to meteorology, oceanography and environmental studies. The method of normal modes and the initial-value problem approach are the two elementary methods that have been used for many years in the study of linear stability of shear flows. It is broadly known that the normal mode analysis is the most commonly used method to study the stability of stability of shear flows but it is not completely successful in determining the complete flow for any flow situation. This technique is useful in the local analysis of perturbations but the greatest difficulty is encountered in determining the solution of these equations at the critical levels or the singular levels. The initial-value problem approach is motivated by the failure of linear stability theory at a critical point. The other method to study the linear stability of any given flow is to treat it as an initial-value problem posed mainly by using Fourier transform in space or by using Laplace transform in time. An initial-value problem solved by operational methods is unlikely to miss the continuous spectrum, if it exists, as a modal analysis might. Many researchers have investigated the stability of stratified shear flows. Case (1960) who though unaware of the work of Eliassen et al, showed that the discrete normal modes of the kind considered by Eliassen et al, for J0 >

1 4

do not constitute a mathematically

complete set of functions. He computed the time evolution of initial disturbances using initial-value problem approach by employing Fourier and Laplace transforms. Miller and Gage (1972) have demonstrated instability for a viscous shear layer at low Prandtl number even with J0 > 41 . Maslowe and Thompson (1971) studied the stability of inviscid stratified shear flow with thermal diffusion numerically and found that the diffusive effects are confined to a neighbourhood of critical layer. Hartmann (1975) considered the case of an unbounded stratified fluid and obtained a formal solution to the initial-value problem of Eliassen et al (1953). Knobloch (1984) considered the stability of thermally stratified Couette flow using initial-value problem approach and concluded that the flow is stable against small disturbances. Criminale and Cordova (1986) investigated the effect of diffusion of heat and mass on the stability of unbounded stratified, viscous shear flow using initial-value problem approach as given by Lord Kelvin (1887) for a constant

THE EFFECTS OF MASS/HEAT, ROTATION ON THE...

285

mean shear flow. They included diffusion of heat and mass and have shown that the asymptotic time behaviour in the inviscid problem is only valid immediately after the initial moment, diffusion of heat can cause destabilization whereas diffusion of mass can stabilize the flow. They have observed that viscosity, however small dominates the final period and decays exponentially independent of Richardson number. Damien Biau and Alessandro Bottaro (2004) studied the effect of stable thermal stratification on shear flow stability. They carried out a linear stability analysis in the spatial framework focusing on both exponential and transient growth and found that in both the cases positive thermal stratification stabilizes the disturbance. In the present paper, we have considered the effects of rotation, mass / heat diffusion on the linearized perturbations in a shear flow using initial-value problem approach as done by Criminale and Cordova (1986). Diffusive effects have long been regarded as small in astrophysical settings (Van Kampen and Felderhof, 1957). Our reasons for including them here are (i) they may act to initiate an instability which might not otherwise occur (e.g. McIntyre, 1970), (ii) if an instability is found, one might estimate the size of the anamolous transport coefficients self consistently by replacing the kinematic values by those needeed to saturate the instability, which may not be small and (iii) if no instability is found, an enhancement of dissipative effects may occur for some class of solutions.

2. Mathematical Formulation We consider xyz-cartesian co-ordinate system about the y-axis, which is rotating with →

uniform angular velocity Ω= (0, f, 0). The equations of motion for a viscous, incom→ pressible, thermally stratified fluid moving with velocity q of density ρ and temperature are: →

∇· q = 0, ρ

! → → → → → → ∂ q + ( q ·∇) q −2 Ω × q = −∇p + ρ g , ∂t

(2.1) (2.2)

→ ∂T + ( q ·∇)T = κT ∇2 T, ∂t

(2.3)

ρ = ρ0 {1 − α1 (T − T0 )}

(2.4)

286


where p is the pressure, α1 is the coefficient of thermal expansion. The constant density ρ0 corresponds to some reference temperature T0 . →

→

In the basic state, q 0 = (U (y) = σy, 0, 0), p = p0 (y), T0 = −γ1 y, where q 0 is the flow velocity, p0 is the pressure and T0 is the temperature. The shear σ and thermal gradient γ1 are constants. To study the stability of the given flow, in the linear stability theory →0

we superimpose a small wave like perturbation upon the mean flow q , p0 , T 0 = γ1 θ0 which are the perturbed quantities of their counter parts. The system of linearized equations for the case of constant shear, constant Brunt V¨ ais¨ al¨ a frequency mean flow conditions allows solutions for the perturbations to be written as v(x, y, z, t) = v0 (t) exp[iαx − i(λ + σαT )y + iγz],

(2.5)

with similar expressions for u, w, p and θ which results in the following equations iαu0 + i(λ + σαt)v0 + iγw0 = 0, du0 + σv0 + f w0 = −iαp0 − v(α2 + v 2 + (λ + σαt)2 )u0 , dt dv0 = i(λ + σαt)p0 + N 2 θ − v(α2 + γ 2 + (λ + σαt)2 )v0 , dt dw0 − f u0 = −iγp0 − v(α2 + γ 2 + (λ + σαt)2 )w0 , dt dθ + v0 = −κT (α2 + γ 2 + (λ + σαt)2 )θ0 . dt

(2.6) (2.7) (2.8) (2.9) (2.10)

where N 2 = −α1 γ1 g. N is the Brunt V¨ ais¨ al¨ a frequency and ρ0 is the equilibrium density. The basic features of this type of solution are (i) the solution at time t = 0 implies that the perturbations have been Fourier decomposed in all spatial directions (ii) the solutions are nonseperable and therefore quite different from the normal mode assumption (iii) the governing ordinary differential equations for the time dependent amplitudes are free from singularities in the inviscid limit and (iv) substitution of equation (2.5) into the full nonlinear equation will show that nonlinearity vanishes identically and therefore equation (2.5) is an exact solution to the problem. Using (i) Squire transformation by defining α = (α2 + γ 2 )1/2 and ϕ = arctan(γ/α), the velocity components in the α and ϕ directions are given by u=

αˆ u + γw ˆ −γ u ˆ + αw ˆ , w= α α

(2.11)

287


and (ii) T =

λ α + τ cos ϕ, cos ϕ = , τ = σt, α α

(2.12)

equations (2.6)-(2.10) becomes d N2 fT w0 = 0, [(1 + T 2 )v0 ] + (1 + T 2 )2 v0 − C0 + dt σ cos ϕ σ cos ϕ dw0 f αT 2 v0 , + (1 + T )w0 = tan ϕ + dt σ cos ϕ

(2.13)

(2.14)

dθ0 v0 + (1 + T 2 )T0 = − . dt Pr σ cos ϕ Eliminating w0 and T0 , equations (2.13)-(2.15) becomes h i h J0 +6 1 6T 2 00 v0000 + 1+T 2 + 1 + P r (1 + T ) v0 + 1+T 2 + 6 2 + (1 + T 2 ) − +2 2 + − Pr

1 1+T 2

1 Pr

h

f T tan ϕ σ

(1+3T 2 ) (1+T 2 )

f T tan ϕ σ

−

+

f 2 αT 2 σ2

+

f 2 αT 2 σ2

2 P r 6T (1

i

ii

(2.15)

1 Pr

1 2 PrT

v00 + (J0 + 2) + 22 2 +

+ T 2) +

3 P r (1

+ T 2 )3 −

tan ϕf σ

+ 2 1 +

1 Pr

−

f 2 αΓ σ2

1 Pr

T (1 + T 2 )

1 (1+T 2 )

v0 = 0 (2.16)

A canonical form for equation (2.16) can be obtained by changing variables. Let v0 (T ) = v1 (T )v2 (T ).

(2.17)

Substituting equation (2.17) into equation (2.16) with 1 1 T3 v1 = exp − 2+ T+ , 1 + T2 3 Pr 3

(2.18)

results in the equation v2000 + Ω11 (T, f, Jo, , P r)v20 + Ω12 (T, f, Jo, , P r)v2 = 0,

(2.19)

where J0 2 Ω11 (T, f, Jo, , P r) = − 1 + T2 3

1 1− Pr

2

2 2

(1 + T ) −

fT f 2T 2α + σ σ2

1 . (1 + T 2 ) (2.20)

288


Ω12 (T, f, Jo, , P r) =

(Jo + 2) − 2

2 27 (1

h

+ T 2 )3 −

f T tan ϕ σ

1 (1+T 2 )

i

f 2 αT 2 σ2

+

f tan ϕ σ

+

3

f 2 αT σ2

1−

1 Pr

− 3 1 −

1 3 Pr

(2.21)

The asymptotic behaviour for large T of v2 can be determined from equation (2.19). Inspection of the expressions for Ω11 and Ω12 from equations (2.20) and (2.21) shows that time scales are mixed, indicating that the only meaningful aymptotic limit as t → ∞ is for finite , no matter how small. The interplay can be seen as follows. For T small, equations (2.20) and (2.21) can be approximated as Ω11

Ω12 ≈

3

2 ≈ Jo − 3 1−

−T 2

1 Pr

1 1− Pr

2

(Jo + 2) −

2f 2 α σ2 3

1−

1 Pr

−T

f tan ϕ σ

−

2

−

f tan ϕ σ

! 1 2 f 2α 2 2 Jo + 1 − − 2 . 3 Pr σ 2 3 27

1−

1 3 Pr

− 29 3 1 −

−T

1 3 Pr

2f tan ϕ σ 3

1−

(2.22)

1 Pr

+

f 2α σ2

. (2.23)

Thus except for the case

1−

1 Pr

< 0, solutions of equation (2.19) will always be

damped. If 0 < P r < ∞, finite, the amplitude of v2 will initially increase. Thus viscosity is damping where heat and mass diffusion can be implemented in a negative manner, atleast for a finite time. Since heat is considered, it is possible to have P r < 1 for real fluids. But if mass is considered instead of heat, Schmidth number, Sc < 1 is unlikely to be met by any reasonable physical fluid.

3. Heat Diffusion If we let P r 1 and consider only = kP rn , k = 0(1) and n is a positive integer and is greater than 2. The new variable is found to be T˜ = P r(n−1) /2T ,

(3.1)

by the analysis for Ω11 and Ω12 changing signs, it is found that the lowest order results in the new scale, with the expansion for the amplitude as V0 = V˜0 (T˜) + P r(n−1) /2 V˜1 (T˜) + · · · .

(3.2)

289


The lowest order equation becomes Jo 0 ˜ V0 (T ) − V0000 (T˜) + T˜2

! ! T˜4 2 tan ϕf˜ + V0 (T˜) = 0. Jo − σ 3

2 T˜3

Solution of equation (3.3) is obtained in the form 11 √ 1 V0 = C1 Hypergeometric P F Q 14 1 − 4Jo, 11 − 14 14 + 2 J0 −

2 tan ϕf˜ σ

T˜7 343

3 1 T˜ 2 − 2

√

2 tan ϕf˜ σ

1−4Jo

2 Jo −

1−

T˜7 343

1 7

√

1 − 4Jo, 17 14 +

1 − 4Jo ,

T˜7 343

1 14

√

1 − 4Jo ,

i

+ C3 Hypergeometric P F Q

2 tan ϕf˜ σ

√

i

+C2 Hypergeometric P F Q 2 Jo −

1 14

(3.3)

i

3 1 T˜ 2 + 2

√

1−4Jo

1 − 71 sq1 − 4Jo, 17 14 +

1 14

√

1 − 4Jo ,

. (3.4)

The first order equation becomes V1000 (T˜)

Jo 0 ˜ + V1 (T ) − T˜2

2 ˜ T3

2 tan ϕf Jo − σ

2T˜6 + 27

Solution of equation (3.2) is obtained in the form √ 1 1 − 4Jo, 56 + V1 = C1 Hypergeometric P F Q 56 − 18 2 J0 −

2 tan ϕf˜ σ

T˜9 6561

3 1 T˜ 2 − 2

√

2 tan ϕf˜ σ

1−4Jo

2 Jo −

T˜9 6561

(3.5)

1 − 4Jo ,

1−

1 9

√

1 − 4Jo, 76 −

T˜9 6561

1 18

√

1 − 4Jo ,

i

i

3 1 T˜ 2 + 2

7 The amplitude will decrease as T˜ 2 (3±

number.


2 tan ϕf˜ σ

√

V1 (T˜) = 0.

i


1 18

!

√

√

1−4Jo

7 6

+

1 18 sq1

− 4Jo, 1 +

1 9

√

1 − 4Jo ,

. (3.6)

1−4jo)

. This limit also depends on the Richardson

290


The generalized hypergeometric function or Barnes extended hypergeometric function h i Hypergeometric P F Q {a1 , a2 , · · · , ap } , {b1 , b2 , · · · , bq } , T˜ has series expansion and

˜ p Fq (a, b, T ) =

∞ X (a1 )k · · · (ap )k T˜k

and (a1 )k = a1 (a1 + 1) · · · (a1 + k − 1).

(b1 )k · · · (bq )k k!

k=0

4. Mass Diffusion In the case of mass diffusion, instead of Prandtl number, P r we have Schmidth number, 1 1, the analysis for Ω11 Sc = κvC , κC is the mass diffusivity. If we let β = 1 − Sc and Ω12 changing signs can be redone. The new variable is found to be T˜ = β 1/2 T and the lowest order equation with V0 = V˜0 (T˜) + β 1/2 V˜1 (T˜) + · · · becomes 2 T˜3

5

−

Jo 0 ˜ V0000 (T˜) + V0 (T ) − T˜2

2 tan ϕf Jo − σ

T˜ + 3

! V0 (T˜) = 0.

(4.1)

Solving equation (4.1), we obtain V0 = C1 Hypergeometric P F Q 2 J0 −

2 tan ϕf˜ σ

T˜4 64

8

3 1 T˜ 2 − 2

√

2 tan ϕf˜ σ

1−4Jo

2 Jo −

T˜4 64

√

1 − 4Jo, 58 +

1−

1 4

√

T˜4 64

√

1 − 4Jo, 11 8 −

1 − 4Jo ,

1 8

√

1 − 4Jo ,

i


2 tan ϕf˜ σ

1 8

i


1 8

i

3 1 T˜ 2 + 2

√

1−4Jo

11 8

+

1 8

√

1 − 4Jo, 1 +

1 4

√

1 − 4Jo ,

. (4.2)

The first order equation becomes

V1000 (T˜)

Jo 0 ˜ + V1 (T ) − T˜2

2 ˜ T3

2 tan ϕf Jo − σ

2T˜3 + 9

! V1 (T˜) = 0.

(4.3)

291


Whose solution is V1 = C1 Hypergeometric P F Q 2 J0 −

2 tan ϕf˜ σ

T˜6 5832

3

1 3 T˜ 2 − 2

√

2 tan ϕf˜ σ

1−4Jo

2 Jo −

1 12

√

1 − 4Jo, 34 +

T˜6 5832

5 4

−

1 12

√

1 − 4Jo, 1 +

2 tan ϕf σ

√

1 − 4Jo ,

T˜6 5832

1 6

√

1 − 4Jo ,

i

+ C3 Hypergeometric P F Q ˜

1 12

i


−

4

i

√ 1

3 T˜ 2 + 2

The amplitude will decrease as T˜

3 ± 12 2

1−4Jo

√

11 8

+

1 8

√

1 − 4Jo, 1 +

1 4

√

1 − 4Jo ,

. (4.4)

1−4Jo

. This limit depends on the Richardson

number. Numerical solutions of equation (2.19) are obtained for different values of J0 and f using the initial conditions given by equation (4.3) as used by Hartmann (1975). (v0 )(0) = −1,

d(v0 ) d2 (v0 ) (0) = 0, (0) = 0. dT dT 2

(4.5)

The constants in equations (3.3) and (4.2) are evaluated using the initial conditions given by equation (4.3).

5. Results and Discussions In this problem, we have considered the effects of rotation, mass / heat diffusion on the linearized perturbations in a shear flow using initial-value problem approach. Figs. 1 (a)-(c) are plots of V0 versus T for f = 0, 1, 2 and Jo = 1, 5, = 0, 0.01 with no mass diffusion (Sc = ∞). Effects resulting from viscosity are clear. As the value of f is increased, velocity decays rapidly. Figs. 2(a)-(c) are plots of V0 versus T for f = 0, 1, 2, J0 =

1 8

and = 0, 0.01. It is seen that the motion has no oscillations and is rapidly

damped even without viscosity. Figs. 3 (a)-(c) displays solutions of equation (3.2) as compared to the exact form of equation (2.26) by considering Sc i,e., mass diffusion instead of P r for f = 0, 1, 2, J0 = 1, = 0.01 and no mass diffusion (Sc = ∞) for the amplitude; differences are very small. All disturbances are ultimately exponentially attenuated. We see that in the presence of large mass diffusivity, rotation effects are

292


stabilizing. In this problem, we have obtained an exact solution for an unbounded stratified shear flow with rotation.


293

294


References [1] Case, Stability of inviscid plane Couette flow, Phys. Fluids, 3 (1960), 143. [2] Criminale Jr W. O. and Cordova J. Q., Effects of diffusion in the asymptotics of perturbations in stratified shear flow, Phys. Fluids, 29(7) (1988), 1949. [3] Damien Biau and Alessandro Bottaro, The effect of stable thermal stratification on shear flow stability, Phys. Fluids, 16(12) (2004), 4742. [4] Eliassen A., Holland E. and Riis E., Two - dimensional perturbation of a flow with constant shear of a stratified fluid, Inst. Weather and Climate Res. Norwegian Acad. Sci. & Lett. Publ. 1 (1953). [5] Hartmann P., Wave propagation in a stratified shear flow, J. Fluid Mech., 71 (1975), 89. [6] Kelvin Lord (W. Thomas), Stability of fluid motion: Rectilinear motion of viscous fluid between two parallel plates, Philos. Mag, 25(5) (1887), 188. [7] Lerner J. and Knobloch E., The stability of dissipative magnetohydro-dynamic shear flow in a parallel magnetic field, Geophysics and Astrophysics, Fluid Dynamics, 33 (1985), 295. [8] Maslowe S. A. and Thompson J. M., Stability of a stratified free shear layer, Phys. Fluids, 14 (1971), 453. [9] Van Kampen N. G. and Felderhof B. U., Theoretical Methods in Plasma Physics, North-Holland Publ. Co., (1957).


MHD FLOW OF AN IONIZED GAS IN A PARALLEL PLATE CHANNEL WITH POROUS LINING A. RAMA DEVI1 , S. SREENADH2 , V. RAMESH BABU3 AND E. SUDHAKARA4 Department of Mathematics, S.V.Arts College, Tirupati -517502, (A.P), India 2,4 Department of Mathematics, Sri Venkateswara University, Tirupati - 517502 (A.P), India 1,3

Abstract Magnetohydrodynamic flow of an ionized gas in a parallel plate channel with porous lining is studied considering one of the walls to be lined with non-erodible porous material. The analysis uses Brinkman model in the porous medium and employs primary and secondary velocity slips at the interface in both partially and fully ionized cases. The influence of Hartmann number M and Hall parameter m on the flow field is studied graphically. It is found that the primary and the secondary velocity distributions in both partially (s = 0) and fully (s = 0.5) ionized cases decrease with increase in Hartmann number M . It is also found that Primary velocity decreases and secondary velocity increases in both partially and fully ionized cases with the increase in the Hall Parameter m.

1. Introduction The study of flow of an electrically conducting fluid has many applications in engineering −−−−−−−−−−−−−−−−−−−−−−−−−−−−

Key Words : Hall Currents, MHD, Ionized Gas, Porous Lining. c http: //www.ascent-journals.com

295

296

A. RAMA DEVI, S. SREENADH, V. RAMESH BABU & E. SUDHAKARA

problems such as MHD generators, plasma studies, nuclear reactors, geothermal energy extraction and the boundary layer control in the field of aerodynamics. MHD has important applications in biomedical engineering in the design of medical devices. The principle of MHD is also used in stabilizing a flow against the transition from laminar to turbulent flow. Another important field of application is electromagnetic propulsion. Basically, an electro-magnetic propulsion system consists of a power source (such as a nuclear reactor), plasma and tube through which the plasma is accelerated by electromagnetic forces. Singh et al. [1] studied free convection in MHD flow of a rotating viscous liquid in porous media. Singh et al. [2] have also studied hydromagnetic oscillatory flow of a viscous liquid past a vertical porous plate in a rotating system. Most of the investigations do not contain the effects of Hall currents on the ionized gas flows. However, in a partially ionized gas, there occurs a Hall current when the strength of the impressed magnetic field is very strong. These Hall effects play a significant role in determining the flow features. Yamanishi [3] have discussed the Hall effects on the steady hydromagnetic flow between two parallel plates. Ionized gas is plasma which is the fourth state of matter. Plasma is a state of matter that starts as a gas and then becomes ionized. Ionizing is the process by which an electrically neutral atom, molecule, or radical loses or gains one or more electrons and becomes an ion. Ionization can occur in gases, liquids, or solids. Physicists often call a highly ionized gas as plasma. Plasma is a gas consisting of charged ions and electrons. The degree of ionization refers to the proportion of neutral particles, such as those in a gas or aqueous solution that are ionized into charged particles. A low degree of ionization is sometimes called partially ionized, and a very high degree of ionization is termed as fully ionized. The Hall effect in the viscous flow of ionized gas between parallel plates under transverse magnetic field has been treated theoretically by Sato [4]. Raju and Rao [5] have studied the Hall effects on temperature distribution in a rotating ionized hydromagnetic flow between parallel walls. Seetaram [6] studied the effects of Hall currents in an ionized hydromagnetic flow between two parallel porous walls for non conducting and conducting cases. In Biomechanics, the blood vessel is often modeled as a circular tube lined with porous material where the finite circular porous layer represents the tissue region surrounding the blood. Channabasappa, Umapathy and Nayak [7] proposed a

MHD FLOW OF AN IONIZED GAS IN A PARALLEL PLATE...

297

theoretical model for the study of flow and heat transfer in a parallel plate channel, one of whose walls is lined with non-erodible porous material. In view of this it will be interest to study the flow of ionized gas in porous lined channels. In this paper, MHD flow of ionized gas in a parallel plate channel with porous lining is investigated. The analysis makes use of the velocity slip boundary conditions of Beavers and Joseph [8] at the interface, with suitable modifications to make into account the finite thickness involved. The influence of physical parameters on the velocity in the channel has been discussed for partially and fully ionized cases. Nomenclature u, w

Velocity components along x and z- directions in free porous region (Zone 1) Velocity components along x and z- directions in porous region (Zone 2)

u ˆ, w ˆ →

B

The magnetic flux density

E J p Ex , E z V0 σ0

The electric field The current density Pressure Electric fields along x and z-directions Suction velocity The coefficient of proportionality between the current density J and collision term in the equation of motion of charged particles The modified conductivities parallel and normal to the direction of electric field Hall parameter

→

σ1 , σ 2 m=

“ we ” 1 + τ1 τ e

we τ, τe

The gyration frequency of electron The mean collision time between electron and ion, electron and neutral particles respectively

σ

√h k

M µ ρ υ K

Thickness of porous lining q 2 2 B 0 h σ0 ρν , Hartmann Number Coefficient of viscosity (µ = ρν) Density Kinematic Viscosity Permeability of the medium

298


s=

pe p

Ionization parameter (the ratio of the electron pressure to the total pressure)

2. Formulation of the Problem The physical model illustrating the problem under consideration is shown in Fig 1. Here we consider the steady viscous flow of an ionized gas with constant properties between two parallel plates of width 2h whose upper plate is impermeable and the lower impermeable plate being lined with a non-erodible porous material of thickness h0 . Thus the flow region is divided into two zones: Zone 1 from the impermeable upper plate to the surface of the porous lining and Zone 2 from the surface of the porous lining to the impermeable lower plate. The x-axis is taken in the direction of hydrodynamic pressure gradient in the plane parallel to the channel walls, but not in the direction of flow. A parallel uniform magnetic field B0 is applied in the y-direction and the Hall currents ∂p . are taken into account while the fluid is driven by a constant pressure gradient − ∂x

Further to simplify the theoretical analysis,the following assumptions are made following Raju and Rao [5]: (i) The density of gas is everywhere constant. (ii) The ionization is in equilibrium which is not affected by the applied electric and magnetic fields. (iii) The effect of space charge is neglected. (iv) The flow is fully developed, that is,

∂( ) ∂x

= 0 and

∂( ) ∂z

∂p = 0 except − ∂x 6= 0.

(v) The magnetic Reynolds number is small (so that the externally applied magnetic field is undisturbed by the flow). The induced magnetic field is small compared with the applied field. Therefore, components in the conductivity tensor are expressed in terms of B0 .

299


Since the walls are infinite in extent, all the physical variables except pressure will depend on y only. The physical configuration and the nature of the flow suggest the →

→

→

following form of velocity vector q , the magnetic flux density B , the electric field E , the current density J. Thus →

q = [u, 0, w],

→

B = [0, B0 , 0],

→

E = [Ex , 0, Ez ],

→

J = [jx , 0, jz ]

(1)

In view of the above assumptions, the governing equations reduce to Zone 1 h − 1−s 1−

σ1 σ0

i

∂p ∂x

2

+ ∂υ ddyu2 + B0 [−σ1 (Ez + uB0 ) + σ2 (Ex − wB0 )] = 0 (2)

s σσ02

∂p ∂x

+

Zone 2 h − 1−s 1−

2 ρυ ddyu2

σ1 σ0

i

+ B0 [σ1 (Ex − wB0 ) + σ2 (Ez + uB0 )] = 0.

∂p ∂x

2

+ ρυ ddyu2ˆ + B0 [−σ1 (Ez + u ˆB0 ) + σ2 (Ex − wB ˆ 0 )] − µk u ˆ=0 (3)

s σσ20

∂p ∂x

+

2 ρυ ddyw2ˆ

+ B0 [σ1 (Ex − wB ˆ 0 ) + σ2 (Ez + u ˆB0 )] −

µ ˆ kw

=0

300


Beavers and Joseph (1967) [3] carried out experimental work for two dimensional channel flow with one porous boundary and imposed pressure gradient. They have postulated a slip condition (B J Condition) at the permeable bed based on their experiments. It may be noted that the thickness of the porous medium in the BJ boundary condition model is infinite. Later Channabasappa et al. [7] proposed a modified condition for problems involving finite thickness of the porous lining. In the BJ model, the slip velocity uB , wB at the nominal surface changes to the constant Darcy velocity Q through a thin layer (called the boundary layer or the velocity slip layer) whose thickness was given by √ √ BJ as of order k. This thickness subsequently has been shown to be equal to k. In view of the use of Brinkman model to determine the velocity field in zone 2, we identify √ Q of the BJ condition with the value of u ˆ and w ˆ at a distance k below the nominal surface and thus modified boundary conditions are used in the problem. Zone 1 : u = 0, w = 0 at y = −h (4) u = uB , w = wB at y = h − h0 Zone 2 :

u ˆ = uB , w ˆ = wB at y = h − h0 (5) u ˆ = 0, w ˆ = 0 at y = h

where uB , wB is given by du dy

= − √αk uB − u ˆ|y=h−h0 +√k

y=h−h0

(6)

dw dy y=h−h0

=

− √αk

wB − w| ˆ y=h−h0 +

√

k

3. Non-dimentionalisation of the Flow Quantities We introduce the following non-dimensional variables and parameters u∗ =

u up ,

w∗ =

=

h0 h,

σ=

σ1 σ0

=

1 , 1+m2

w up ,

√h , k

σ2 σ0

=

u ˆ∗ = mx =

m , 1+m2

u ˆ up ,

w ˆ∗ =

w ˆ∗ up ,

Ex B 0 up ,

mz =

M2 =

B02 h2 σ0 ρυ .

x∗ = hx , ‘ y ∗ = hy ,

Ez B 0 up ,

up = −

∂p ∂x

h2 ρυ ,

(7)

In view of the above dimensionless quantities, Eqs. (2) to (6) take the following form: Neglecting the asterisks (∗), we get,

301


Zone 1 : L1 +

d2 u dy 2

L2 +

d2 w dy 2

−

σ1 2 σ0 M (mz

+

σ1 2 σ0 M (mx

+ u) +

σ2 2 σ0 M (mx

− w) = 0 (8)

Zone 2 : L1 +

d2 u ˆ dy 2

L2 +

d2 w ˆ dy 2

−

σ1 2 σ0 M (mz

+

σ1 2 σ0 M (mx

− w) +

+u ˆ) +

σ2 2 σ0 M (mz

σ2 2 σ0 M (mx

+ u) = 0

− w) ˆ − σ 2 (ˆ u) = 0 (9)

− w) ˆ +

σ2 2 σ0 M (mz

+u ˆ) −

σ 2 (w) ˆ

=0

The boundary conditions are Zone 1 : u = 0, w = 0 at

y = −1 (10)

u = uB , w = wB at y = 1 − Zone 2 : u ˆ = uB , w ˆ = wB at y = 1 − (11) u ˆ = 0, w ˆ = 0 at y = 1 where uB , wB is given by

du dy y=1−

= −ασ (uB − u ˆ)|y=1−+ 1

σ

(12)

dw dy y=1−

= −ασ (wB − w)| ˆ y=1−+ 1

σ

Denoting q = u + iw, qˆ = u ˆ + iw, ˆ L = L1 + iL2 , E = mx + imz . Equations (9) to (12) can be written in complex form as Zone 1 : d2 q + dy 2

−σ1 σ0

2

M +i

σ2 σ0

M

2

q = −L − i

σ1 σ0

2

M E−

σ2 σ0

M 2E

(13)

Zone 2 : d2 qˆ −σ1 σ2 σ1 σ2 2 2 2 2 + M +i M qˆ − σ qˆ = −L − i M E− M 2 E. (14) 2 dy σ0 σ0 σ0 σ0

4. Solution of the Problem If the side walls are made up of conducting material and short circuited by an external conductor, the induced electric current flows out of the channel. In this case no electric

302


potential exists between the side walls. If we assume zero electric field also in the xand z-directions, then mx = 0, mz = 0. In this case equations (13) and (14) become to d2 q + aq = −L dy 2

(15)

d2 qˆ + dˆ q = −L dy 2

(16)

where a = a1 + ia2 ; d = d1 + id2 . Solving equations (15) and (16) subject to the boundary conditions (10) to (12), the expression q and qˆ is obtained as follows q = (A1 Cosha3 yCosa4 y − A2 Sinha3 ySina4 y + A3 Sinha3 yCosa4 y −A4 Cosha3 ySina4 y − f14 ) (17) +i(A1 Sinha3 ySina4 y + A2 Cosha3 yCosa4 y + A3 Cosha3 ySina4 y +A4 Sinha3 yCosa4 y − g14 ) qˆ = (B1 Coshd3 yCosd4 y − B2 Sinhd3 ySind4 y + B3 Sinhd3 yCosd4 y −B4 Coshd3 ySind4 y − f15 ) (18) +i(B1 Sinhd3 ySind4 y + B2 Coshd3 yCosd4 y + B3 Coshd3 ySind4 y +B4 Sinhd3 yCosd4 y − g15 ) Expanding real and imaginary parts, we get solutions for Zone1and Zone 2. On using (17) and (18) in (12) we obtain uB , wB . They are all depend on s. The value of s is 0.5 for neutral fully-ionized gas and s = 0 for a weakly ionized gas. The primary and secondary velocities for both Zones (u, w and u ˆ, w) ˆ are obtained as follows: Zone 1 : (free flow region) u = (A1 Cosha3 yCosa4 y − A2 Sinha3 ySina4 y + A3 Sinha3 yCosa4 y (19) −A4 Cosha3 ySina4 y − f14 ) w = (A1 Sinha3 ySina4 y − A2 Cosha3 yCosa4 y + A3 Cosha3 ySina4 y (20) −A4 Sinha3 yCosa4 y − g14 )


303

Zone 2: (porous flow region) u ˆ = (B1 Coshd3 yCosd4 y − B2 Sinhd3 ySind4 y + B3 Sinhd3 yCosd4 y (21) −B4 Coshd3 ySind4 y − f15 ) w ˆ = (B1 Sinhd3 ySind4 y + B2 Coshd3 yCosd4 y + B3 Coshd3 ySind4 y (22) +B4 Sinhd3 yCosd4 y − g15 ) where L1

= 1−s 1−

1 −sm −M 2 M 2m , L = , a = , a = , 2 1 2 1 + m2 1 + m2 1 + m2 1 + m2

d1 = a1 − σ 2 , d2 = a2 sp sp sp 2 2 2 2 a1 + a2 + a1 a1 + a2 − a1 d21 + d22 + d1 a3 = , a4 = , d3 = , 2 2 2 sp d21 + d22 − d1 d4 = 2 f1 = Sinha3 Cosa4 , g1 = Cosha3 Sina4 , f2 = Cosha3 Cosa4 , g2 = Sinha3 Sina4 f3 = Sinha3 (1 − )Cosa4 (1 − ), g3 = Cosha3 (1 − )Sina4 (1 − ) f4 = Cosha3 (1 − )Cosa4 (1 − ), g4 = Sinha3 (1 − )Sina4 (1 − ) f5 = Sinha(2)Cosa4 (2 − ), g5 = Cosha3 (2 − )Sina4 (2 − ) f6 = Sinhd3 Cosd4 , g6 = Coshd3 Sind4 , f7 = Coshd3 Cosd4 , g7 = Sinhd3 Sind4 f8 = Sinhd3 Cosd4 , g8 = Coshd3 Sind4 , f9 = Coshd3 Cosd4 , g9 = Sinhd3 Sind4 f10 = Sinhd3 (1 − )Cosd4 (1 − ), g10 = Coshd3 (1 − )Sind4 (1 − ) f11 = Coshd3 (1 − )Cosd4 (1 − ), g11 = Sinhd3 (1 − )Sind4 (1 − ) 1 1 f12 = Sindh3 1 − + Cosd4 1 − + , σ σ 1 1 g12 = Coshd3 1 − + Sind4 1 − + σ σ 1 1 f13 = Coshd3 1 − + Cosd4 1 − + , σ σ 1 1 g13 = Sinhd3 1 − + Sid4 1 − + . σ σ

304

f14 = f16 = F18 = f20 = f22 = f24 = g25 = f27 = f29 =


L1 a1 + L2 a2 L2 a1 − L1 a2 L1 d1 + L2 d2 L2 d1 − L1 d2 ; g14 = ; f15 = ; g15 = ; 2 2 2 2 2 2 a1 + a2 a1 + a2 d1 + d2 d21 + d22 f1 f5 + g1 g5 g1 f5 − f1 g5 f2 f5 + g2 g5 g2 f5 − f2 g5 ; g16 = ; f17 = ; g17 = ; 2 2 2 2 2 2 f5 + g5 f5 + g5 f5 + g5 f52 + g52 f6 f8 + g6 g8 g6 f8 − f6 g8 f7 f8 + g7 g8 g7 f8 − f7 g8 ; g18 = ; f19 = ; g19 = ; 2 2 2 2 2 2 f8 + g8 f8 + g8 f8 + g8 f82 + g82 g20 f5 − f20 g5 f20 f5 + g20 g5 ; g21 = ; (f4 − f2 ); , g20 = g4 − g2 ; f21 = 2 2 f5 + g5 f52 + g52 f14 f2 + g14 g2 g14 f2 − f14 g2 f1 f21 − g1 g21 ; g22 = f1 g21 + g1 f21 ; f23 = ; g23 = ; 2 2 f2 + g2 f22 + g22 f15 f7 + g15 g7 g15 f1 − f15 g7 ; g24 = ; f25 = f23 (1 − f22 ) + g23 g22 ; f72 + g72 f72 + g72 g23 (1 − f22 − f23 g22 ; f26 = f14 f21 − g14 g21 ; g26 = f14 g21 + g14 f21 ; g27 f8 − f27 g8 f27 f8 + g27 g8 ; g28 = f7 − f11 ; g27 = g7 − g11 ; f28 = f82 + g82 f82 + g82 f8 f28 − g8 g28 ; g29 = f8 g28 + g8 f28 ; f30 = f24 (1 + f29 ) − g24 g29 ;

g30 = g24 (1 + f29 + f24 g29 ); f31 = f15 f28 − g15 g28 ; g31 = f15 g28 + g15 f28 ; f32 = f30 f13 − g30 g13 ; g32 = f30 g13 + g30 g13 ; f33 = f31 f12 − g31 g12 ; g33 = f31 g12 + g31 f12 ; f34 = f25 f3 − g25 g3 ; g34 = f25 g3 + g25 f3 ; f35 = f26 f4 − g26 g4 ; g35 = f26 g4 − g26 f4 ; f36 = f16 f3 − g16 g3 ; g36 = f16 g3 + g16 f3 ; f37 = f17 f4 − g17 g4 ; g37 = g17 f4 + f17 g4 ; f38 = f18 f13 − g18 g13 ; g38 = f18 g13 + g18 f13 ; f39 = f19 f12 − g19 g12 ; g39 = f19 g12 + g12 ; f40 = ασ(f32 − f33 − f15 ); g40 = ασ(g32 − g33 − g15 ); f41 = a3 (f34 − f35 ) − a4 (g34 − g35 ); g41 = a3 (g34 − g35 ) − a4 (f34 − f35 ); f42 = a3 (f36 + f37 ) − a4 (g36 + g37 ); g42 = a3 (g36 + g37 ) + a4 (f36 + f37 ); f43 = ασ(1 − f38 + f39 ); g43 ) = ασ(−g38 + g39 ); f44 = f40 − f41 ; g44 = g40 − g41 ; f45 = f42 + f43 ; g45 = g43 ; g44 f45 − f44 + g45 f44 f45 + g44 g45 f46 = ; g46 = ; 2 + f2 2 + g2 f45 f45 45 45 A1 = f46 f16 − g46 g16 + f25 ; a2 = f46 g16 + g46 f16 = g25 ; A3 = f46 f17 − g46 g17 − f26 ; ‘ A4 = g46 f17 + f46 g17 − g26 ; B1 = f46 f18 − g46 g18 + f30 ; B2 = g46 f18 + f46 g18 + g30 ; B3 = −f46 f19 + g46 g19 − f31 ; B4 = −f46 g19 − g46 f19 − g31 .


305

5. Numerical Results and Discussions The solutions for both primary and secondary velocities in both Zones: Zone 1 from the impermeable upper plate to the surface of the porous lining and Zone 2 from the surface of the porous lining to the impermeable lower plate are obtained. The graphs for velocity distribution, such as both primary and secondary velocity distributions are shown in Fig. 2 to 9. These are given for two cases, viz., partially ionized (s=0) and fully ionized (s = 0.5) cases. The variation in primary and secondary velocities are shown in Figures 2 and 3 for different values of Hartmann number M and for fixed α, , m and σ. It is noticed that for partially ionized (s = 0) case, primary and secondary velocities decrease with the increase in the Hartmann number M . This is because the increase in the magnetic field gives rise to reduction in velocity in the channel. The same phenomenon is observed for fully ionized (s = 0.5) case, as can be seen from figures 4 and 5. The variation in primary and secondary velocities in both cases. viz., partially ionized (s = 0) and fully ionized (s = 0.5) cases are shown in Figures 6 to 9 for different values of Hall parameter m and for fixed α, , M and σ. It is noticed that the primary velocity decreases as can be seen from figures 6 and 8. The secondary velocity increases as can be seen from figures 7 and 9 in both cases with the increase in the Hall parameter m. From Table 1, it is observed that the effect of α is to decrease primary and secondary velocities for non porous and porous zones in both partially and fully ionized cases. Table 2, it is observed that the effect of σ is to decrease primary and secondary velocities for non porous and porous zones in partially ionized case and increase primary and secondary velocities for non porous and porous zones in fully ionized case.

306



307

308


Table 1 Velocity distributions for different values of α with M = 7, σ = 35, = 0.2, m = 2. S. No.

α

1 2 3 4 5 6

0 0.5 1 1.5 2 2.5

S=0 Primary Secondary Zone 1 Zone 2 Zone 1 Zone 2 0.00010 0.01612 -0.00105 -0.00791 0.00003 0.01590 -0.00124 -0.00814 -0.00008 0.01553 -0.00155 -0.00853 -0.00033 0.01480 -0.00214 -0.00925 -0.00116 0.01259 -0.00370 -0.01106 -0.02001 -0.01798 -0.00763 -0.00517

S = 0.5 Primary Secondary Zone 1 Zone 2 Zone 3 Zone 4 -0.00517 -0.00041 -0.00910 -0.01345 -0.00231 -0.00182 0.00806 -0.00135 -0.00167 -0.00105 0.00975 -0.00054 -0.00138 -0.00075 0.01034 -0.00028 -0.00122 -0.00059 0.01063 -0.00014 -0.00112 -0.00049 0.01063 -0.00007

Table 2 Velocity distributions for different values of σ with M = 7, α = 2, = 0.2, m = 2. S. No.

σ

1 2 3 4 5 6

34.0 34.2 34.4 34.6 34.8 35.0

S=0 Primary Secondary Zone 1 Zone 2 Zone 1 Zone 2 -0.00762 0.00043 -0.00798 0.00197 -0.00766 0.00015 0.01618 0.00166 -0.00774 -0.00011 0.01609 0.00130 -0.00784 -0.00042 0.01601 0.00089 -0.00798 -0.00078 0.01595 0.00046 -0.00798 -0.00124 0.01590 0.00003

S = 0.5 Primary Secondary Zone 1 Zone 2 Zone 3 Zone 4 -0.00014 -0.00059 0.01063 -0.00122 0.00010 -0.00044 0.01069 -0.00113 0.00037 -0.00034 0.01076 -0.00106 0.00062 -0.00026 0.01083 -0.00101 0.00086 -0.00019 0.01090 -0.00099 0.00108 -0.00012 0.01097 -0.00098

References [1] Singh N. P, Gupta S. K. and Singh Atul Kumar, Free convection in MHD flow of a rotating viscous liquid in porous medium past a vertical porous plate, Proc. Nat. Acad. Sci, India, 71A (2001), 149-157. [2] Singh N. P., Singh Ajay Kumar, Yadav M. K. and Singh Atul Kumar, Hydromagnetic oscillatory flow of a viscous liquid past a vertical porous plate in a rotating system, Ind. Theor. Phy, 50 (2002), 37-43. [3] Yamanishi T., Hall effects on hydromagnetic flow between two parallel plates, Phy. Soc., Japan, Osaka, 5 (1962), 29. [4] Sato H., The Hall effects in the viscous flow of ionized gas between parallel plates under transverse magnetic field, J. Phys. Soc. Japan, 16 (1961), 1427-1433. [5] Raju L.T., Rao R.V.V., Hall effects on temperature distribution in a Rotating Ionized Hydromagnetic flow between parallel walls, Int. J. Eng. Sci., 31(7) (1993), 1073. [6] Seetaram P., The Effect of Hall currents on an ionized hydromagnetic flow between two parallel porous walls. M.Phil Dissertation, Andhra University, Visakapatnam, (A.P.), India. [7] Channabasappa M. N., Umapathy K. G., Nayak I. V., Convective Heat Transfer in a parallel Plate Channel with porous Lining,warme-und Stoffubertragung, 7 (1983), 211-216. [8] Beavers G. S. and Joseph D. D., Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30 (1967), 197.


ON BINARY QUADRATIC EQUATION 2x2 − y 2 + 2(2x − 2y − 1) = 1 P. THIRUNAVUKARASU1 AND S. SRIRAM2 1 Assistant Professor, PG and Research Department of Mathematics, Periyar E.V.R. College, Tiruchirappalli - 620 023, Tamil Nadu, S. India 2 Assistant Professor, PG and Research Department of Mathematics National College, Tiruchirappalli - 620 001, Tamil Nadu, S. India

Abstract The binary quadratic equation 2x − y 2 + 2(2x − 2y − 1) = 1 is studied for its non-trivial integral solutions. A few interesting relations among the solution are presented. 2

1. Introduction Binary Quadratic Diophantine equations are rich in variety. For an extensive review of sizable literature and various problems refer [1-9]. In this communications, we consider yet another interesting binary quadratic equation 2x2 − y 2 + 2(2x − 2y − 1) = 1 and obtain infinitely many non-trivial integral solutions. The recurrence relations satisfied by the values of x and y are given. Also, a few interesting relations between x and y are obtained. −−−−−−−−−−−−−−−−−−−−−−−−−−−−

Key Words : Binary Quadratic, Diophantine equation, Integral solutions. c http: //www.ascent-journals.com

309

310

P. THIRUNAVUKARASU & S. SRIRAM

2. Methods of Analysis The Diophantine equation under consideration is 2x2 − y 2 + 2(2x − 2y − 1) = 1

(1)

It is to be noted that (1) represents hyperbola. By taking the Linear Transformation. x = U + T, y = U + 2T, we get U 2 = 2(T + 1)2 + 1.

(2)

U 2 = 2X 2 + 1

(3)

X = T + 1.

(4)

Equation (2) can be written as

where

Equation (3) is known as Pell’s Equation. The general integral solution are given by Us +

√

√ 2Xs = (3 + 2 2)s+1

(5)

where s = 0, 1, 2, · · · . Since irrational roots occurs in Pairs, Us −

√

√ 2Xs = (3 − 2 2)s+1 .

(6)

Thus we get Us = Xs =

i √ √ 1h (3 + 2 2)s+1 + (3 − 2 2)s+1 2 i √ √ 1 h √ (3 + 2 2)s+1 − (3 − 2 2)s+1 . 2 2

Here the solutions of equation (1) is given by i √ √ 1h xs = (3 + 2 2)s+1 + (3 − 2 2)s+1 + 2 i √ √ 1h ys = (3 + 2 2)s+1 + (3 − 2 2)s+1 + 2 for s = 0, 1, 2, · · ·

i √ √ 1 h √ (3 + 2)s+1 − (3 − 2 2)s+1 − 1 2 2 i √ √ 1 h √ (3 + 2)s+1 − (3 − 2 2)s+1 − 2 2

311

ON BINARY QUADRATIC EQUATION...

The above values of xs and ys satisfy the following recurrence relations respectively. xs+2 − 6xs+1 + xs = 4 ys+2 − 6ys+1 + ys = 8. A few examples are given below

s=0 s=1 s=2 s=3 s=4 s=5

xs 4 28 168 984 5740 33460

ys 5 39 237 1391 8117 47319

A few properties among the solutions are given below. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

It is observed that y values are odd and the x values are even. 6(ys − xs )2 , 6(2xs − ys )2 are nasty numbers. 12(ys − xs + 1)2 is a nasty number (2xs − ys )2 + 2(ys − xs + 1)2 is the sum of two perfect square 660(xs+2 − 6xs+1 + xs ) is a R2 number 330(xs+2 − 6xs+1 + dz ) is a R2 number xs+2 − 6xs+1 + xs is a perfect square 6(xs+2 − 6xs+1 + xs ) is a nasty number 12(2x2s+1 − y2s+1 + 1) is a nasty number ys+2 − 6ys+1 + ys + 8 is a perfect square 1729 4 (xs+2 − 6xs+1 + xs ) is a R3 number 1729 8 (ys+2 − 6ys+1 + ys ) is a R3 number.

It is observed that y values are odd and that x values are even.

3. Conclusion In conclusion one may search for other patterns of solutions.

312


References [1] Carmichael R. D., The Theory of Numbers and Diophantine Analysis, Dover Publications, New York (1959). [2] Dickson L. E., History of the Theory of Numbers, Vol.II, Chelisa Publishing Co., New York (1952). [3] Mollin R. A., All Solution of the Diophantine equation x2 − Dy 2 = n, for East J. Math. Sci., Social Volume, Part III, (1998), 257-293. [4] Mordell L. J., Diophantine Equations, Academic Press, London (1969). [5] Telang S. G., Number Theory, Tata McGraw - Hill Publishing Company, New Delhi (1996). [6] Nigel P. Smart, The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, London (1999). [7] Bhanumurthy T. S., A Modern Introduction to Ancient Indian Mathematics, Wiley Eastern Limited, London (1995). [8] Goplan M. A. and Ganapathy R., Note On The Diophantine Equation x2 − 3xy + y 2 + 10x − 10y − 21 = 0. Acta cienica Indica, XXVIIM(2) (2001), 267. [9] Gopalan M. A., Gokila K. and Vidhayalakashmi S., On The Diophantine Equation x2 − 4xy + y 2 − 2x + 2y − 6 = 0, Act acienica Indica, XXXIII M.(2) (2007), 567.


MODIFIED SIGMOID FUNCTION IN UNIVALENT FUNCTION THEORY O. A. FADIPE-JOSEPH1 , A. T. OLADIPO2 AND A. UZOAMAKA EZEAFULUKWE3 1 Department of Mathematics, University of Ilorin, P. M. B. 1515, Ilorin, Nigeria E-mail : [email protected], 2 Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, P. M. B. 4000, Ogbomoso, Nigeria E-mail: atlab− [email protected] 3 Department of Mathematics, University of Nigeria, Nsukka, Nigeria. E-mail: [email protected]

Abstract The properties of sigmoid function in relation to univalent functions theory is investigated. It was established that the modified sigmoid function belongs to the class of Caratheodory function.

1. Introduction and Preliminaries Special functions is a branch of Mathematics which is of utmost importance to scientist

−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Key Words : Sigmoid function, Caratheodory function, Univalent function, Special function. 2000 AMS Subject Classification : Primary 30C45, Secondary 33E99. c http: //www.ascent-journals.com

313

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O. A. FADIPE-JOSEPH, A. T. OLADIPO & A. UZOAMAKA EZEAFULUKWE

and engineers who are concerned with actual mathematical calculations. It has applications in specific problems of physics, engineering and computer science. Special functions are some particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics or other applications. However, there is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. In particular, elementary functions, especially trigonometric functions are also considered as special functions. The theory of special functions has been developed essentially in the nineteenth century due to the contribution of C. F. Gauss, C. G. J Jacobi, F. Klein and many others. However, in the twentieth century the theory of special functions has been overshadowed by other fields such as real and functional analysis, topology, algebra and differential equations. Special functions play an important role in geometric function theory. Example of special function is activation function. Activation function acts as a squashing function, such that the output of a neuron in a neural network is between certain values (Usually 0 and 1, or -1 and 1). There are three types of activation function, namely: threshold function, piecewiselinear function and sigmoid function. The most popular activation function in the hardware implementation of artificial neural networks (ANN) is the sigmoid function. Sigmoid function is often used with gradientdescendent type learning algorithms. There are different possibilities for evaluating this function, such as truncated series expansion, look-up tables, or piecewise approximation. Thus, it is of some interest to explore its characteristics. The sigmoid function, g(z) =

1 1+e−z

is useful because it is differentiable, which is im-

portant for the weight-learning algorithms. The sigmoid function will increase the size of the hypothesis space that the network can represent. Neural networks can be used for complex learning tasks. The sigmoid function has very important properties, including (1) It outputs real numbers between 0 and 1. (2) It maps a very large input domain to a small range of outputs. (3) It never loses information because it is a one-to- one function. (4) It increases monotonically.

MODIFIED SIGMOID FUNCTION IN UNIVALENT...

315

These properties enable us to use sigmoid function in univalent function theory. Caratheodory functions [1] : Let g(z) = 1 + C1 z + C2 z 2 + ... be analytic in the unit disc D and satisfying the conditions g(0) = 1, Re g(z) ≥ 0, then the function g(z) is called the Caratheodory function. The class of these functions is denoted by P. Lemma 1.1 [2,3] : If g(z) = 1 + C1 z + c2 z 2 + ...Cn z n + ... is holomorphic on the unit disk D, and has positive real part, then |Cn | ≤ 2, n = 1, 2, .... This inequality is sharp for each n. Lemma 1.2 [4] : Let g(z) be univalent in D. Then f (z) ≺ g(z) if and only if f (0) = g(0) and f (D) ⊂ g(D).

2. Main Results Theorem 2.1 : Let g(z) be a sigmoid function and G(z) = 2g(z) then G(z) ∈ P , |z| < 1. Proof : Suppose G(z) =

2 1+e−z

G(0) = 1

G(z) + G(z) =

4 (1 + Re (e−z )) |1 + e−z |2

This shows that Re(e−z ) depends only on the sign of cosy. Since z is in the unit disc, x and y vary from −1 to 1, since x2 + y 2 < 1. Also, since π (−1, 1) ⊂ −π 2 , 2 , , cosy should be positive. Therefore, ReG(z) > 0. Hence, the function g under investigation maps the unit disc into the half plane (with G(0) = 1) and is therefore in the Caratheodory class. Theorem 2.2 : If G(z) ∈ P then f (z) is a normalized univalent function. Proof : We need only to show that f (0) = 0 and f 0 (0) = 1. Suppose G(z) ∈ P , we can write G(z) = f (0) = 21 (0) = 0 f 0 (0) = 21 (2) = 1. Hence, the desired result.

zf 0 (z) f (z)

316

O. A. FADIPE-JOSEPH, A. T. OLADIPO & A. UZOAMAKA EZEAFULUKWE

Theorem 2.3 : Let g(z) =

1 , 1+e−z

g is of bounded turning.

Proof : We need to show that Re g 0 (z) > 0. e−z (1 + e−z )2

g 0 (z) =

Since e−x > 0 and cosy > 0 (from Theorem 2.1). Hence, Re g 0 (z) > 0. Remark [2] : Some special classes of univalent functions defined by simple geometric properties are established. Apart from their intrinsic interest they serve as test cases to the much more difficult full classes of univalent functions. These classes can be completely characterized by simple inequalities. They are closely connected with functions of positive real part and with subordination. The above remark would be implored to establish more results on modified sigmoid function. We first introduce the series form of the sigmoid function. The series form of a modified sigmoid function Let 2 1 + e−z

G(z) = then

∞ P

G(z) = 1 +

m=1

(−1)m 2m

∞ P

n=1

(−1)n n n! z

Theorem 2.4 : Let Gn,m (z) = 1 +

m .

P

(−1)m ∞ m=1 2m

hP

∞ (−1)n n n=1 n! z

im

|Gn.m (z)| < 2. Proof : ∞ X 1 |Gn,m (z)| ≤ 1 + 2m m=1

∞ X (−1)n m n z n! n=1

∴ |Gn,m (z)| < 2. Taking m = 1 in Theorem 2.4, we have ∞ P Theorem 2.5 : Let G(z) = 1+ Cn z n where Cn = n=1

This inequality is sharp for each n. Proof : The proof follows from Lemma 1.1.

−(−1)n 2n!

then |Cn | ≤ 2, n = 1, 2, · · · .

317

MODIFIED SIGMOID FUNCTION IN UNIVALENT...

Subordination : Let f (z) and g(z) be analytic in D = {z : |z| < 1}. We say that f (z) is subordinate to g(z) if there exists a function ϑ(z) analytic (not necessarily univalent) in D satisfyting ϑ(0) = 0 and G(0) = 1 = N (0) such that f (z) = g (ϑ(z)) (|z| < 1)

(1)

Subordination is denoted by f (z) ≺ g(z). Theorem 2.6 : G(z) ∈ P if and only if G(z) ≺

1+z 1−z

= N (z).

Proof : It follows from [2, page 35] and [5, page 136]. In particular |G0 (0)| < |N 0 (0)| i.e 1 < 2. Hence the desired result.

Acknowledgements The first author acknowledged: the Abdus Salam International Centre for Theoretical Physics (ICTP), Italy for the associateship award received, the useful interactions with Fabio Vlacci and Ramadas Ramakrishnan in ICTP.

References [1] Polatoˇ g lu Y., Bolcal M. and Sen A., Some Results of the Caratheodory’s Class, Journal of Istanbul K¨ ult¨ ur University, (2002), 57-58. [2] Gong S., Bieberbach Conjecture, American Mathematical Society International Press, (1999). [3] Bansal D., Upper bound of second Hankel determinant for a new class of analytic functions, Applied Mathematics Letters, 26 (2013), 103-107. [4] Pommerenke C., Univalent functions, Vandenhoeck and Ruprecht in G¨ ottingen, (1975). [5] Ahlfors L. V., Complex Analysis, McGraw-Hill Book Company, (1966).


ON BINARY QUADRATIC EQUATION 3x2 − 2y 2 + 6(2x − 2y − 1) = 1 P. THIRUNAVUKARASU1 AND S. SRIRAM2 1 Assistant Professor, PG and Research Department of Mathematics, Periyar E.V.R. College, Tiruchirappalli - 620 023, Tamil Nadu, S. India 2 Assistant Professor, PG and Research Department of Mathematics National College, Tiruchirappalli - 620 001, Tamil Nadu, S. India

Abstract The binary quadratic equation 3x2 − 2y 2 + 6(2x − 2y − 1) = 1 is studied for tis non-trivial integral solutions. A few interesting relations among the solution are presented.

1. Introduction Binary Quadratic Diophantine equations are rich in variety. For an extensive review of sizable literature and various problems refer [1-9]. In this communications, we consider yet another interesting binary quadratic equation 3x2 − 2y 2 + 6(2x − 2y − 1) = 1 and obtain infinitely many non-trivial integral solutions. The recurrence relations satisfied by the values of x and y are given. Also, a few interesting relations between x and y are obtained. −−−−−−−−−−−−−−−−−−−−−−−−−−−−

Key Words : Binary Quadratic, Diophantine equation, Integral solutions. c http: //www.ascent-journals.com

319

320


2. Methods of Analysis The Diophantine equation under consideration is 3x2 − 2y 2 + 6(2x − 2y − 1) = 1

(1)

It is to be noted that (1) represents hyperbola. By taking the Linear Transformation. x = u + 2T, y = u + 3T, we get U 2 = 6(T + 1)2 + 1.

(2)

U 2 = 6X 2 + 1

(3)

X = T + 1.

(4)

Equation (2) can be written as

where

Equation (3) is known as Pell’s Equation. The general integral solution are given by Us +

√

√ 6Xs = (5 + 2 6)s+1

(5)

where s = 0, 1, 2, · · · . Since irrational roots occurs in Pairs, Us −

√

√ 6Xs = (5 − 2 6)s+1 .

(6)

Thus we get Us = Xs =

i √ √ 1h (5 + 2 6)s+1 + (5 − 2 6)s+1 2 i √ √ 1 h √ (5 + 2 6)s+1 − (5 − 2 6)s+1 . 2 6

Here the solutions of equation (1) is given by i i √ √ √ √ 1h 1 h xs = (5 + 2 6)s+1 + (5 − 2 6)s+1 + √ (5 + 6)s+1 − (5 − 2 6)s+1 − 2 2 2 6 i √ s+1 √ s+1 i √ √ 1h 3 h + (5 − 2 6) ys = (5 + 2 6) + √ (5 + 2 6)s+1 − (5 − 2 6)s+1 − 3 2 2 6 ‘ for s = 0, 1, 2, · · ·

321

ON BINARY QUADRATIC EQUATION...

The above values of xs and ys satisfy the following recurrence relations respectively. xs+2 − 10xs+1 + xs = 16 ys+2 − 10ys+1 + ys = 24. A few examples are given below

s=0 s−1 s=1 s=2 s=3 s=4 s=5

xs 7 87 879 8719 86327 854567 8459359

ys 8 106 1076 10678 105728 1046626 10360556

A few properties among the solutions are given below. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

It is observed that y values are even and the x values are odd. (3xs − 2ys )2 − 1 is a nasty number 6(3xs − 2ys )2 − 36(ys − xs + 1)2 is a nasty number (3xs − 2ys )2 + 6(ys − xs + 1)2 is the sum of two perfect square 66[xs+2 + ys+2 ] − 660[xs+1 + ys+1 ] + 66(xs + ys ) is a R2 -number 7[xs+2 − 10xs+1 + xs ][ys+2 − 10xs+1 + xs ] − 48 is a R2 -number 6x2s+1 − 4y2s+1 + 2 is a perfect square 6x2s+1 − 4y2s+1 − 2 is a nasty number 12[3x2s+1 − 2ys+1 ] + 12 is a nasty number 12[3x2s+1 − 2ys+1 ] − 12 is a perfect square.

It is observed that y values are even and that x values are odd.

3. Conclusion In conclusion one may search for other patterns of solutions.

322


References [1] Carmichael R. D., The Theory of Numbers and Diophantine Analysis, Dover Publications, New York (1959). [2] Dickson L. E., History of the Theory of Numbers, Vol.II, Chelisa Publishing Co., New York (1952). [3] Mollin R. A., All Solution of the Diophantine equation x2 − Dy 2 = n, for East J. Math. Sci., Social Volume, Part III, (1998), 257-293. [4] Mordell L. J., Diophantine Equations, Academic Press, London (1969). [5] Telang S. G., Number Theory, Tata McGraw - Hill Publishing Company, New Delhi (1996). [6] Nigel P. Smart, The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, London (1999). [7] Bhanumurthy T. S., A Modern Introduction to Ancient Indian Mathematics, Wiley Eastern Limited, London (1995). [8] Goplan M. A. and Ganapathy R., Note On The Diophantine Equation x2 − 3xy + y 2 + 10x − 10y − 21 = 0. Acta cienica Indica, XXVIIM(2) (2001), 267. [9] Gopalan M. A., Gokila K. and Vidhayalakashmi S., On The Diophantine Equation x2 − 4xy + y 2 − 2x + 2y − 6 = 0, Acta cienica Indica, XXXIII M.(2) (2007), 567.


DETERMINATION AND STABILIZATION OF THE ATTITUDE OF AN UNDERACTUATED MOBILE ROBOT USING QUATERNION AND LA SALLE INVARIANCE PRINCIPLE 1 ´ B. B. SALMERON-QUIROZ , G. VILLEGAS-MEDINA2 , S. A. 3 4 ´ RODRIGUEZ-PAREDES , J. R. AGUILAR SANCHEZ AND

1,2,3,4,5

5 ˜ P. NIN0-SUAREZ The Instituto Polit´ecnico Nacional (I.P.N.), SEPI ESIME UA, D.F., M´exico E-mail: 1 [email protected]

Abstract In the last decade, the problem of underactuated attitude control has received considerable interests. Underactuated attitude control means to use one or two actuators to realize three axes stabilization or tracking problem. This stems not only from a theoretical point of view but also from the practical value of the results. On the theoretical perspective, it has been established that underactuated attitude system has strong nonlinear properties and hence solution based on a linearization are not practical; on other side, underactuated attitude control strategy only needs less than three actuators to perform control algorithms, which can also be used to deal with the failure of one actuator. The present paper deals with the development of simple control laws for an embedded implementation. The rigid body orientation is modeled with quaternion, which eliminates attitude estimation singularities.This paper deals with the global asymptotical stabilization of a rigid body through a bounded quaternion-based feedback and torque estimation. It includes the gyro bias model. A quaternion measurement model is introduced. It avoids the linearization

−−−−−−−−−−−−−−−−−−−−−−−−−−−−

Key Words : Quaternion, Attitude, Stabilization, La Salle Invariance, Robot Mobile, MEM’s. c http: //www.ascent-journals.com

323

324

´ ´ ˜ B. B. SALMERON-QUIROZ, G. VILLEGAS-MEDINA, S. A. RODRIGUEZ-PAREDES, J. R. AGUILAR SANCHEZ & P. NIN0-SUAREZ

step that induces undesirable effects. The global convergence of the proposed technique is proved. Simulations and validation with some robustness tests are performed in a robot mobile.

1. Introduction The control of underactuated vehicles has received increasing interests in relation with various robotic applications. Underactuated vehicles are abundant in real life and appear in a broad range of applications including Robotics, Aerospace Systems, Marine Systems, Mobile Systems, Locomotive Systems, etc.. They are characterized by the fact that they have more degrees of freedom than actuator(s), so that the control design of these vehicles is often challenging. Underactuated mechanical systems are systems that have fewer control inputs than configuration variables. Underactuated systems appear in a broad range of applications including Robotics, Aerospace Systems, Marine Systems, Flexible Systems, Mobile Systems, and Locomotive Systems. The “underactuation” property of under actuated systems is due to the following four reasons [2]: (i) dynamics of the system (e.g. aircraft, spacecraft, helicopters, underwater vehicles, locomotive systems without wheels), (ii) by design for reduction of the cost or some practical purposes (e.g. satellites with two thrusters and flexible-link robots), (iii) actuator failure (e.g. in a surface vessel or aircraft), (iv) imposed artificially to create complex low-order nonlinear systems for the purpose of gaining insight in control of high-order underactuated systems (e.g. the Acrobot, the Pendubot, the Beam-and-Ball system, the Cart-Pole system, the Rotating Pendulum, the TORA system). The attitude estimation of an autonomous vehicle is a subject that has attracted a strong interest the last years. This is due to the fact that can be applied to multiple applications, such as spacecrafts, satellites, tactical missiles, where the attitude is essential for control or monitoring purposes. In the last decade the application of MicroElectro-Mechanical-Systems (MEMS) has gained a strong of interest. In addition to traditional attitude estimation in aerospace and automobile communities, the reduced cost of MEMS inertial sensors has spurred new applications in robotics [2], virtual reality [4] and biomechanics [3]. Furthermore, the increasingly interest in Unmanned Aerial Vehicles (UAVs) [5] has motivated the development of low cost, lightweight and low-power consumption Attitude and Heading Reference Systems (AHRS) and backup

DETERMINATION AND STABILIZATION OF THE ATTITUDE OF AN...

325

attitude indicators. An AHRS is composed of inertial and magnetic sensors, namely, three rate gyros, three accelerometer and three magnetometers, orthogonally mounted such that the sensor frame axes coincide with the body frame in question. In fact, an AHRS is an attitude estimator since the signal sensors are coupled with a proper mathematical background. This attitude estimation problem is described as following: Rate gyros provide continuous attitude information with good short-term stability when their measurements are integrated. The first one deals with a constraint least-square minimization problem proposed firstly by Wahba [6],(see [7] pages 9-11) and [7]. The second approach is within the framework of the Extended Kalman Filter [8] (EKF) or Additive Extended Kalman Filter (AEKF) [9]. The third approach issues from nonlinear theory, and non linear observers are applied to the attitude determination problem [11], [12], [13]. In this approach, the convergence of the error to zero is proved in a Lyapunov sense. In this paper, an attitude estimator using quaternion representation is studied. Here a novel method for solving Wahba’s problem is used. This method allows to find an quaternion from the measures provided by an Attitude and Heading Reference Systems (AHRS), the error between current and desired orientations is directly determined thanks to the measurements of the reference vectors delivered by the body’s sensors. The technique for the determination and stabilization of the attitude is designed assuming that the angular velocity sensors and the reference vectors sensors are embedded and the law uses only the measurements of the reference vectors and the necessary damping that is introduced by the angular velocity feedback is replaced by a linear filter of the measurement of at least two reference vectors. The present paper is organized as follows. In section II a quaternion-based formulation of the orientation of rigid body is given. The problem statement is formulated in section III. The control law and attitude’s estimation is presented in section IV. Simulation results are given in section V. The paper ends with some concluding remarks given in section VI.

326


2. Mathematical Background The attitude of a rigid body can be represented by a unit quaternion, consisting of a →

unit vector e , known as the Euler axis, and a rotation angle β about this axis. The quaternion q is then defined as follows:  cos β2 q= →

e sin

where



q0 → q

= β 2

! ∈H

(1)

→T →

H = {q|q02 + q q = 1 (2) →T

q = [q0 q ]T , q0 ∈ R,

→

q ∈ R3 }

→

q = [q1 q2 q3 ]T and q0 are known as the vector and scalar parts of the quaternion

respectively. In attitude control applications, the unit quaternion represents the rotation from an inertial coordinate system N (xn , yn , zn ) located at some point in the space (for instance, the earth NED frame (North, East, Down)), to the body coordinate system B(xb , yb , zb ) located on the center of mass of a rigid body. →

If r is a vector expressed in N , then its coordinates in B are expressed by: b=q⊗r⊗q →T

(3) →

→T

→

where b = [0 b ]T and r = [0 r ]T are the quaternions associated to vectors b and r

respectively. ⊗ denotes the quaternion multiplication and q is the conjugate quaternion of q, defined as: →T

q = [q0 − q ]T .

(4)

The rotation matrix C(q) corresponding to the attitude quaternion q, is computed as: →T →

→→T

C(q) = (q02 − q q )I3 + 2( q q

→x

−q0 [ q ])

(5)

where I3 is the identity matrix and [ξ x ] is a skew symmetric tensor associated with the axial vector ξ :

x   ξ1 0 −ξ3 ξ2 0 −ξ1  . [ξ x ] =  ξ2  =  ξ3 ξ3 −ξ2 ξ1 0 

(6)

→

Thus, the coordinate of vector r expressed in the B frame is given by: →

→

b = C(q) r

(7)


327

Figure 1 : Mobile Robot The attitude error is used to quantify the mismatch between two attitudes. If q defines the current attitude quaternion and qd is the reference quaternion, i.e. the desired orientation, then the error quaternion that represents the attitude error between the current orientation and the desired one is given by qe = q ⊗ qd−1 .

(8)

⊗ denotes the quaternion multiplication and qd−1 is the complementary rotation of the quaternion qd , which is the quaternion conjugate (see ([1]) for more details). The attitude dynamics of a rigid body is described by → → → ˙ J ω = − ω ×J ω +Γ

(9)

where J ∈ R3×3 is the symmetric positive definite constant inertial matrix of the rigid body expressed in the B frame and Γ ∈ R3 is the vector of control torques. Note that the torque also depend on the environmental disturbance (aerodynamic, gravity gradient, etc.).

3. Problem Statement In the case of the attitude estimation, one seeks to estimate the attitude and accelerations of a rigid body. From now on, it is assumed that the system is equipped with a tri-axis accelerometer, three magnetometer and three rate gyros mounted orthogonally. In this section we describe the bodys kinematic of the model [18], our typical capture configuration relies primarily on the Robot of figure 1 equipped with six 1 inch diameter

328


wheels driven by 1 DC gear head motors. The mechanical model (seen in figure 1, is based on single pinion architecture suitable for light vehicles and consists of following elements: a steering rack, a steering column coupled to the steering rack through a pinion gear, and the assist motor. Tie-rods connect the steering rack to the tires. The controller was able to set the power level to each motor independently, but there was no feedback loop based on tachometers or current sensing. In order to estimate the attitude robot position with respect to an inertial frame, a module containing three rate gyros, three accelerometer and three magnetometer assembled in tri-axis, are positioned near the centre of robot. The combination of this information jointly to knowledge a priori of the dynamic of the movement of the robot makes possible to obtain information of the attitude of the robot respect to the base. The equation describing the relation between the quaternion and the bodys kinematic →

is given in introducing the angular variation ω= [ωx ω ωz ]T from this, it follows. 1 1 → → q˙ = Ω( ω)q(t) = Ξ(q) ω (t) 2 2

(10)

→

where Ω( ω y Ξ(q) are defined as:   →  Ω( ω) =  

. → −[ ω x] .. ω .. ··· . ··· .. →T −ω . 0 →

    

(11)

 → q0 I3×3 + [ q ×]   ··· Ξ(q) =   →T − q

(12)



→

→

→

→

→

The matrix [ ω ×] and [ q ×] are obtained by the cross product issue of a × b = [ a ×]b →

con [ a ×] ∈ R3×3 : 

 0 −a3 a2 → 0 −a1  . [ a ×] =  a3 −a2 a1 0

(13)

The quaternion must be: →

q T q = q T q +q02 = 1.

(14)


329

In the other hand, the matrix Ξ(q) has the relation: ΞT (q)Ξ(q) = q T qI3×3 Ξ(q)ΞT (q) = q T qI4×4 − q T q ΞT (q)(q)

(15)

= 03×1

Generally ΞT (q)λ = −ΞT (λ)q, for any λ ∈ H. →T →

→→T

C(q) = (q02 − q q )I3×3 + 2 q q

→

−2q0 [ q ×]

(16)

that is denoted like the orientation matrix 3 − D of dimension 3 × 3. →

The angular velocity ω is obtained by finite differences from equation (10) at the instants k and k −1 (k estimation instant). →

ω= 2ΞT (q)q˙

→

ω= 2ΞT (q) ∗

q(k) − q(k − 1) Ts

(17) .

(18)

A. Modeling sensors In application of inertial and magnetic sensors, the inertial coordinate frame N is chosen to be the N ED (North, East, Down) coordinate frame. In this work the origin of N is located at Mexico City, (GP S : 190 290 5200 N, 990 70 3700 W (19 : 4977780 , −99 : 1269440 )). The “reference vectors” are the gravitational and magnetic vectors, which are well known. The “vectors observations”, i.e. the gravitational and magnetic vectors expressed in the body frame B, are obtained from an tri-axis accelerometer and a triaxis magnetometer sensors. The angular velocity is obtained from three rate gyros orthogonally mounted. →

(1) Rate Gyros : The angular velocity ω= [ω1 ω2 ω3 ]T is measured by the rate gyros, which are supposed to be orthogonally mounted. The output signal of a rate gyro is influenced by various factors, such as bias drift and noise. In the absence of rotation, the output signal can be modeled as the sum of a white Gaussian noise and of a slowly varying function, an integration process is required in order to obtain the current attitude quaternion.

330


The kinematics equation is given by:   q˙0     = → ˙ q

→T

 1 2



− q

 

→

I3 q0 + [ q ×]

→ ω

.

(19)

→ 1 ω 2 Ξ(q)

=

Even the smallest variation of the rate gyro measurement will produce a wrong estima→

tion of the attitude. The bias is denoted by ν , belonging to space R3 . The rate gyro measurements are modeled by [17]: → → ω G=ω

→

→

+ ν + ηG

→ ˙

(20)

→

→

ν = −T −1 ν + η ν

→

(21)

→

where η G and η ν ∈ R3 are supposed by Gaussian white noises and T = τ I3 is a diagonal matrix of time constants. In this case, the constant τ which has been set to τ = 100s. →

The bias vector ν will be estimated online, using the observer presented in the following section. (2) Accelerometers : Since the 3 − axis accelerometer is fixed to the body, the measurements are expressed in the body frame B. Thus, the accelerometer output can be written as: → b A= →

→

→

→

C(q)( a − g )+ η A

(22)

→

where g = [0 0 g]T and a ∈ R3 are the gravity vector and the inertial accelerations of the body respectively. Both are expressed in frame N.g = 9.81m/sec2 denotes the →

gravitational constant and η A ∈ R3 is the vector of noises that are supposed to be white Gaussian. →

(3) Magnetometers : The magnetic field vector h M is expressed in the N frame it →

is supposed to be h M = [hMx 0 hMz ]T . Since the measurements take place in the body frame B, they are given by: → b M=

→

→

C(q) h M + η M

→

(23)

where η M ∈ R3 , denotes the perturbing magnetic field. This perturbation vector is supposed to be modeled by Gaussian white noises.


331

4. Control Law Formulation In this section, a control law that stabilizes the system described by (19) and (9) is proposed. The goal is to design a control torque that is bounded. This is achieved with the use of the saturation function σM defined by the saturation function 1. Definition 1 (Saturation function) : Let σM : R → R denote the classical saturation function defined by : (1) σM (s) = s if |s| < M ; (2) σM (s) = sign(s)M elsewhere. With the above definition, our main result is the following: Theorem 1 : Consider the rigid body rotational dynamics described by (19) and (9) with the following bounded control inputs Γ = [Γ1 Γ2 Γ3 ]T defined by: Γi = −αi σMi (λi [ωi + ρi qi ])

(24)

σMi with i ∈ {1, 2, 3} are saturation functions with Mi ≥ 3λi ρi , αi , λi and ρi are strictly positive real parameters. αi Mi = Γi where Γi represents the physical bound on the i-th torque Γi . Then the inputs (24) almost globally asymptotically stabilize the rigid body →

→

to the origin (q0 = 1, q = 0 and ω= 0). Proof : Consider the candidate Lyapunov function V , V

→T →

=

T 1 → ω 2

K −1 J ω +((1 − q0 )2 + q q )

=

T 1 → ω 2

→ K −1 J ω

→

(25) +2(1 − q0 )

where J is defined as before and K = diag(κ1 , κ2 , κ3 ), with κi > 0 which must be determined. Furthermore, K −1 = γ1 J + γ2 I3 with γ1,2 > 0. Since K −1 J is positive definite, the function V is positive definite, radially unbounded and which belongs to the class C 2 . The derivative of (25) after using (19) and (9) is given by

332


V˙

→T → ˙ = ω K −1 J ω − 2q˙0 →T

→

→T →

→

= ω (γ1 J + γ2 I3 )(− ω ×J ω +Γ)+ q ω →T

→

→

→T

→

→

= γ1 ω J(− ω ×J ω + γ2 ω (− ω ×J ω) {z } | {z } | =0

(26)

=0

→T →

→T

+ ω (γ1 J + γ2 I3 )Γ+ q ω →T →

→T

→

→T →

= ω K −1 Γ+ q ω= (K −1 ω)T Γ+ q ω . → → ˜ Let us consider the following linear application ω = K −1 ω. In this case, V˙ becomes:

V˙

→T → → ˜T ˜ = ω Γ+ q K ω

˜ Γ +κ k ω ˜ +ω ˜ Γ + κ3 Γ3 + κ3 q3 ω ˜3 = ω ˜ +κ q ω ˜ +ω {z } | 1 {z1 1 }1 | 2 2 {z 2 2 }2 | 3 3 V˙ 2

V˙ 1

(27)

V˙ 3

V˙ is the sum of three terms V˙ 1 , V˙ 2 and V˙ 3 . Analyzing Vi for i ∈ {1, 2, 3}, one gets from Γi in (24) and equation (27). ˜i. V˙ i = −αi ω ˜ i σMi (λi [ωi + ρi qi ]) + κi qi ω

(28)

Assume that |ωi | > 2ρi , that is |ωi | ∈]2ρi , +∞[. Since |qi | ≤ 1, it follows that |ωi +ρi qi | ≥ ρi + for some strictly positive sufficiently small. Thus ωi + ρi qi has the same sign as ωi and since K is positive definite, ωi has the same sign as ω ˜ i . From equation (28) and the norm condition on the quaternion, V˙ i can be bounded as follows: V˙ i = αi ω ˜ i σMi (λi [ωi + ρi qi ]) + κi qi ω ˜i (29) ≤ −αi [˜ ωi |σMi (λi (ρi + )) + κi |˜ ωi | Taking κi < αi min(Mi , λi (ρi + ))

(30)

one can assure the decrease of Vi , more precisely: V˙ i ≤ −2[αi min(Mi , λi (ρi + )) − κi ]ρi .

(31)


333

Consequently, ωi enters Φi = {ωi : |ωi | ≤ 2ρi } in finite time t1 and remains in it thereafter. In this case, ωi + ρi qi ∈ [−3ρi , 3ρi ]. During that time, the quaternion can not diverge in finite time since it is structurally bounded. Let Mi be such that Mi ≥ 3λi ρi , equation (30) then becomes: κi < αi λi (ρi + ).

(32)

For t2 > t1 , the argument of σMi will be bounded as follows |λi (ωi + ρi qi )| ≤ 3λi ρi ≤ Mi .

(33)

Consequently, σMi operates in a linear region Γi = −αi λi [ωi + ρi qi ].

(34)

V˙ i = −αi λi ω ˜ i ωi − αi λi ρi ω ˜ i q i + κi ω ˜ i qi .

(35)

As a result, (28) becomes

Choosing κi = αi λi ρi which satisfies inequality (32), one obtains −1 2 2 V˙ i = −αi λi ω ˜ i ωi = −αi λi κ−1 i ωi = −ρi ωi ≤ 0.

(36)

The same argument is applied to V˙ 2 and V˙ 3 , (27) becomes V˙ = V˙ 1 + V˙ 2 + V˙ 3

(37)

−1 2 −1 2 2 = −(ρ−1 1 ω1 + ρ2 ω2 + ρ3 ω3 ) ≤ 0.

(38)

In order to complete the proof, the LaSalle Invariance Principle is invoked. All the trajectories converge to the largest invariant set Ω in Ω → →

→ →

→

Ω = {( q , ω) : V˙ = 0} = {( q , ω) : ω= 0}.

(39)

→ → ˙ In the invariant set, J ω = −[α1 λ1 ρ1 q1 α2 λ2 ρ2 q1 α3 λ3 ρ3 q3 ]T = A q = 0 with A ∈ R3×3

such that aji = αj λj ρj for i = j and aji = 0 otherwise and therefore to remain in Ω, →

one must satisfy q = 0 and from normality condition q0 = ±1. Actually, the points →

→

(q0 = ±1, q = 0, ω= 0) correspond, respectively, to a minimum (V = 0) and a local maximum (V = 4) of the Lyapunov function (25). Consequently, V˙ = 0 at these

334


equilibrium points. If at t0 = 0, the closed-loop system lies to local maximum, it remains in this point for t > t0 . Nevertheless, if at t0 the closed-loop system is away from these equilibrium points, and since V˙ < 0 outside the two equilibrium points, the →

→

system state will converge to the equilibrium point (q0 = 1, q = 0, ω= 0) and it will remain there for all subsequent time, since in this point V = V˙ = 0. This ends the 2

proof of the almost global asymptotic stability.

From the proof of the Theorem 1, if the closed-loop system is away from the equilibrium →

→

points q0 = ±1, q = 0, ω= 0), the system will asymptotically approach to the point →

→

→

(q0 = 1, q = 0, ω= 0), which can be considered an attractor point, whereas (q0 = −1, q = →

0, ω= 0) can be considered a repeller point. However, the repeller point becomes an attractor using the control law Γ= − αi σMi (|lai [ωi − ρi qi ]) instead of (24). In a practical context, it is essential to select the equilibrium point to be achieved in order to minimize the angular path. Therefore, applying Γi = −αi σMi (λi [ωi + sing(q0 )ρi qi ])

(40)

ensures that, of the two rotations of angle β and 2π − β, the one of smaller angle is chosen. Then, the control law (40) stabilizes globally asymptotically the two-point set in the quaternion space. This can be demonstrated by adapting the previous proof using the following Lyapunov function  2(1 − q0 ), if q0 ≥ 0 1 →T −1 →  V = ω K J ω+  2 2(1 + q0 ), if q0 < 0

(41)

Note that the stability analysis has been carried out considering the asymptotic condition q → qd = [1 0 0 0]T and with (40) it is possible to achieved qd = [±1 0 0 0]T . In the case where the asymptotic condition q → qd with qd 6= [±1 0 0 0]T is considered, the control law to be applied becomes Γi = −αi σMi (λi [ωi + sign(qe0 )ρi qei ])

(42)

where qe represents the attitude error between the current orientation and the desired one. Since it is assumed that the body-fixed control axes coincide with the principal axes of


335

inertia, it becomes: K = diag(κ1 , κ2 , κ3 ) (43) = diag(α1 λ1 ρ1 , α2 λ2 ρ2 , α3 λ3 ρ3 ). Observe that Jxx 6= Jyy 6= Jzz implies κ1 6= κ2 6= κ3 . Actually, (43) gives the conditions for the feedback gains selection. Consequently, the torque (control) bounds in each axis can be chosen differently on condition that the principal moments of inertia are different. Furthermore, note that γ1 = 0 implies K = I3 γ2−1 , in this case the gains are restricted to be identical in each axis, which is the case of the classic PD control.

5. Validation A. Simulation Results In this section, some simulation results are presented in order to show the performance of the proposed control laws. A rigid body with low moment of inertia is taken as the experimental system. In fact, the low moment of inertia makes the system vulnerable to high angular accelerations which proves the importance to apply the control.

Figure 2 : Estimation and Prediction of the Acceleration The proposed technique is compared to the existing methods (namely, the Multiplicative Extended Kalman Filter (MEKF) and the Additive Kalman Filter (AEKF)). Initial

336


conditions are set to extreme error values in order to assess the effectiveness of attitude estimation. These results are depicted in figures 2 and 3. The proposed method performances are similar to those of the Extended Kalman Filter (Multiplicative and Additive) (see figure 4). However, for extreme errors the convergence rate for our estimation - prediction is higher.

Figure 3 : Estimation and Prediction of the Quaternion

Figure 4 : Method proposed Vs. variant of kalman filter

337


B. Experimental results The estimation methodology proposed in this work is implemented and evaluated in real time in the mobile robot (figure 1, in order to assess its effectiveness. For this purpose an embedded system was designed, and developed. Special attention was paid to the low power consumption requirements and weight, leading to the selection of the Digital Signal Controller dsPIC which was used with a clock speed of 4MHz. It contains extensive Digital Signal Processor (DSP) functionality with a high performance 16 − bit microcontroller (MCU) architecture but without floating point unit. For this part of the work, numerical implementation is made. The discrete-time model that describes the attitude kinematics is expressed by:

Figure 5 : Experimental Data

→ q k+1

→

= e ξ k Ts (44)

→ q0

=

→

q (0).

→

The vector q k is the quaternion at the time instant kTs , where Ts is the systems →

sampling interval. Equation 44 is valid provided that the angular velocity ω k measured at the time instant kTs is assumed to be constant in the time interval [kTs , Ts ]. The

338


→

process of computing the initial condition q 0 is usually called alignment. Once 44 is solved, the attitude matrix can be updated via the equation 5. The attitude estimate is used twice during positioning estimation. The accelerometer senses the bodys own (non-gravitational) acceleration and the projection of the gravitational acceleration (equation 22). →N

The next step requires integration of a estimated. →N

p

Z (t) =

t

dt

0

Z

t0

→

→

(t) = C[ q (t)]T b A (t) to derive the position →

→

C T [ q (t00 )] b A (t00 )dt00 .

(45)

0

0

A numerical integration routine such as the trapezoidal rule can be used to solve 45 numerically. →

→

→

The normalized quaternion q i moves the unit sphere along an arc connecting q 0 to q e . Finally, the desired expression of the interpolated quaternion, which fulfils the initial and end conditions for a stride, is as follows: →u →i q k= q k

→

⊗ q k , k = 0, · · · , M.

(46)

The gravity-compensated acceleration is then single time- integrated; the initial and end conditions are imposed to avoid drift (null velocity), before positioning estimation. →

The angular velocity ω is obtained by finite differences at the instants k and k − 1 (k estimation instant). →

ω= 2ΞT (q)q˙ q(k) − q(k − 1) → T ω= 2Ξ (q) ∗ . Ts

(47) (48)

All sensors outputs are analog except the Micromag 3 which is digital and uses the Serial Peripheral Interface (SPI) bus system as underlying physical communication layer. The total system supply voltage is 3.3 V . The dimension and weight are 60×40×15mm and 60g, respectively. For purpose of validation A Commercial AHRS [19] is used to acquire the data instead of the MEMS sensors presented in section (III-A )( all embebed in the robot of figure 1). This AHRS also provides the Euler angles. The methodology is implemented in real-time using the LabView environment. Remember that the attitude estimate is computed using a unit quaternion formulation. For comparison purpose, the estimate quaternion is converted into Euler angles.


339

As can be shown, after large angular velocity change over a long period, the AHRS has a low convergence rate (approx. 1 min) compared to the one achieved with our proposed methodology. On the other hand, this system doesn’t provide the acceleration of the body so for validation we have done slowly movement and abrupt movement to appreciate the effect of the acceleration in our method. In figure 4 the global convergence of the estimation techniques is proved.

6. Conclusion and Future Works In this paper, a control law for the global stabilization of a rigid body was proposed. The presented methodology is especially simple. It is based on quaternion error and a nonlinear observer the attitude is parameterized by the unit quaternion. Furthermore, the proposed approach can be extended to the stabilization of a pico-satellite or a micro-satellite. Remain to perform several validations in the robot mobil and to those provided by a vision-based human motion capture system that will be used as a reference attitude estimation system and embebed in robots to assist people and improve human performance in daily and task-related activities, focusing in particular on populations with special needs, including those convalescing from trauma, rehabilitating from cognitive and/or physical injury, aging in place or in managed care, and suffering from developmental or other social, cognitive, or physical disabilities. Another application desired for the presented approach is the stabilization of micro-satellite and UAV, simulations using the dimension and the actuator characteristics of a pico-satellite and a micro-satellite and to compare the proposed approach with other control schemes.

Acknowledgement The authors would like to thank to the Instituto Polit´ ecnico Nacional (IPN) - SEPI ESIME Unidad Azcapotzalco and the SIP for the projects 20130784, 20130853 and 20130845, the CONACyT and the B.U.A.P-F.C.E.

340


References [1] Shuster M. D., A survey of attitude representations, Journal of the astronautical sciences, (1993), 439-517. [2] Roumeliotis S. I., Sukhatme G. S., and Bekey G. A., Smoother based 3D attitude estimation for mobile robot localization. In IEEE International Conference on Robotics and Automation, ICRA’99, (1999). [3] Fourati H., Manamanni N., Afilal L. and Handrich H., A nonlinear filtering approach for the attitude and dynamic body acceleration estimation based on inertial and magnetic sensors: Bio-logging application. IEEE Sensor Journal, 11(1) (2011), 233-243. [4] Hol J. D., Sch¨ on, Gustafsson F. and Slycke P. J¿, Sensor fusion for augmented reality. In The 9th International Conference on Information Fusion, (2006). [5] Hamel T., Mahony R. and Tayebi A., Introduction to the special issue on aerial robotics. Control Engineering Practice, 18(17), (2010). [6] Whaba G., A Least Squares Estimate of Spacecraft Attitude, SIAM Review, (1965), 409. [7] Choukroun D., Novel methods for attitude determination using vector observations, Israel Institute of Technology, Haifa, Israel, (2003). [8] Lefferts E. J. and Markley F. L. and Shuster M. D., Kalman Filtaring for Spacecraft Attitude Estimation,Journal of Guidance, Control, and Dynamics, (1982), 417-429. [9] Markley F. L., Attitude Error Representations for Kalman Filtering,Journal of Guidance, Control, and Dynamics, (2003), 311-317. [10] Markley F. L., Crassidis J. L. and Cheng Y., Nonlinear Attitude Filtering Methods,AIAA Guidance, Navigation, and Control Conference, (2005). [11] Salcudean S., A globally convergent velocity obsever for rigid body motion,IEEE Transactions on Automatic Control, (1991), 493-1497. [12] Vik B. and Fossen T. I., An Nonlinear Observer for GPS and INS Integration, 40th IEEE Conference on Decision and Control, (2001). [13] Thienel J. and Sanner R. M., A coupled nonlinear spacecraft attitude controller and observer with an unknown constant gyro bias and gyro noise, IEEE Transactions on Automatic Control, (2003), 2011-2014. [14] Beeby S., Ensell G., Kraft M. and White N., MEMS Mechanical Sensors, Artech House Inc., (2004). [15] Salmeron-Quiroz Bernardino Benito, Guerrero-Castellanos Jose Fermi, VillegasMedina Gerardo, Rodriguez-Paredes Salvador Antonio and Lam-Farias Luis, Estimation of the Attitude in an Industrial Robot: An Unit Quaternion Approach, Journal of Communication and Computer, 9(11) (November 2012), 1287-1292, ISSN: 1548-7709. [16] Salmeron-Quiroz B. B., Castellanos J.F.G., Paredes S.A.R., Villegas Medina G., Global estimation of robots attitude via quaternion and data fusion, Industrial Electronics and Applications (ICIEA), 2012 7th IEEE Conference on , (18-20 July 2012), 524-529.


[17] Brown R. G. and Hwang P.Y.C., Introduction to Random Signal and Applied Kalman Filtering, Wiley, New, (1997). [18] Crassidis J. L. F. and Markley L. A Minimum Model Error Approach for Attitude Estimation. Catholic University of America. Washington, USA, (2004). [19] microstrain http://www.microstrain.com/3dm-gx1.aspx August (2005).

341


CERTAIN FAMILY OF ANALYTIC AND MULTIVALENT FUNCTIONS WITH NORMALIZED CONDITIONS PRAVIN GANAPAT JADHAV Research Scholar, Shri Jagdishprasad Jhabarmal Tibrewala University, Jhunjhunu, Rajasthan, India Assistant Professor in Mathematics, Hon. Balasaheb Jadhav Arts, commerce and Science College, Ale, Pune (M.S.), India E-mail: [email protected]

Abstract There are many subclasses of analytic and multivalent functions. The aim of this paper is to define new classes of starlike and convex functions. We have attempted to obtain coefficient estimate, distortion theorem, radius of star likeness, convexity and closure theorem for the classes S ∗ (α, β, ξ, γ, p) and K ∗ (α, β, ξ, γ, p).

1. Introduction Let A denote the class of functions given by, ∞ X

f (z) = z p +

ak z k

(1.1)

k=1+p

which are analytic in the unit disc E = {z : |z| < 1}. Let S be the subclass of A consisting of analytic and multivalent functions of the form (1.1) . We denote by S ∗ (α) and K(α) the subclasses of S consisting of all functions which are, respectively starlike and convex of order α in E with 0 ≤ α < 1. Thus, −−−−−−−−−−−−−−−−−−−−−−−−−−−−

c http: //www.ascent-journals.com

343

344

PRAVIN GANAPAT JADHAV

∗

S (α) = and

K(α) =

f ∈ S : Re

zf 0 (z) f (z)

> α : 0 ≤ α < 1, z ∈ E

zf 0 (z) > α : 0 ≤ α < 1, z ∈ E . f ∈ S : Re 1 + f (z)

We say that a function f (z) is in the class S(α, β, ξ, γ, p) if and only if, zf 0 (z) − p f (z) 0, θ > −p, z ∈ E). The integral operator L(z) =

d+θ+p−1 θ+p−1

d zθ

Z 0

z

x d−1 xθ−1 1 − f (x)dx z

(7.1.3)

(d > 0, θ > −p, z ∈ E). We have from (7.1.2) and (7.1.3) ∞ X θ+p d K(z) = z − ak z k θ+k

(7.1.4)

∞ X Γ(θ + k)Γ(d + θ + p) L(z) = z − ak z k . Γ(d + θ + k)Γ(θ + p)

(7.1.5)

p

k=1+p

p

k=1+p

Theorem 12 : Let f ∈ S ∗ (α, β, ξ, γ, p) then K(z) ∈ S ∗ (α, β, ξ, γp). Proof : We have

∞ X θ+p d K(z) = z − ak z k . θ+k p

k=1+p

358

PRAVIN GANAPAT JADHAV

We need to prove that ∞ X k − p + β[(p − α)ξ + αγ − γk] θ + p d ak ≤ 1. β[(p − α)ξ + αγ − γp] θ+k

k=1+p

Since f ∈ S ∗ (α, β, ξ, γ, p) then from Theorem 1 we have ∞ X k − p + β[(p − α)ξ + αγ − γk] ak ≤ 1. β[(p − α)ξ + αγ − γp]

k=1+p

But

θ+p θ+k

d

< 1 therefore Theorem 12 holds and the proof is over.

Theorem 13 : Letf ∈ S ∗ (α, β, ξ, γ, p) then L(z) ∈ S ∗ (α, β, ξ, γ, p). Proof : ∞ X Γ(θ + k)Γ(d + θ + p) ak z k . L(z) = z − Γ(d + θ + k)Γ(θ + p) p

k=1+p

We need to prove that ∞ X k − p + β[(p − α)ξ + αγ − γk] θ + p d ak ≤ 1. β[(p − α)ξ + αγ − γp] θ+k

k=1+p

Since f ∈ S ∗ (α, β, ξ, γ, p) then from Theorem 1 we have ∞ X k − p + β[(p − α)ξ + αγ − γk] ak ≤ 1. β[(p − α)ξ + αγ − γp]

k=1+p

But

Γ(θ+k)Γ(d+θ+p) Γ(d+θ+k)Γ(θ+p)

< 1 therefore Theorem 13 holds and the proof is over.

Theorem 14 : Let f ∈ S ∗ (α, β, ξ, γ, p) then K(z) is starlike of order 0 ≤ σ < 1 in |z| < r1 where  r1 = inf  k

p−α k−α

θ+k θ+p

d

k − p + β[(p − α)ξ + αγ − γk] β[(p − α)ξ + αγ − γp]

1 ! k−p

 .

359

CERTAIN FAMILY OF ANALYTIC AND MULTIVALENT FUNCTIONS WITH...

d ∞ P θ+p

Proof : K(z) = z p −

k=1+p

θ+k

ak z k . It is sufficient to prove

(K(z))0

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