Applied Mathematical Sciences, Vol. 6, 2012, no. 82, 4075 - 4080
Fractional Calculus Model for Damped Mathieu Equation: Approximate Analytical Solution Abdelhalim Ebaid1 , Doaa M. M. ElSayed2 and Mona D. Aljoufi1 Department of Mathematics, Faculty of Science, Tabuk University P.O. Box 741, Tabuk 71491, Saudi Arabia1
[email protected] Department of Mathematics, Faculty of Education, Ain Shams University Roxy, Cairo, Egypt2 doaa−
[email protected] Abstract The classical damped Mathieu differential equation was firstly introduced by Mathieu in 1868 to determine the vibration modes of a stretched membrane having an elliptical clamped boundary. In this paper, we consider the fractional calculus model of damped Mathieu equation and introduce the approximate analytical solution by using two different methods, namely, Adomian decomposition method and a series method.
Keywords: : Adomian decomposition method; Damped Mathieu equation; Fractional calculus
1
Introduction
Mathieu equation was first introduced by Mathieu [1] in 1868 when he determined the vibration modes of a stretched membrane having an elliptical clamped boundary. The subject of infinite determinants to obtain the transition curves was subsequently introduced in 1886 [2]. Before launching into the main idea of the present work, we introduce the damped Mathieu equation as given in [1] by y¨(t) + y(t) ˙ + [a − 2q cos(2t)] y(t) = 0, (1) with y(0) = 1,
y(0) ˙ = 0.
(2)
Equation (1) is known as the classical damped Mathieu equation which is present in almost all physical systems. Our objective in this paper is to present
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a fractional calculus model to Eq. (1) and obtaining the approximate analytical solution using two different methods, Adomian decomposition method (ADM) and a series method.
2
Fractional calculus model
During the past 10 years or so, the fractional calculus starts to attract increasing attention of physicists and engineers from an application point of view. It was found that many systems in interdisciplinary fields can be elegantly described with the help of fractional derivatives such as viscoelastic systems, dielectric polarization, electromagnetic waves, quantitative finance, and quantum evolution of complex system. In an historical account, Srivastava and Saxena [3] cite applications of fractional calculus in fluid flow, rheology, diffusive transport, electrical networks, probability, viscoelasticity, electrochemistry of corrosion, and chemical physics. Reference [4] cites numerous others that apply fractional calculus to problems in viscoelasticity. Another historical account of fractional calculus is in the collection of papers on applications in physics edited by Hilfer [5]. The fractional derivative approach provides a powerful tool for modeling damped systems [6]. Accordingly, the fractional calculus model for equation (1) is assumed in the form 1/2
Dt2 y(t) + Dt y(t) + [a − 2q cos(2t)] y(t) = 0,
(3)
with y(0) = 1,
Dt y|t=0 = 0.
(4) 1/2
The damped term of the conventional model is replaced by Dt y(t). The mathematical definition of fractional calculus has been the subject of several different approaches [6]. The most frequently encountered definition of an integral of fractional order is the Riemann-Liouville integral, in which the fractional order integral is defined as I q f (x) =
d−q f (x) 1 = −q dx Γ(q)
x 0
f (t)dt , (x − t)1−q
(5)
while the definition of fractional order derivative is defined as Dxq f (x)
dm dq f (x) 1 = = dxq Γ(m − q) dxm
x 0
f (t)dt , (x − t)1+q−m
(6)
where q is the order of the operation which is positive real number and m is an integer that satisfies m − 1 ≤ q < m. In the next section, the Adomian decomposition method (ADM) shall be used to obtain the approximate solution for equation (3). The main advantage
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of the decomposition method is that it does not change the problem into a convenient one for the use of linear theory, see for examples [7-13]. Moreover, no linearization or perturbation is required and therefore it can reduce the labor of perturbation method. The solution of a fractional differential equation has been obtained through the ADM by [14].
3
Application of the ADM
The ADM is adopted in this section for solving Eq. (3). To do that, we may rewrite Eq. (3) as 1/2
Ly(t) = [2q cos(2t) − a] y(t) − Dt y(t),
(7)
1/2
where L = d2 /dt2 is an easily invertible linear operator, Dt (.) is the RiemannLiouville fractional derivative of order 1/2. According to the ADM, Eq. (7) can be written as 1/2
y(t) = y(0) + Dt y|t=0 + L−1 [2q cos(2t) − a] y(t) − L−1 [Dt y(t)], where L−1 = becomes
tt 0 0
(8)
dtdt. On using the given initial conditions, equation (8)
−1
y(t) = 1 + 2qL
∞
(−1)k (2t)2k 1/2 y(t) − aL−1 y(t) − L−1 [Dt y(t)], (2k)! k=0
(9)
In the light of the ADM the solution is expressed in a series form given by y(t) =
∞ n=0
yn (t).
(10)
Substituting (10) into (9), the following recurrence scheme is established for n ≥ 0: −1
yn+1 = 2qL
∞
(−1)k (2t)2k 1/2 yn−k (t) − aL−1 yn (t) − L−1 [Dt yn (t)], (11) (2k)! k=0
where y0 = 1. In view of (11) and by the help of the following fractional derivative formula Γ(α + 1) α−q Dtq tα = t , (12) Γ(α − q + 1) we can easily obtain the first few components as 2
y1 (t) = (2q − a) t2! − 4
t3/2 , Γ(5/2) 7/2
3
t y2 (t) = [(2q − a)2 − 8q] t4! − 2(2q − a) Γ(9/2) + 2 t3! ,
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y3 (t) = [(2q − a)3 − 56q(2q − a) + 32q] t6! − [3(2q − a)2 − 43q] × t11/2 Γ(13/2)
5
9/2
t + 32 (2q − a) t5! − 3 Γ(11/2) , 8
y4 (t) = [(2q − a)4 − 176q(2q − a)2 + 512q(2q − a) + 960q 2 − 128q] t8! − [4(2q − a)3 − 297q(2q − a) + 7
1283q t15/2 ] Γ(17/2) 4
13/2
+ 2 [6(2q − a)2 −
6
t + 4 t6! . 123q] t7! − 43 (2q − a) Γ(15/2)
(13)
4
Therefore, the five term-approximate solution φ5 (t) = tional damped Mathieu differential equation is given by 2
φ5 (t) = 1 + (2q − a) t2! −
t3/2 Γ(5/2)
n=0
yn (t) of the frac-
4
7/2
t + [(2q − a)2 − 8q] t4! − 2(2q − a) Γ(9/2) +
3
6
2 t3! + [(2q − a)3 − 56q(2q − a) + 32q + 4 ] t6! − [3(2q − a)2 − 43q] 11/2
5
9/2
t t + 32 (2q − a) t5! − 3 Γ(11/2) + ................ × Γ(13/2)
(14)
It is clear that all the components contain mixed terms with degrees not lees than 3/2, for each. So it may be difficult to obtain a closed form in general. Consequently, the practical solution may be taken as five-term approximation to y(t) as given by Eq. (14). Here, it is important to note that the approximate solution given above by the five-terms has been obtained by the ADM without and need to linearization or perturbation. The ADM has been rigorously proven to converge by [15] and applications have shown very accurate results.
4
Application of a series method
We can look for the solution y(t) of the equation (3) in the form of the fractional power series [3]: y(t) =
∞
n=0
n
yn (t)t 2 .
(15)
Substituting Eq. (15) into Eq. (3), it then follows ∞
n n=0 2
n 2
n
− 1 t 2 −1 yn + 2q
∞
n=0
∞
) (n−1) Γ( n+2 2 2 yn n=0 Γ( n+1 ) t 2
[n/4] (−1)k 22k k=0
(2k)!
+a
∞
n=0
n
yn (t)t 2 −
n
yn−4k (t)t 2 = 0.
(16)
Note that y0 = 1, by comparing the coefficients of the resulting fractional power series in equation (16) we obtain y1 = y2 = 0, y3 = y8 = y12 =
(2q−a)2 −8q , 4!
− , Γ( 52 )
y9 =
y4 =
(2q−a) , 2!
3 − Γ(11 ) , 2
4 +(2q−a)3 −56q(2q−a)+32q , 6!
y10 =
y5 = 0, y6 = 32 (2q−a) , 5! 3
2 , 3!
y11 =
2
2 −43q] − [3(2q−a) , Γ( 13 ) 2
y13 = − 4 Γ((2q−a) , y14 = 15 ) 2
y7 = − 2(2q−a) , Γ( 9 ) 2 [6(2q−a)2 −123q] . 7!
(17)
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Fractional calculus model for damped Mathieu equation
The results obtained above lead to a series solution of Eq. (3) as y(t) = 1 −
3
t 2 Γ( 52 )
9
3 t 2 Γ( 11 ) 2
+
+
(2q−a)t2 2!
32 (2q−a)t5 5!
+
−
2 t3 3!
−
7
2(2q−a)t 2 Γ( 52 )
[3(2q−a)2 −43q]t Γ( 13 ) 2
11 2
+
[(2q−a)2 −8q]t4 4!
+ .........,
− (18)
which is the same series solution obtained by using the ADM.
5
Conclusions
The fractional calculus model of the damped Mathieu differential equation is presented and discussed in the present paper. Two different methods are applied to search for the approximate analytical solution. It is found that the series solution obtained by applying the Adomain decomposition method is identical to the series solution obtained by using the series method. Although the series method is found easier to deal with the linear fractional model of the damped Mathieu equation, it is not so if a nonlinear term is involved. In the later case, the Adomian’s method can be applied easily whatever the form of the nonlinearity.
References [1] E. Mathieu, Memoire sure le Mouvement Vibratiore d’une Membrane de Forme Elliptique, J. Math., 13 (1868), 137 - 203. [2] A.H. Nayfeh, D.T. Mook Nonlinear Oscillations, New York, Wiley, 1979. [3] H. M. Srivasta, R.K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput., 118 (2001), 1 - 52 [4] R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201 - 210 [5] R. Hilfer (ED), Applications of Fractional Calculus in Physics, Applications of Fractional Calculus in Physics, 2000. [6] I. Podlubny, Fractional Differential Equations, Academic press, San Diego, California, 1999. [7] N. T. Eldabe, E. M. Elghazy, A. Ebaid, Closed form solution to a second order boundary value problem and its application in fluid mechanics, Phys. Lett. A, 363 (2007), 257 - 259.
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[8] A. Ebaid, A new numerical solution for the MHD peristaltic flow of a bio-fluid with variable viscosity in a circular cylindrical tube via Adomian decomposition method, Phys. Lett. A, 372 (2008), 5321 - 5328. [9] A. Ebaid, Exact solutions for a class of nonlinear singular two-point boundary value problems: The decomposition method, Z. Natur. A, 65 (2010), 145 - 150. [10] A. Ebaid, Modification of Lesnic’s approach and new analytic solution for second-order boundary value problems with Dirichlet boundary conditions, Z. Natur. A, 372 (2010), 692 - 696. [11] A. Ebaid, A new analytical and numerical treatment for singular twopoint boundary value problems via the Adomian decomposition method, J. Comput. Appl. Math. , 235 (2011), 1914 - 1924. [12] A. Ebaid, Approximate analytical solution of a nonlinear boundary value problem and its application in fluid mechanics, Z. Natur. A, 66 (2011), 423 - 426. [13] E.H. Ali, A. Ebaid, R. Rach, Advances in the Adomian decomposition method for solving two-point nonlinear boundary value problems with Neumann boundary conditions, Comput. Math. Applic., 63 (2012), 10561065. [14] S.S. Ray, R.K. Bera, Analytical solution of the Bagley Torvik equation by Adomian decomposition method, Appl. Math. Comput. , 168 (2005), 398 - 410. [15] T. Mavoungou and Y. Cherruault, Convergence of Adomian’s method and applications to nonlinear partial differential equations, Kybernetes 21 (6) (1992), 13 - 25. Received: March, 2012
