Comparison the two Methods for Solving Partial Differential Equations

Vol 65, No. 1;Jan 2015

Comparison the two Methods for Solving Partial Differential Equations in Electric N.F. Oyman Serteller1, D. Ustundag2 1

Faculty of Technology, Department of Electrical Engineering, Marmara University, Istanbul, Turkey. 2 Faculty of Art and Science, Department of Mathematics, Marmara University, Istanbul, Turkey

Abstract In this paper, Adomian Decomposition Method (ADM) with modified structure is applied for linear or non-linear partial differential equations (PDE), where mostly used in electrical systems and the results are compared with Finite Difference Methods (FDM). Three examples are presented: Lossless (nondissipating) long line system (one dimesion wave equation), a linear telegraph equation and nonlinear Klein Gordon equation. The study outlines the significant features of both methods. The results show that these methods are very efficient, convenient and can be applied to a large class of problems. Keywords: Adomian decomposition method, Finite Difference Method, linear-nonlinear PDE, Computational Algebra.

1. Introduction Modeling, simulation, control and design of many electrical systems involve dealing with linear or non-linear partial differential equation. Common analytical techniques, based on assuming that nonlinearities are relatively insignificant, sometimes strongly affect the solution with respect to the real physics of the phenomenon. Therefore, seeking exact and approximate solutions is still significant problem and it becomes increasingly important to be familiar with all traditional and recently developed methods for finding exact and approximate solutions and the implementations of these methods. Computer Algebra systems gives possibility to carry out not only numerical but, also symbolic computations. Many traditional algorithms can therefore be improved, sometimes considerably, via embedding symbolical parts into the numerical algorithms. so that those hybrid techniques involving numeric as well as symbolic manipulations provide arbitrary precision in defeating instability problems and reduce number of iterations in general. Solving linear or nonlinear partial differential equations has already been an interesting task for scientists. Numerical methods which are traditionally used such as finite difference and characteristic approaches need large consumption of computational work and usually the effect of rounding –off error causes loss of accuracy in the results (Cerri G.,et.al.,2008; Chapra S. et.al 2010; Leitao V.M.,2010; Oyman Serteller,N.F., et.al.,2006, Sadiku M.,2009). However, the Adomian decomposition method (Adomian, 1998; Machadol J. M., et.al, 2005; Khalifa A.K., et.al,2008) has been applied to a various class of stochastic and deterministic problems in different branches of applied science. For nonlinear models, the method has shown reliable results in supplying analytical approximation that converges rapidly. The advantage of this method is that it provides a direct scheme for solving nonlinear equations seen in transient phenomena in electric machine, wave equation and transient analysis in transmission lines without the need linearization, perturbation and any transformation (Haldar K. 2009; Kaya D., 2000; Khalifa et.al.,2008; Machado J.M. et. al, 2005; Mamis M.S.et.al., 1999;Michalik 207

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M., 2008.; Oyman Serteller N.F.,et.al, 2014; Srivatava V.K. et. al, 2013; Khasseinov K.A., 2013). In particular, the relevant three cases which have difficulties in resolving problems in electrical systems, such as structure of nonlinearity, inaccurate approximation and inrapidly convergent are studied. The main objective of this contribution is to introduce a comparative study for an accuracy of the methods and to show that both methods can be applied effectively and efficiently to those problems either linear or non-linear cases. Calculations are performed through Mathematica making some useful contribution to computer program and leveraging symbolic evaluations of ADM. 2. Adomian Decomposition Method The principal algorithm for the Adomian decomposition method when applied to a general nonlinear equation is in the form: (1)   Lu  Ru  Nu  g   . The linear terms are decomposed into L  R , while the nonlinear terms are represented by N . L is taken as to be term with the highest order derivative in order to avoid difficult integration involving complicated functions, and R is the remainder of the other linear terms. By multiplying both sides of Eq. (1) by L1 , that is a highest linear operator,

  L1 Lu  L1 g  L1 Ru  L1 Nu   

(2)

and solving Eq. (2) for u we get:

u  A  Bt  L1 g  L1 R u  L1 N u ,

(3)

where A and B are found from given boundary or initial conditions. This equation appears in diverse phenomena such as electromagnetic wave equation, thermal or acoustic equation, and transient equation in electrical system, electromagnetic wave equation in plasma and so on in other electrical system (Srivatava V.K. 2013; Pourghali, R., et.al.2010; Yudianto D.et.al.2010; Mohammed M.S.et.al., 2014; Raslan K.R., 2013; Krajewski W., 2004). Let u be expanded into infinite series as 

u   uk .

(4)

k 0

Then, the nonlinear term N will be decomposed by an infinite series of Adomian polynomials such as 

Nu   Ak ,

(5)

k 0

where Ak is so-called Adomian polynomial (Adomian, 1998; Raslan & Zain, 2013) formally introduce formulas 1  dn      Ak   k  N   λ i ui    , n  0,1, 2,3,... (6) k  dλ   k 0    λ 0 that can generate Adomian polynomial for all forms of nonlinearity. For a given initial and boundary conditions, the initial solution u0 becomes u0  A  Bt  L1 ( g ) . 208

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By substituting Eq. (7), (5) and (4) into Eq.(3) the solution can then be written as 

u k 0

k





k 0

k 0

 u0  L1 R  uk  L1  Ak .

(8)

Finally, all of uk ( x, t ) is calculable, and given in the following form:  p 1  (9) u ( x, t )  lim   uk  x, t   . p  k  0   The discussion of the convergence analysis of this decomposition series is not subject to this paper, but we refer to Adomian’s studies.

3. Numerical results and discussion In this section, we demonstrate the feasibility and efficiency of ADM and FDM through three cases. All of the numerical codes have been written in Mathematica 8.04 software and executed on a work station with four processors which of each has got Intel Core 2 Quad CPU with 2.4 GHz speed. Case1: The wave equation in one dimension comes across in Lossless (non-dissipating) long line system (Khalifa A.K., et. al. 2008; KhasseinovK.A.2013; Yudianto D. et.al.2010). As mathematically speaking, it is homogeneous linear two-dimensional wave equation  2 u( x, t ) 1  2 u( x, t )   u( x, t ) . (10)  t 2 x 2 defined on a region R  0,10,1 with specified initial conditions:

u (0, t )  0, u (0, t )  t

(11)

The parameter  is defined by  

1

where L and C is the inductance and LC capacitance of the line, respectively. To solve it by using ADM, we simply took Eq. (10) in an operator form in the same manner as given in Eq. (3) to compute u ( x, t ) . In order to start a recursive scheme defined in Eq. (8), we need to compute u 0 ( x, t ) by using Eq. (6) and found it as u 0 ( x, t )  x t . (12) Then, remaining components u1 , u2 , u3 ,... are successively determined using the recursive scheme either directly by hand or using symbolic computation with Mathematica: x 3  t u1  x, t    Lxx1  f  x, t   u0  x, t    Lxx1 Ltt  u0  x, t      3! x 5  t 1 1 u2  x, t    Lxx  f  x, t  u1  x, t     Lxx Ltt   u1  x, t      (13) 5! 









so on. Thus the components which constitute u ( x, t ) are written like this, 209

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u ( x, t )  x t  t (

x3 x5 x7 x9    ) 3! 5! 7! 9!

(14)

Factoring of Eq. (14) gives approximated solution of Eq. (10) as the exact answer in advance u ( x, y )  y Sinh ( x) graphically seen in Figure 1 ( a).

(a)

(b)

(c )

Figure 1. Solution of case 1 obtained by using Exact Solution, ADM, and IFDM (a), (b) and ( c) versus with grid points, respectively.

Let us reconsider now “case 2” which is a linear telegraph equation or linearized differential length of machine winding equation. Both of them use same differential equation with tiny differences (Oyman Serteller & Ustundag, 2014; Srivatava et. al.., 2013; Mamis et. al. , 1999; Krajewski, 2004) on a region defined by R  0,1 0,1 :

 2 u ( x, t )  2 u ( x, t ) u ( x, t )   u ( x, t )  2 . 2 2 t x t

(15)

Under the initial conditions,

u ( x,0)  Sin x , u t ( x ,0 )  0

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with the same manner as given in Eq. (6) we find the zeroth component of u0 ( x, t ) as u 0 ( x, t )  Sin x (17) and compute the remaining components, u1 , u2 , u3 ,... , recursively:

u1 ( x, t )  - t 2 Sin ( x) 2 3 1 t Sin (x)  t 4 Sin (x) 3 6 1 2 5 1 6 u 3 ( x, t )  - t 4 Sin xt Sin (x)t Sin(x) 3 15 90  By this way, we obtain the first few components of u ( x, t ) : u( x, t )  u( x, t )  u0 ( x, t )  u1 (x, t )  u2 ( x, t )  u3 ( x, t )  u 2 ( x, t ) 

2 1 2 1  Sin( x)  t 2Sin( x)  t 3Sin( x)  t 4Sin( x)  t 5Sin( x)  t 6Sin( x) 3 6 15 90

(18)

(19)

u( x, t )  u0 ( x, t )  u1 ( x, t )  .... 2 1 1  Sin( x)  y2 Sin( x)  y3Sin( x)  y4 Sin( x)  y4 (30  y(12  y))Sin( x) 3 6 90

(a)

(19)

(b)

Figure 2. Solution of case 2, obtained by using ADM and FDM. (a)and (b), respectively.

Lastly, we reconsider “case 3” in which one-dimensional Klein Gordon equation, which is a nonlinear partial differential equation u ( x, t )  2 u ( x, t )   (u ( x, t )) 2 , (20) t 2 x 2 defined on a region R   0,1  0,1 211

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u  x, 0   1  Sin  x 

. ut  x, 0   0 Using Eq (21) we get u0  x, t   u  x, 0   t  ut  x, 0   1  Sin  x  

(21)

(22)

and then

u1 ( x, t )  (1  Sin( x)) 2 u2 ( x, t )  

y2 1  y 2Sin(x)  y 2Sin(x) 2 2

(23)

 Continuing to this process, we find

u(x,t)u0 (x,t)u1 (x,t)u2(x,t) 

. (24) 1 (1Sin(x))(14472y2 18y4 5y6  y2(7224y2 7y4)Sin(x)) y4Cos(2x)(63y2  y2Sin(x))) 144

(a)

(b)

Figure 3. Solution of case 3, obtained by using ADM and FDM. (a) and (b), respectively.

In Figure 3, which is the most difficult equation among others, are studied successfully and shown in graphs. It is nonlinear and to solve it either by FDM or ADM is stressful and complex. It is clear that better approximations can be obtained by evaluating more components of u ( x, t ) . By using FDM for a  50  50  lattice points, since absolute error is so small. As studied, the results, shown in Figures 1 (a), (b), (c), Fig.2(a) and (b), Fig.3(a) and (b) indicated that using of few terms of ADM series give more accurate solution than that of FDM because of round off errors. Even though ADM and FDM looks like giving 212

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an exact solution of cases 1, 2 and 3. Additionally it is seen that absolute error degrees increase if we move to grid points away from initial boundary, but decreases by adding new terms of the series. In the programming, ADM’s absolute error is shown with red points; it is seen very small size on the right bottom of the figure 5. Moreover in figure 6 and figure 7, it is clear that since ADM does not require discretization of the variables its solution is not affected by computation round off errors therefore there is no need to use large consumption of CPU time and computer memory comparing with FDM.

Absolute Errors for ADM and IFDM

CPU Time

Error 18

5.  10 12

16

4.  10 12

12

Time (s)

14

3.  10 12

10

ADM

8

IFDM

6

2.  10 12

4 2

1.  10 12

0

20

40

60

80

100

Non-linear

Grid Points

Fig.5 Absolute error for cases

linear(Exp.1)

linear(Exp.2)

Fig.6 Consumption of CPU time

Storage Capacity 40 30

ADM

20

FDM

10 0 Example 1

Example 2

Example 3

Fig.7 Used storage need for program

4. Conclusion Preliminary studies on the linear, nonlinear and even inhomogeneous PDEs samples in electrical engineering are presented. Three complex and most encountered problems are solved by using ADM and FDM. Overall results indicate that ADM and FDM are reliable and efficient for studying analytically and numerically solution of transient and 213

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wave equations in electric. Both can solve various equations without failure or invalidation. However, ADM has merits and advantages over FDM because it doesn’t need any discretization and length of program coding is short and flexible. In addition, accuracy of result can be increased with use of larger Adomian polynomial terms but, ADM has only difficulties arising in calculation of Adomian Polynomials in nonlinear cases. Symbolic code in Mathematica offers promising and systematic way calculating Adomian polynomials for all forms of nonlinearity. Therefore, it deserves further investigations. References Adomian G., 1998. Solutions of nonlinear PDE. Appl Math Lett, 11, 121-123. Chapra, S.C., Canale R., P. 2010. Numerical Methods for Engineers ,6.edition , McGraw-Hill,852pp. Cerri, G., Moglie, F., Montesi R., Russo,.Vecchioni , P.E, 2008. FDTD Solution of the Maxwell Boltzman System for Electromagnetic Wave Propogation in Plasma, IEEE Transaction on Antennas and Propagation 56, 2584-2588. Haldar K., 2009. Application of Adomian’s approximation to Blood flow through arteiıes in the presence of a magnetic field, Applied Mathematics, 1, 17-28. Kaya D., 2000, An Application of the decomposition method second order wave equations. Int. J. Comp. Math. 75 235-245. Khalifa A.K., Aslan K.R., Alzubaidi H.M., 2008, Numerical Study Using ADM for the modified regularized Long wave equation",. App. Mathematical Modelling, 32, pp. 2962-2972. Leitao, V. M. A. 2010,Generalized finite differences using fundamental solutions, Int J Numer Meth Eng, 81, 564-583. Khasseinov, K. A.,2013. Stepwise reduci bility of the n - th order linear differential equations with variable coefficients. Jökull , 63,23-34. Krajewski W. 2004.Numerical modelling of the electric Field in HV sunstations", IEE Proc. SciMeas .Tech. 151, 267-272. Machado1 J. M., Verardi1 S.L. L., and Shiyou Y.,2005. An Application of the Adomian’s DecompositionMethod to the Analysis of MHD Duct FlowsBorner ", IEEE Transactıons on Magnetıcs, 41, 1588-1892. Mamis MS., Abbosov T., Herdem, Köksal M., ,1999 Tranient Analysis of Electrical Machine by Differantial Taylor Equation ,IPT'99,325-328. Michalik Dr M., Simulation and Analysis of Power System Transients, Lecture Notes e-mail: michalik @ pwr.wroc.pl Wrocław, February, 2008 Mohamed M.S., Malki F.Al & Altalhi N.. 2014. Analytic and approximate solution of time and space fractional nonlinear cubic equation via laplace transform.Jokull Journal,64 490-503. Oyman Serteller N.F. Atalay A., 2006.Thermal Analysis of Ferromagnetic Actuator by Using Finite Element Method, Elsevier Physica B, Condensed Matter, 3, 366-368. Oyman Serteller N.F ,Ustundag D. 2014, Numerical Solution of Energy Transmission Lines Equivalent Circuit Equations with Adomian Decomposition Method , CDSC 2014,3rd International Conference on Complex Dynamical System &Their Application, Ankara ,Turkey. Pourgholi, R., Rostamian, M. and Emamjome, M. 2010, A numerical method for solving a nonlinear inverse parabolic problem". Inverse Probl Sci En, 18, 1151-1164. Raslan K.R, Zain ,F.A.S. 2013, Comparison study between differential transform method and Adomian Decomposition method for some delay differential equation. Int. J. Phys. Sci.8 ,1880-1884. Sadiku M.N.O, 2009.Numerical Techniques in Electromagnetics with Matlab. Third edition, CRC press, 119pp. Srivatava V. K, Mukesh K.Awasthi, R.K. Charasia, M. Tamsir,The telegraph Equation and Its solution by reduced differantial transform method, Hindawi modelling and simulation,Agust 2013.pp. 16. Yudianto, D. and Xie, Y. B. 2010. A comparison of some numerical methods in solving 1-D steadystate advection dispersion reaction equation". Civ Eng Environ Syst, 27, 2, 155-172.

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