Research Article Approximation Algorithm for a System of Pantograph Equations

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1 Applied Mathematics Volume 01 Article ID pages doi:101155/01/ Research Article Approximation Algorithm for a System of Pantograph Equations Sabir Widatalla 1 and Mohammed Abdulai Koroma 1 1 Department of Mathematics Harbin Institute of Technology Harbin China Department of Physics and Mathematics College of Education Sinnar University Singa Sudan Correspondence should be addressed to Sabir Widatalla sabirtag@yahoocom Received 3 November 011; Accepted January 01 Academic Editor: Yuantong Gu Copyright q 01 S Widatalla and M A Koroma This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited We show how to adapt an efficient numerical algorithm to obtain an approximate solution of a system of pantograph equations This algorithm is based on a combination of Laplace transform and Adomian decomposition method Numerical examples reveal that the method is quite accurate and efficient it approximates the solution to a very high degree of accuracy after a few iterates 1 Introduction The pantograph equation: u t f u t u ( qt t 0 u 0 u 0 11 where 0 <q<1 is one of the most important kinds of delay differential equation that arise in many scientific models such as population studies number theory dynamical systems and electrodynamics among other In particular it was used by Ockendon and Tayler 1 to study how the electric current is collected by the pantograph of an electric locomotive from where it gets its name The primary aim of this paper is to develop the Laplace decomposition for a system of multipantograph equations:

2 Applied Mathematics u 1 t β 1u 1 t f 1 ui t u i ( qj t u t β u t f ui t u i ( qj t 1 u n t β n u n t f n ui t u i ( qj t u i 0 u i0 i 1n j 1 where β i u i0 Candf i are analytical functions and 0 <q j < 1 In 001 the Laplace decomposition algorithm LDA was proposed by khuri in who applied the scheme to a class of nonlinear differential equations In this method the solution is given as an infinite series usually converging very rapidly to the exact solution of the problem A major advantage of this method is that it is free from round-off errors and without any discretization or restrictive assumptions Therefore results obtained by LDA are more accurate and efficient LDA has been shown to easily and accurately to approximate a solutions of a large class of linear and nonlinear ODEs and PDEs 4 Ongun 5 for example employed LDA to give an approximate solution of nonlinear ordinary differential equation systems which arise in a model for HIV infection of CD4 T cells Wazwaz 6 also used this method for handling nonlinear Volterra integro-differential equations Khan and Faraz 7 modified LDA to obtain series solutions of the boundary layer equation and Yusufoglu 8 adapted LDA to solve Duffing equation The numerical technique of LDA basically illustrates how Laplace transforms are used to approximate the solution of the nonlinear differential equations by manipulating the decomposition method that was first introduced by Adomian 9 10 Adaptation of Laplace Decomposition Algorithm We illustrate the basic idea of the Laplace decomposition algorithm by considering the following system: L t u 1 R 1 u 1 u n N 1 u 1 u n g 1 L t u R u 1 u n N u 1 u n g 1 L t u n R n u 1 u n N n u 1 u n g n With the initial condition u i 0 u i0 i 1n where L t is first-order differential operator R i and N i i 1n are linear and nonlinear operators respectively and g i i 1n are analytical functions

3 Applied Mathematics 3 The technique consists first of applying Laplace transform denoted throughout this paper by L to the system of equations in 1 to get L L t u 1 L R 1 u 1 u n L N 1 u 1 u n L [ g 1 ] L L t u L R u 1 u n L N u 1 u n L [ g ] 3 L L t u n L R n u 1 u n L N n u 1 u n L [ g n ] Using the properties of Laplace transform and the initial conditions in to get L u 1 H 1 s 1 s L R 1 u 1 u n 1 s L N 1 u 1 u n L u H s 1 s L R u 1 u n 1 s L N u 1 u n 4 L u n H n s 1 s L R n u 1 u n 1 s L N n u 1 u n where H i s 1 s ( ui 0 L [ g i ] i 1n 5 The Laplace decomposition algorithm admits a solution of u i t in the form u i t u ij t i 1n 6 where the terms u ij t are to be recursively computed The nonlinear operator N i is decomposedasfollows: N i u 1 u n A ij 7

4 4 Applied Mathematics and A ij are the so-called Adomian polynomials that can be derived for various classes of nonlinearity according to specific algorithms set by Adomian 9 10 A i0 f u i0 A i1 u i1 f u i0 A i u i f u i0 1! u i1 f u i0 8 A i3 u i3 f u i0 u i1 u i f u i0 1 3! u i1 f u i0 Substituting 6 and 7 into 4 and Using the linearity of Laplace transform we get L [ ] u 1j H1 s 1 L [ ( ] 1 R 1 u1j u nj L [ ] A 1j s s L [ ] u j H s 1 L [ ( ] 1 R u1j u nj L [ ] A j s s L [ u n j ] H n s 1 s L [ ( ] 1 R n u1j u nj s L [ ] Anj 9 We thus have the following recurrence relations from corresponding terms on both sides of 9 : L u io t H i s L u i1 t 1 s L R u 10u n0 1 s L A i0 L u i t 1 s L R u 11u n1 1 s L A i Generally L [ u i j 1 t ] 1 s L[ R ( u 1j u nj ] 1 s L[ A ij ] 13 Applying the inverse Laplace transform to 10 gives the initial approximation u i0 t L 1 H i s i 1n 14

5 Applied Mathematics 5 Substituting these values of u i0 into the inverse Laplace transform of 11 gives u i1 The other terms u i u i3 can be obtained recursively in similar fashion from [ 1 u i j 1 t L 1 s L[ R ( ] ] 1 u 1j u nj s L[ ] A ij j To provide clearly a view of the analysis presented above three illustrative systems of pantograph equations have been used to show the efficiency of this method 3 Test Problems All iterates are calculated by using Matlab 7 The absolute errors in Tables 1 3 are the values of u i t n u ij t those at selected points Example 31 Consider the two-dimensional pantograph equations: u 1 u 1 t u t u 1 u u 1 t u t u e t/ e t e t/ e t 31 u u 0 1 Applying the result of 14 gives us u 10 t 4 e t/ e t u 0 t e t/ e t 3 The iteration formula 15 for this example is [ ( ( ] 1 t u 1 j 1 L 1 s L u 1j t u j t u 1j [ ( ( ] 1 t u j 1 L 1 s L u 1j t u j t u j 33

6 6 Applied Mathematics Table 1: Comparison of the absolute errors for Example 31 t Exact solution u 1 u 1 e t n n 4 n E 3 110E 5 154E E 438E E E E E E E 4460E E 1 536E 59E 3 u e t u n n 4 n E E 519E E E E 1668E E E E 1 166E 7E E E E 1668E E E 0 163E E 3 Starting with an initial approximations u 10 t and u 0 t and use the iteration formula 33 We can obtain directly the other components as u 11 t 14 6t e t e t e t/ 4e t/ 8e t/4 u 1 t 1 8t e t e t 4e t/ e t/ 8e t/4 u 1 t t 17 t e t 14e t/ 6e t/ 48e t/4 4e t/4 64e t/8 34 u t t 17 t e t 14e t/ 6e t/ 48e t/4 4e t/4 64e t/8 Table 1 shows the absolute error of LDA with n 4 and 6 Example 3 Consider the system of multipantograph equations: ( ( t t u 1 t u 1 t e t cos u e 3/4 t cos u t et u 1 ( ( t t u sin 4 ( t u u u 0 0 Let us start with an initial approximation: u 10 t 1 u 0 t 0 36

7 Applied Mathematics 7 Table : Comparison of the absolute errors for Example 3 t Exact solution u 1 u 1 e t cos t n 1 n n E E 443E E E E 474E E E E E 119E E 1 171E 1 474E 740E E E 1 895E 1960E u sin t u n 1 n n E 1 73E 5174E E E 1 104E E E E 1 575E 1 630E 38E E 1 508E 1 80E 176E E E E E The iteration formula 15 for this example is [ ( ( ] 1 t u 1 j 1 L 1 s L u 1j t u j t u 1j u j 1 L 1 [ 1 s L( e t A 1j A j ] 37 where A i0 u i0 ( t A i1 u i0 A i u i1 A i3 u i1 ( t u i1 ( ( ( t t t u i0 u i ( ( ( t t t u i u i0 u i3 i 1 38 Table shows the absolute error of LDA with n 1 and 3 Example 33 Consider the three-dimensional pantograph equations: u 1 t u ( t u 3 t t cos u t 1 t sin t u 3

8 8 Applied Mathematics u 3 t u t u 1 t t cos t u u 0 0 u By 14 our initial approximation is ( t u 10 t 3 4 cos t sin u 0 t sin t t cos t t 310 u 30 t cos t t sin t 1 The iteration formula 15 for this example is [ ( ( ] 1 t u 1 j 1 L 1 s L u j u 3j t u j 1 L 1 [ 1 s L( A j ] 311 u 3 j 1 L 1 [ 1 s L( u j t u 1j t ] where A 0 u 0 A 1 u 0 A u 1 A 3 u 1 ( t u 1 ( ( t t u 0 u ( ( ( t t t u u 0 u 3 31 Table 3 shows the absolute error of LDA with n 1 and 3 4 Conclusion The main objective of this paper is to adapt Laplace decomposition algorithm to investigate systems of pantograph equations We also aim to show the power of the LAD method by reducing the numerical calculation without need to any perturbations discretization or/and other restrictive assumptions which may change the structure of the problem being

9 Applied Mathematics 9 Table 3: Comparison of the absolute errors for Example 33 t Exact solution u 1 u 1 cos t n 1 n n E 1 14E 155E E E E 474E E E 1 074E 1 367E 8051E E E 1 404E 665E E 1 590E E E u tcos t u n 1 n n E 1 139E E E E 1 105E 584E E E E 1139E 1408E E E 31E 6069E E 1 158E E E u 3 sin t u 3 n 1 n n E 1 785E E E E 1 45E 1666E 99595E E E 45691E 48397E E E E E E E E E solved LDA method gives rapidly convergent successive approximations through the use of recurrence relations We believe that the efficiency of the LDA gives it a much wider applicability References 1 J R Ockendon and A B Tayler The dynamics of a current collection system for an electric locomotive Proceedings of the Royal Society of London A vol 3 no 1551 pp S A Khuri A Laplace decomposition algorithm applied to a class of nonlinear differential equations Applied Mathematics vol 1 no 4 pp S A Khuri A new approach to Bratu s problem Applied Mathematics and Computation vol 147 no 1 pp M Khan M Hussain H Jafari and Y Khan Application of laplace decomposition method to solve nonlinear coupled partial differential equations World Applied Sciences Journal vol 9 pp M Y Ongun The Laplace Adomian Decomposition Method for solving a model for HIV infection of CD4 T cells Mathematical and Computer Modelling vol 53 no 5-6 pp A-M Wazwaz The combined Laplace transform-adomian decomposition method for handling nonlinear Volterra integro-differential equations Applied Mathematics and Computation vol 16 no 4 pp Y Khan and N Faraz Application of modified Laplace decomposition method for solving boundary layer equation King Saud University Science vol 3 no 1 pp E Yusufoğlu Numerical solution of Duffing equation by the Laplace decomposition algorithm Applied Mathematics and Computation vol 177 no pp G Adomian Solving Frontier Problems of Physics: The Decomposition Method vol 60 of Fundamental Theories of Physics Kluwer Academic Publishers Dordrecht The Netherlands G Adomian A review of the decomposition method in applied mathematics Mathematical Analysis and Applications vol 135 no pp

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