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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 714681, 9 pages
Research Article

Approximation Algorithm for a System of Pantograph Equations

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Department of Physics and Mathematics, College of Education, Sinnar University, Singa, Sudan

Received 23 November 2011; Accepted 2 January 2012

Academic Editor: Yuantong Gu

Copyright © 2012 Sabir Widatalla and Mohammed Abdulai 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(𝑡,𝑢(𝑡),𝑢(𝑞𝑡)),𝑡0,𝑢(0)=𝑢0,(1.1) 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:𝑢1(𝑡)=𝛽1𝑢1(𝑡)+𝑓1𝑡,𝑢𝑖(𝑡),𝑢𝑖𝑞𝑗𝑡,𝑢2(𝑡)=𝛽2𝑢2(𝑡)+𝑓2𝑡,𝑢𝑖(𝑡),𝑢𝑖𝑞𝑗𝑡,𝑢𝑛(𝑡)=𝛽𝑛𝑢𝑛(𝑡)+𝑓𝑛𝑡,𝑢𝑖(𝑡),𝑢𝑖𝑞𝑗𝑡,𝑢𝑖(0)=𝑢𝑖0,𝑖=1,,𝑛,𝑗=1,2,,(1.2) where 𝛽𝑖,𝑢𝑖0𝒞, and 𝑓𝑖 are analytical functions, and 0<𝑞𝑗<1.

In 2001, the Laplace decomposition algorithm (LDA) was proposed by khuri in [2], 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 [24]. 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].

2. Adaptation of Laplace Decomposition Algorithm

We illustrate the basic idea of the Laplace decomposition algorithm by considering the following system:𝐿𝑡𝑢1=𝑅1𝑢1,,𝑢𝑛+𝑁1𝑢1,,𝑢𝑛+𝑔1,𝐿𝑡𝑢2=𝑅2𝑢1,,𝑢𝑛+𝑁2𝑢1,,𝑢𝑛+𝑔2,𝐿𝑡𝑢𝑛=𝑅n𝑢1,,𝑢𝑛+𝑁𝑛𝑢1,,𝑢𝑛+𝑔𝑛.(2.1)

With the initial condition𝑢𝑖(0)=𝑢𝑖0,𝑖=1,,𝑛,(2.2) where 𝐿𝑡 is first-order differential operator, 𝑅𝑖 and 𝑁𝑖, 𝑖=1,,𝑛, are linear and nonlinear operators, respectively, and 𝑔𝑖, 𝑖=1,,𝑛, are analytical functions.

The technique consists first of applying Laplace transform (denoted throughout this paper by ) to the system of equations in (2.1) to get𝐿𝑡𝑢1𝑅=1𝑢1,,𝑢𝑛𝑁+1𝑢1,,𝑢𝑛𝑔+1,𝐿𝑡𝑢2𝑅=2𝑢1,,𝑢𝑛𝑁+2𝑢1,,𝑢𝑛𝑔+2,𝐿𝑡𝑢𝑛𝑅=𝑛𝑢1,,𝑢𝑛𝑁+𝑛𝑢1,,𝑢𝑛𝑔+𝑛.(2.3)

Using the properties of Laplace transform, and the initial conditions in (2.2) to get𝑢1=11(𝑠)+𝑠𝑅1𝑢1,,𝑢𝑛+1𝑠𝑁1𝑢1,,𝑢𝑛,𝑢2=21(𝑠)+𝑠𝑅2𝑢1,,𝑢𝑛+1𝑠𝑁2𝑢1,,𝑢𝑛,𝑢𝑛=𝑛1(𝑠)+𝑠𝑅𝑛𝑢1,,𝑢𝑛+1𝑠𝑁𝑛𝑢1,,𝑢𝑛,(2.4) where𝑖1(𝑠)=𝑠𝑢𝑖𝑔(0)+𝑖,𝑖=1,,𝑛.(2.5)

The Laplace decomposition algorithm admits a solution of 𝑢𝑖(𝑡) [2] in the form𝑢𝑖(𝑡)=𝑗=0𝑢𝑖𝑗(𝑡),𝑖=1,,𝑛,(2.6) where the terms 𝑢𝑖𝑗(𝑡) are to be recursively computed. The nonlinear operator 𝑁𝑖 is decomposed as follows:𝑁𝑖𝑢1,,𝑢𝑛=𝑗=0𝐴𝑖𝑗.(2.7) and 𝐴𝑖𝑗 are the so-called Adomian polynomials that can be derived for various classes of nonlinearity according to specific algorithms set by Adomian [9, 10].𝐴𝑖0𝑢=𝑓𝑖0,𝐴𝑖1=𝑢𝑖1𝑓𝑢𝑖0,𝐴𝑖2=𝑢𝑖2𝑓𝑢𝑖0+1𝑢2!2𝑖1𝑓𝑢𝑖0,𝐴𝑖3=𝑢𝑖3𝑓𝑢𝑖0+𝑢𝑖1𝑢𝑖2𝑓𝑢𝑖0+1𝑢3!2𝑖1𝑓𝑢𝑖0,.(2.8) Substituting (2.6) and (2.7) into (2.4),and Using the linearity of Laplace transform, we get𝑗=0𝑢1𝑗=1(1𝑠)+𝑠𝑗=0𝑅1𝑢1𝑗,,𝑢𝑛𝑗+1𝑠𝑗=0𝐴1𝑗,𝑗=0𝑢2j=2(1𝑠)+𝑠𝑗=0𝑅2𝑢1𝑗,,𝑢𝑛𝑗+1𝑠𝑗=0𝐴2𝑗,𝑗=0𝑢𝑛𝑗=𝑛1(𝑠)+𝑠𝑗=0𝑅𝑛𝑢1𝑗,,𝑢𝑛𝑗+1𝑠𝑗=0𝐴𝑛𝑗.(2.9)

We thus have the following recurrence relations from corresponding terms on both sides of (2.9): 𝑢𝑖𝑜(𝑡)=𝑖𝑢(𝑠),(2.10)𝑖1=1(𝑡)𝑠𝑅𝑢10,,𝑢𝑛0+1𝑠𝐴𝑖0𝑢,(2.11)𝑖2=1(𝑡)𝑠𝑅𝑢11,,𝑢𝑛1+1𝑠𝐴𝑖1,.(2.12)

Generally, 𝑢𝑖(𝑗+1)=1(𝑡)𝑠𝑅𝑢1𝑗,,𝑢𝑛𝑗+1𝑠𝐴𝑖𝑗.(2.13)

Applying the inverse Laplace transform to (2.10) gives the initial approximation𝑢𝑖0(𝑡)=1𝑖(𝑠),𝑖=1,,𝑛.(2.14) Substituting these values of 𝑢𝑖0 into the inverse Laplace transform of (2.11) gives 𝑢𝑖1. The other terms 𝑢𝑖2,𝑢𝑖3, can be obtained recursively in similar fashion from𝑢𝑖(𝑗+1)(𝑡)=11𝑠𝑅𝑢1𝑗,,𝑢𝑛𝑗+1𝑠𝐴𝑖𝑗,𝑗=0,1,2,.(2.15)

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 13 are the values of |𝑢𝑖(𝑡)𝑛𝑗=0𝑢𝑖𝑗(𝑡)|, those at selected points.

Table 1: Comparison of the absolute errors for Example 3.1.
Table 2: Comparison of the absolute errors for Example 3.2.
Table 3: Comparison of the absolute errors for Example 3.3.

Example 3.1. Consider the two-dimensional pantograph equations: 𝑢1=𝑢1(𝑡)𝑢2(𝑡)+𝑢1𝑡2𝑒𝑡/2+𝑒𝑡,𝑢2=𝑢1(𝑡)𝑢2(𝑡)𝑢2𝑡2+𝑒𝑡/2+𝑒𝑡𝑢1(0)=1,𝑢2(0)=1.(3.1) Applying the result of (2.14) gives us 𝑢10(𝑡)=42𝑒𝑡/2𝑒𝑡,𝑢20(𝑡)=22𝑒𝑡/2+𝑒𝑡.(3.2) The iteration formula (2.15) for this example is 𝑢1(𝑗+1)=11𝑠𝑢1𝑗(𝑡)𝑢2𝑗(𝑡)+𝑢1𝑗𝑡2,𝑢2(𝑗+1)=11𝑠𝑢1𝑗(𝑡)𝑢2𝑗(𝑡)𝑢2𝑗𝑡2.(3.3) Starting with an initial approximations 𝑢10(𝑡) and 𝑢20(𝑡) and use the iteration formula (3.3). We can obtain directly the other components as 𝑢11(𝑡)=14+6𝑡+𝑒𝑡𝑒𝑡2𝑒𝑡/24𝑒𝑡/28𝑒𝑡/4,𝑢21(𝑡)=128𝑡𝑒𝑡𝑒𝑡4𝑒𝑡/2+2𝑒𝑡/28𝑒𝑡/4,𝑢12(𝑡)=158+16𝑡+172𝑡22𝑒𝑡14𝑒𝑡/26𝑒𝑡/248𝑒𝑡/424𝑒𝑡/464𝑒𝑡/8,u22(𝑡)=158+16𝑡+172𝑡22𝑒𝑡14𝑒𝑡/26𝑒𝑡/248𝑒𝑡/424𝑒𝑡/464𝑒𝑡/8.(3.4)
Table 1 shows the absolute error of LDA with 𝑛 = 2, 4, and 6.

Example 3.2. Consider the system of multipantograph equations: 𝑢1(𝑡)=𝑢1(𝑡)𝑒𝑡𝑡cos2𝑢2𝑡22𝑒(3/4)𝑡𝑡cos2𝑡sin4𝑢1𝑡4,𝑢2(𝑡)=𝑒𝑡𝑢21𝑡2𝑢22𝑡2,𝑢1(0)=1,𝑢2(0)=0.(3.5) Let us start with an initial approximation: 𝑢10𝑢(𝑡)=1,20(𝑡)=0.(3.6) The iteration formula (2.15) for this example is 𝑢1(𝑗+1)=11𝑠𝑢1𝑗(𝑡)𝑢2𝑗(𝑡)+𝑢1𝑗𝑡2,𝑢2(𝑗+1)=11𝑠𝑒𝑡𝐴1𝑗𝐴2𝑗,(3.7) where 𝐴𝑖0=𝑢2𝑖0𝑡2,𝐴𝑖1=2𝑢𝑖0𝑡2𝑢i1𝑡2,𝐴𝑖2=𝑢2𝑖1𝑡2+2𝑢𝑖0𝑡2𝑢𝑖2𝑡2,𝐴𝑖3=2𝑢𝑖1𝑡2𝑢𝑖2𝑡2+2𝑢𝑖0𝑡2𝑢𝑖3𝑡2,,𝑖=1,2.(3.8)
Table 2 shows the absolute error of LDA with 𝑛=1,2, and 3.

Example 3.3. Consider the three-dimensional pantograph equations: 𝑢1(𝑡)=2𝑢2𝑡2+𝑢3𝑡(𝑡)𝑡cos2,𝑢2(𝑡)=1𝑡sin(𝑡)2𝑢23𝑡2,𝑢3(𝑡)=𝑢2(𝑡)𝑢1𝑢(𝑡)𝑡cos(𝑡),1(0)=1,𝑢2(0)=0,𝑢3(0)=0.(3.9) By (2.14) our initial approximation is 𝑢10𝑡(𝑡)=34cos2𝑡2𝑡sin2,𝑢20(𝑢𝑡)=sin(𝑡)+𝑡cos(𝑡)+𝑡,30(𝑡)=cos(𝑡)𝑡sin(𝑡)+1.(3.10)

The iteration formula (2.15) for this example is𝑢1(𝑗+1)=11𝑠2𝑢2𝑗𝑡2+𝑢3𝑗,𝑢(𝑡)2(𝑗+1)=11𝑠2𝐴2𝑗,𝑢3(𝑗+1)=11𝑠𝑢2𝑗(𝑡)𝑢1𝑗,(𝑡)(3.11) where𝐴20=𝑢220𝑡2,𝐴21=2𝑢20𝑡2𝑢21𝑡2,𝐴22=𝑢221𝑡2+2𝑢20𝑡2𝑢22𝑡2,𝐴23=2𝑢21𝑡2𝑢22𝑡2+2𝑢20𝑡2𝑢23𝑡2,(3.12)

Table 3 shows the absolute error of LDA with 𝑛=1,2, 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 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.


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