Approximation Algorithm for a System of Pantograph Equations

Abstract

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(𝑡𝑢,𝑢(0(𝑡))=,𝑢𝑢(₀𝑞𝑡)),𝑡≥0,,(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:𝑢𝑢^1(𝑡)=𝛽₁𝑢₁(𝑡)+𝑓₁𝑡,𝑢_𝑖(𝑡),𝑢_𝑖𝑞_𝑗𝑡_2,𝑢(𝑡)=𝛽₂𝑢₂(𝑡)+𝑓⋮₂𝑡,𝑢_𝑖(𝑡),𝑢_𝑖𝑞_𝑗𝑡_𝑛,𝑢(𝑡)=𝛽_𝑛𝑢_𝑛(𝑡)+𝑓_𝑛𝑡,𝑢_𝑖(𝑡),𝑢_𝑖𝑞_𝑗𝑡_𝑖,(0)=𝑢_𝑖0,𝑖=1,…,𝑛,𝑗=1,2,…,(1.2) where 𝛽_𝑖,𝑢_𝑖0∈𝒞, and 𝑓_𝑖 are analytical functions, and 0<𝑞_𝑗<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 [2–4]. 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].

2. Adaptation of Laplace Decomposition Algorithm

We illustrate the basic idea of the Laplace decomposition algorithm by considering the following system:𝐿𝐿_𝑡𝑡𝑢₁=𝑅₁𝑢₁,…,𝑢_𝑛+𝑁₁𝑢₁,…,𝑢_𝑛+𝑔₁,𝐿_𝑡𝑢₂=𝑅₂𝑢₁,…,𝑢_𝑛⋮+𝑁₂𝑢₁,…,𝑢_𝑛+𝑔₂,𝑢_𝑛=𝑅_n𝑢₁,…,𝑢_𝑛+𝑁_𝑛𝑢₁,…,𝑢_𝑛+𝑔_𝑛.(2.1)

With the initial condition𝑢_𝑖(0)=𝑢_𝑖0,𝑖=1,…,𝑛,(2.2) where 𝐿_𝑡 is first-order differential operator, 𝑅_𝑖 and 𝑁_𝑖, 𝑖=1,…,𝑛, are linear and nonlinear operators, respectively, and 𝑔_𝑖, 𝑖=1,…,𝑛, are analytical functions.

The technique consists first of applying Laplace transform (denoted throughout this paper by ℒ) to the system of equations in (2.1) to getℒℒ𝐿𝐿_𝑡𝑡𝑢₁=ℒ𝑅₁𝑢₁,…,𝑢_𝑛+ℒ𝑁₁𝑢₁,…,𝑢_𝑛+ℒ𝑔₁,ℒ𝐿_𝑡𝑢₂=ℒ𝑅₂𝑢₁,…,𝑢_𝑛⋮+ℒ𝑁₂𝑢₁,…,𝑢_𝑛+ℒ𝑔₂,𝑢_𝑛=ℒ𝑅_𝑛𝑢₁,…,𝑢_𝑛+ℒ𝑁_𝑛𝑢₁,…,𝑢_𝑛+ℒ𝑔_𝑛.(2.3)

Using the properties of Laplace transform, and the initial conditions in (2.2) to getℒ𝑢₁=ℋ₁(𝑠)+1𝑠ℒ𝑅¹𝑢₁,…,𝑢_𝑛+1𝑠ℒ𝑁¹𝑢₁ℒ𝑢,…,𝑢_𝑛₂,ℒ𝑢=ℋ₂(𝑠)+1𝑠ℒ𝑅²𝑢₁,⋮…,𝑢_𝑛+1𝑠ℒ𝑁²𝑢₁,…,𝑢_𝑛_𝑛,=ℋ_𝑛(𝑠)+1𝑠ℒ𝑅^𝑛𝑢₁,…,𝑢_𝑛+1𝑠ℒ𝑁^𝑛𝑢₁,…,𝑢_𝑛,(2.4) whereℋ_𝑖(𝑠)=1𝑠𝑢^𝑖(0)+ℒ𝑔_𝑖,𝑖=1,…,𝑛.(2.5)

The Laplace decomposition algorithm admits a solution of 𝑢_𝑖(𝑡) [2] in the form𝑢_𝑖(𝑡)=^∞_𝑗=0𝑢_𝑖𝑗(𝑡),𝑖=1,…,𝑛,(2.6) where the terms 𝑢_𝑖𝑗(𝑡) are to be recursively computed. The nonlinear operator 𝑁_𝑖 is decomposed as follows:𝑁_𝑖𝑢₁,…,𝑢_𝑛=^∞_𝑗=0𝐴_𝑖𝑗.(2.7) 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].𝐴𝐴_𝑖0𝑖1=𝑓𝑢_𝑖0,𝐴_𝑖2=𝑢_𝑖1𝑓^′𝑢_𝑖0,=𝑢_𝑖2𝑓^𝑢_𝑖0+12!𝑢^2𝑖1𝐴_𝑖3𝑓^𝑢_𝑖0,=𝑢_𝑖3𝑓^𝑢_𝑖0+𝑢_𝑖1𝑢_𝑖2𝑓^𝑢_𝑖0+13!𝑢^2𝑖1𝑓^𝑢_𝑖0,….(2.8) Substituting (2.6) and (2.7) into (2.4),and Using the linearity of Laplace transform, we get^∞_𝑗=0ℒ𝑢_1𝑗=ℋ₁(𝑠)+1𝑠^∞_𝑗=0ℒ𝑅₁𝑢_1𝑗,…,𝑢_𝑛𝑗+1𝑠^∞_𝑗=0ℒ𝐴_1𝑗,_∞_𝑗=0ℒ𝑢_2j=ℋ₂(𝑠)+1𝑠^∞_𝑗=0ℒ𝑅₂𝑢_1𝑗,…,𝑢_𝑛𝑗+1𝑠^∞_𝑗=0∞⋮ℒ𝐴_2𝑗,_𝑗=0ℒ𝑢_𝑛𝑗=ℋ_𝑛(𝑠)+1𝑠^∞_𝑗=0ℒ𝑅_𝑛𝑢_1𝑗,…,𝑢_𝑛𝑗+1𝑠ℒ^∞_𝑗=0𝐴_𝑛𝑗.(2.9)

We thus have the following recurrence relations from corresponding terms on both sides of (2.9): ℒ𝑢ℒ𝑢_𝑖𝑜(𝑡)=ℋ_𝑖(𝑠),(2.10)_𝑖1(𝑡)=1𝑠ℒ𝑅𝑢¹⁰,…,𝑢_𝑛0+1ℒ𝑢𝑠ℒ𝐴^𝑖0,(2.11)_𝑖2(𝑡)=1𝑠ℒ𝑅𝑢¹¹,…,𝑢_𝑛1+1𝑠ℒ𝐴^𝑖1,….(2.12)

Generally, ℒ𝑢_𝑖(𝑗+1)(𝑡)=1𝑠ℒ𝑅𝑢^1𝑗,…,𝑢_𝑛𝑗+1𝑠ℒ𝐴^𝑖𝑗.(2.13)

Applying the inverse Laplace transform to (2.10) gives the initial approximation𝑢_𝑖0(𝑡)=ℒ⁻¹ℋ_𝑖(𝑠),𝑖=1,…,𝑛.(2.14) Substituting these values of 𝑢_𝑖0 into the inverse Laplace transform of (2.11) gives 𝑢_𝑖1. The other terms 𝑢_𝑖2,𝑢_𝑖3,… can be obtained recursively in similar fashion from𝑢_𝑖(𝑗+1)(𝑡)=ℒ⁻¹1𝑠ℒ𝑅𝑢^1𝑗,…,𝑢_𝑛𝑗+1𝑠ℒ𝐴^𝑖𝑗,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 1–3 are the values of |𝑢_𝑖(𝑡)−∑^𝑛𝑗=0𝑢_𝑖𝑗(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: 𝑢^1=𝑢₁(𝑡)−𝑢₂(𝑡)+𝑢₁𝑡2−𝑒^𝑡/2𝑢₂+𝑒^−𝑡,′=−𝑢₁(𝑡)−𝑢₂(𝑡)−𝑢₂𝑡2+𝑒^𝑡/2+𝑒^𝑡𝑢₁(0)=1,𝑢₂(0)=1.(3.1) Applying the result of (2.14) gives us 𝑢𝑢₁₀₂₀(𝑡)=4−2𝑒^𝑡/2−𝑒^−𝑡,(𝑡)=2−2𝑒^−𝑡/2+𝑒^𝑡.(3.2) The iteration formula (2.15) for this example is 𝑢_1(𝑗+1)=ℒ⁻¹1𝑠ℒ𝑢^1𝑗(𝑡)−𝑢_2𝑗(𝑡)+𝑢_1𝑗𝑡𝑢2_2(𝑗+1),=ℒ⁻¹1𝑠ℒ−𝑢^1𝑗(𝑡)−𝑢_2𝑗(𝑡)−𝑢_2𝑗𝑡2.(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 𝑢𝑢₁₁₂₁(𝑡)=14+6𝑡+𝑒^−𝑡−𝑒^𝑡−2𝑒^−𝑡/2−4𝑒^𝑡/2−8𝑒^𝑡/4,𝑢₁₂(𝑡)=12−8𝑡−𝑒^−𝑡−𝑒^𝑡−4𝑒^−𝑡/2+2𝑒^𝑡/2−8𝑒^−𝑡/4,(𝑡)=158+16𝑡+172𝑡²u₂₂−2𝑒^−𝑡−14𝑒^𝑡/2−6𝑒^−𝑡/2−48𝑒^𝑡/4−24𝑒^−𝑡/4−64𝑒^𝑡/8,(𝑡)=158+16𝑡+172𝑡²−2𝑒^−𝑡−14𝑒^𝑡/2−6𝑒^−𝑡/2−48𝑒^𝑡/4−24𝑒^−𝑡/4−64𝑒^𝑡/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: 𝑢^′₁(𝑡)=−𝑢₁(𝑡)−𝑒^−𝑡cos𝑡2𝑢²2𝑡−2𝑒^−(3/4)𝑡cos𝑡𝑢_22sin4𝑡𝑢¹4𝑡,(𝑡)=𝑒^𝑡𝑢²¹𝑡𝑢₁2−𝑢²²2𝑡,(0)=1,𝑢₂(0)=0.(3.5) Let us start with an initial approximation: 𝑢𝑢₁₀(𝑡)=1,₂₀(𝑡)=0.(3.6) The iteration formula (2.15) for this example is 𝑢_1(𝑗+1)=ℒ⁻¹1𝑠ℒ𝑢^1𝑗(𝑡)−𝑢_2𝑗(𝑡)+𝑢_1𝑗𝑡𝑢2_2(𝑗+1),=ℒ⁻¹1𝑠ℒ𝑒^𝑡𝐴_1𝑗−𝐴_2𝑗,(3.7) where 𝐴_𝑖0=𝑢^2𝑖0𝑡𝐴_𝑖12,=2𝑢_𝑖0𝑡𝐴_𝑖22𝑢ⁱ¹2𝑡,=𝑢^2𝑖1𝑡𝐴_𝑖32+2𝑢^𝑖02𝑡𝑢^𝑖22𝑡,=2𝑢_𝑖1𝑡2𝑢^𝑖22𝑡+2𝑢^𝑖02𝑡𝑢^𝑖32𝑡,…,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: 𝑢^′₁(𝑡)=2𝑢₂𝑡2+𝑢³(𝑡)−𝑡cos𝑡𝑢_22,(𝑡)=1−𝑡sin(𝑡)−2𝑢²³𝑡𝑢_32,𝑢(𝑡)=𝑢₂(𝑡)−𝑢₁(𝑡)−𝑡cos(𝑡),₁(0)=−1,𝑢₂(0)=0,𝑢₃(0)=0.(3.9) By (2.14) our initial approximation is 𝑢₁₀(𝑡)=3−4cos𝑡𝑢₂₀2−2𝑡sin2𝑡,𝑢(𝑡)=−sin(−𝑡)+𝑡cos(𝑡)+𝑡,₃₀(𝑡)=−cos(𝑡)−𝑡sin(𝑡)+1.(3.10)

The iteration formula (2.15) for this example is𝑢_1(𝑗+1)=ℒ⁻¹1𝑠ℒ2𝑢^2𝑗2𝑡+𝑢^3𝑗𝑢(𝑡)_2(𝑗+1),=ℒ⁻¹1𝑢_3(𝑗+1)𝑠ℒ−2𝐴^2𝑗,=ℒ⁻¹1𝑠ℒ𝑢^2𝑗(𝑡)−𝑢_1𝑗(𝑡),(3.11) where𝐴₂₀=𝑢²²⁰𝑡𝐴₂₁2,=2𝑢₂₀𝑡𝐴₂₂2𝑢²¹2𝑡,=𝑢²²¹𝑡𝐴₂₃2+2𝑢²⁰2𝑡𝑢²²2𝑡,=2𝑢₂₁𝑡⋮2𝑢²²2𝑡+2𝑢²⁰2𝑡𝑢²³2𝑡,(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.