Table of Contents
ISRN Computational Mathematics
Volume 2012, Article ID 704184, 6 pages
Research Article

A Comparative Study on the Stability of Laplace-Adomian Algorithm and Numerical Methods in Generalized Pantograph Equations

1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2Department of Mathematics, Sinnar University, Singa 107, Sudan

Received 31 May 2012; Accepted 29 July 2012

Academic Editors: T. Allahviranloo and K. T. Miura

Copyright © 2012 Sabir Widatalla. 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.


The main objective of this paper is to examine the stability and convergence of the Laplace-Adomian algorithm to approximate solutions of the pantograph-type differential equations with multiple delays. This is done by comparatively investigating it with other methods.

1. Introduction

Delay differential equations (DDEs) are a large and important class of differential equations in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. They often arise in wide and diverse range of applications in population studies [1, 2], economics [3], medical biology [4, 5], controls of mechanical systems [6], and so forth. The pantograph equation is one of the most important kinds of DDEs. The name pantograph was used by Iserles and Liu [7] to study how the electric current is collected by the pantograph of an electric locomotive, from where it gets its name.

In recent years, some promising approximate analytical solutions have been proposed, such as the Taylor collocation method [8], Bernstein polynomials [9], spline method [10, 11], and other methods are reviewed in [2, 12, 13].

In this work we consider the following problems

Problem 1. 𝑢𝑞(𝑡)=𝛽𝑢(𝑡)+𝑓𝑡,𝑢(𝑡),𝑢1𝑡𝑞,𝑢2𝑡𝑞,,𝑢𝑡,𝑢(0)=𝑢0.(1)

Problem 2. 𝑢(𝑚)𝑞(𝑡)=𝑓𝑡,𝑢(𝑡),𝑢1𝑡𝑞,𝑢2𝑡𝑞,,𝑢𝑡,𝑚1𝑘=0𝐶𝑖𝑘𝑢(𝑘)(0)=𝜆𝑖,𝑖=0,1,,𝑚1,(2)
where 𝑓 is an analytical function, 𝐶𝑖𝑘,𝜆𝑖, and 𝛽𝐶; 0<𝑞𝑖<1,   𝑖=1,2,,.

The aim of this paper is to employ and examine the stability of the Laplace-Adomian algorithm (LAA) for solving Problems 1 and 2. This method was first proposed by Khuri [14], 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 LAA are more accurate and efficient. LAA has been shown to accurately and easily approximate solutions of large class of linear and nonlinear ODEs and PDEs [1416]; for example, Ongun [17] employed LAA to give an approximate solution of nonlinear ordinary differential equation systems, such as a model for HIV infection of CD4+ T cells, Wazwaz [18] also used this method for handling the nonlinear Volterra integro-differential equations, Khan and Faraz [19] modified LAA to obtain series solutions of the boundary layer equation, and Yusufoglu [20] adapted LAA to solve the Duffing equation.

The numerical technique of LAA basically illustrates how the Laplace transforms are used to approximate the solution of the nonlinear differential equations by manipulating the decomposition method that was first introduced by Adomian [21, 22].

2. Laplace-Adomian Algorithm

To illustrate the basic idea of the Laplace-Adomian algorithm, we consider the following nonlinear operator: 𝑢(𝑡)=𝑅𝑢+𝑁𝑢+𝑓(𝑡),(3) with the initial condition 𝑢(0)=𝑢0,(4) where 𝑅 is a linear operator, 𝑁 is a nonlinear operator, and 𝑓(𝑡) is a given analytical function.

The technique consists first of applying the Laplace transform (denoted throughout this paper by ) to both sides of (3), to get [][][][].𝑢(𝑡)=𝑅𝑢+𝑁𝑢+𝑓(𝑡)(5) Applying the formulas of the Laplace transform, we obtain []1𝑢(𝑡)=(𝑠)+𝑠[]+1𝑅𝑢𝑠[],𝑁𝑢(6) where 1(𝑠)=𝑠[](𝑢(0)+𝑓(𝑡)).(7) Suppose the answer to (3) is as follows: 𝑢(𝑡)=𝑛=0𝑢𝑛(𝑡),(8) where the terms 𝑢𝑛(𝑡) are to be recursively computed, the nonlinear operator 𝑁𝑢 is decomposed as follows: 𝑁𝑢(𝑡)=𝑛=0𝐴𝑛,(9) where 𝐴𝑛is an infinite series of the Adomian polynomials 𝑢0,𝑢1,,𝑢𝑛, calculated by the formula [21] 𝐴𝑛=1𝑑𝑛!𝑛𝑑𝜆𝑛𝑁𝑛𝑖=0𝜆𝑖𝑢𝑖𝜆=0,𝑛=0,1,2,.(10)

Substituting (8) and (9) into (6) leads to 𝑛=0𝑢𝑛1(𝑡)=(𝑠)+𝑠𝑅𝑛=0𝑢𝑛+1𝑠𝑛=0𝐴𝑛.(11)

Using the linearity of the Laplace transform gives (11) as 𝑛=0𝑢𝑛(1𝑡)=(𝑠)+𝑠𝑛=0𝑅𝑢𝑛+1𝑠𝑛=0𝐴𝑛.(12)

Matching both sides of (11) yields 𝑢0=1(𝑡)𝑠[]𝑢(𝑢(0)+𝑓(𝑡))=(𝑠),(13)1=1(𝑡)𝑠𝑅𝑢0+1𝑠𝐴0,𝑢(14)2=1(𝑡)𝑠𝑅𝑢1+1𝑠𝐴1.(15)

Generally 𝑢𝑛+1=1(𝑡)𝑠𝑅𝑢𝑛+1𝑠𝐴𝑛,𝑛0.(16)

Applying the inverse Laplace transform to (13) gives the initial approximation 𝑢0(𝑡)=11𝑠[])(𝑢(0)+𝑓(𝑡)=𝐻(𝑡).(17)

Substituting this value of 𝑢0 into the inverse Laplace transform of (14) gives 𝑢1. The other terms 𝑢2,𝑢3, can be obtained recursively in similar fashion from 𝑢𝑛+1(𝑡)=11𝑠𝑅𝑢𝑛+1𝑠𝐴𝑛,𝑛0.(18)

Using (18), we can rewrite them as follows to obtain 𝑢: 𝑢0=11𝑠[])𝑢(𝑢(0)+𝑓(𝑡)=𝐻(𝑡),1=11𝑠𝑅𝑢0+1𝑠𝐴0=1𝑠1𝑢𝑅0+1𝑠1𝐴0𝑢2=11𝑠𝑅𝑢1+1𝑠𝐴1=1𝑠1𝑅1𝑠1𝑢𝑅0+1𝑠1𝐴0+𝑠1𝐴1=1𝑠1𝑅2𝑢0+1𝑠1𝑅1𝑠1𝐴0+1𝑠1𝐴1𝑢3=11𝑠𝑅𝑢2+1𝑠𝐴2=1𝑠1𝑅3𝑢0+1𝑠1𝑅21𝑠1𝐴0+1𝑠1𝑅1𝑠1𝐴1+1𝑠1𝐴2,𝑢𝑛+1=11𝑠𝑅𝑢𝑛+1𝑠𝐴𝑛=1𝑠1𝑅𝑛+1𝑢0+1𝑠1𝑅𝑛1𝑠1𝐴0+1𝑠1𝑅𝑛11𝑠1𝐴1++1𝑠1𝐴𝑛,.(19)

Subsiutiting the values of 𝑢0,𝑢1,𝑢2, into (8) gives 𝑢(𝑡)=𝐻(𝑡)+1𝑠1𝑢𝑅0+1𝑠1𝐴0+1𝑠1𝑅2𝑢0+1𝑠1𝑅1𝑠1𝐴0++1𝑠1𝑅𝑛𝑢0+1𝑠1𝑅𝑛1×1𝑠1𝐴0+1𝑠1𝑅𝑛21𝑠1𝐴1++1𝑠1𝐴𝑛=+𝑚=01𝑠1𝑅𝑚𝐻+(𝑡)𝑚=1𝑚1𝑘=01𝑠1𝑅𝑚𝑘11𝑠1𝐴𝑘.(20)

The sum of 𝑚 terms 𝑢0,𝑢1,𝑢2,,𝑢𝑚1 in (8), that is, 𝑄𝑚=𝑚1𝑛=0𝑢𝑛, where 𝑚, 𝑄𝑚 tends to 𝑢. This means that 𝑄𝑚 is an appropriate approximation of 𝑢. The terms in the above series soon tend to zero where 1/(𝑚𝑛)! has been the coefficients of calculations derived from the operation (1𝑠1), 𝑚 is the number of terms, and 𝑛 is the order of the operation derivation. Therefore, it has a rapid convergence.

3. Test Problems

In this section, we will apply Laplace-Adomian algorithm to solve the pantograph delay equation. The absolute errors in Tables 14 are the values of |𝑢(𝑡)𝑛𝑖=0𝑢𝑖(𝑡)| at selected points. All iterates are calculated by using Matlab 7.

Table 1: Comparison of the absolute errors for Example 1.
Table 2: Comparison of the absolute errors for 𝑞=0.2 for Example 2.
Table 3: Absolute errors for Example 3.
Table 4: Absolute errors for Example 4.
3.1. Stability of the Laplace-Adomian Algorithm

A method is said to be stable when the obtained solution undergoes small variations as there are slight variations in inputs and parameters and when probable perturbations in parameters that are effective in equations and conditions prevailing them do not introduce, in comparison to the physical reality of the problem, any perturbations in what is returned. We propose here to compare the Laplace-Adomian algorithm with other numerical methods by offering examples and examining the stability of the Laplace-Adomian algorithm.

Example 1 (Evans and Raslan [13]). Consider the following pantograph equation: 𝑢1(𝑡)=21𝑢(𝑡)+2𝑒(1/2)𝑡𝑢𝑡2,0𝑡1,𝑢(0)=1.(21)
The exact solution is 𝑢(𝑡)=𝑒𝑡. Applying the result of (17) gives us 𝑢0(𝑡)=11𝑠=1.(22)
The iteration formula (18) for this example is 𝑢𝑛+1(𝑡)=11𝑢2𝑠𝑛+𝑒(𝑡/2)𝑡𝑢𝑛𝑡2,𝑛0.(23)

In Table 1 we make a comparison between the absolute errors obtained by the Bernstien series [9] in column 2, together with the spline method [10, 11] in columns 3 and 4, and finally, the Laplace-Adomian algorithm with 𝑛=5,6, and 7 in the last three columns. Given the following absolute errors that were obtained from these methods, we can conclude that the rate of convergence of LAA is higher than the other methods. However, taking a closer look at the errors of LAA shows that the error increases significantly more than the other methods in the bottom of the columns. This means that the stability of LAA is decreasing more than the other methods.

Example 2 (Muroya et al. [23]). Consider the following pantograph equation: 𝑢𝑞(𝑡)=𝑢(𝑡)+2𝑢𝑞(𝑞𝑡)2𝑒𝑞𝑡,0𝑡1,𝑢(0)=1.(24)
The exact solution is 𝑢(𝑡)=𝑒𝑡. Applying the result of (17), gives us 𝑢0(𝑡)=11𝑠𝑞12𝑒𝑞𝑡=12+12𝑒𝑞𝑡.(25)
The other terms of the sequences 𝑢𝑛 can be obtained directly from (18) as in general 𝑢𝑛+1(𝑡)=11𝑠𝑢𝑛+𝑞2𝑢𝑛(𝑞𝑡),𝑛0.(26)
Table 2 compares the results of the Laplace-Adomian algorithm and the Taylor collocation method [8]. It can be seen that the Laplace-Adomian algorithm is weaker in the stability than Taylor collocation method and yet is stronger in the convergence.

Example 3 (Liu and Li [12]). Consider the multipantograph delay equation with variable coefficients 𝑢(𝑡)=𝑢(𝑡)+𝜇1(𝑡)𝑢(0.5𝑡)+𝜇2(𝑡)𝑢(0.25𝑡),0𝑡1,𝑢(0)=1.(27) Here 𝜇1(𝑡)=𝑒0.5𝑡sin(0.5𝑡), and 𝜇2(𝑡)=2𝑒0.75𝑡cos(0.5𝑡)sin(0.25𝑡). The initial approximation for this example is 𝑢0(𝑡)=11𝑠=1.(28) And from (18), we obtain 𝑢𝑛+1(𝑡)=11𝑠𝑢𝑛𝑒(1/2)𝑡𝑡sin2𝑢𝑛𝑡22𝑒(3/4)𝑡𝑡cos2𝑡sin4𝑢𝑛𝑡4,𝑛0.(29) Comparisons of approximate solutions for few terms with exact solution 𝑢(𝑡)=𝑒𝑡cos(𝑡) are illustrated in Table 3.

Example 4. Consider the pantograph equation of third order 𝑢(𝑡)=1sin(𝑡)2𝑢2𝑡2𝑡+2cos2𝑢𝑡2,0𝑡1,𝑢(0)=0.(30) Let us start with an initial approximation 𝑢0(𝑡)=11𝑠[]1sin(𝑡)=𝑡1+cos(𝑡).(31)
The other terms of the sequences 𝑢𝑛 can be obtained directly from (18) 𝑢𝑛+1(𝑡)=12𝑠𝐴𝑛𝑡2𝑡+cos2𝑢𝑛𝑡2,𝑛0,(32) where 𝐴𝑛 is the Adomian polynomials that represent the nonlinear term 𝑢2(𝑥/2). The Adomian polynomials for 𝐴(𝑢(𝑡))=𝑢2(𝑡/2) are given by 𝐴0=𝑢20𝑡2,𝐴(33)1=2𝑢0𝑡2𝑢1𝑡2,𝐴(34)2=𝑢21𝑡2+2𝑢0𝑡2𝑢2𝑡2,(35)(36)
Table 4 shows the absolute error of the numerical approximation by using a few iterations.

The results for Examples 34 in Figures 1 and 2, respectively, show that the approximations converge rapidly and only a few iterations are sufficient to obtain accurate solutions.

Figure 1: Error 𝐸(𝑡𝑖) for 𝑛=1,2 and 3 for Example 3.
Figure 2: Error 𝐸(𝑡𝑖) for 𝑛=1,2 and 3 for Example 4.

Example 5. Consider the multipantograph equation 5𝑢(𝑡)=6𝑡𝑢(𝑡)+4𝑢2𝑡+9𝑢3+𝑡21,0𝑡1,𝑢(0)=1.(37) Following the procedures in the previous examples gives us 𝑢0(𝑡)=11𝑠+2𝑠41𝑠2𝑡=1𝑡+33.(38)
According to (18), we obtain 𝑢1(𝑡)=736𝑡25𝑡122,𝑢2(𝑡)=1825𝑡722175𝑡2163,𝑢3(𝑡)=12775𝑡12963,𝑢𝑛(𝑡)=0,𝑛4.(39) Thus 𝑢(𝑡)=𝑛=0𝑢𝑛(𝑡)=1+676𝑡+1675𝑡722+12157𝑡12963(40) which is the exact solution.

4. Conclusion

In this paper, the Laplace-Adomian algorithm has been successfully applied to the generalized pantograph equations with multiple delays to obtain high approximate solutions or exact solutions with little iterations used. Moreover, the Laplace-Adomian algorithm is simple and easy to use and stronger in convergence compared with other methods. Despite these advantages, it is seen that its stability is lower than other numerical methods.


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