A numerically or theoretically. In the present study,

 

A Comparative Study Numerical Methods for Solving

 Integro – Differential Equations

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       Shams E. Ahmes#1

[email protected]

#Department of Mathematic, Faculty of Sciences and Arts-Tubarjal, Aljouf University, Saudi Arabia

. #Department of Mathematic, University of Gezira, Sudan

    

 

Abstract

           Our aim of this paper, is to introduce a comparative study to solve integro-differential equations by using different numerical methods, namely; the Adomian decomposition Sumudu transform method (ADSTM), the homotopy perturbation method (HPM), the Adomian decomposition Sumudu transform method with the Pade approximant (ADST – PA method), and the variational iteration method (VIM).

 

Keywords:  the Adomian decomposition Sumudu transform method; the homotopy perturbation method; the Pade approximant; the variational iteration method; integro-differential equation.

 

1.      Introduction

The nonlinear integro-differential equation has a major deal of purpose in many science applications such as glass- forming process, heat transformer, diffusion process in general, neutron diffusion, and biological spaces. In addition, it also can be found in applied mathematics, physics, and engineering applications. The wide use of these equations is the most important reason why they have drawn mathematician’s attention. Despite this, they are not easy to find an answer, either numerically or theoretically.

       In the present study, we consider the nonlinear integro- differential equation of the following type:

    ,                                                         (1)

With the initial condition;

.                                                                            (2)

Where  is known as the source term and  is a nonlinear function.

 

The main objective of this paper  is to introduce a comparative study to solve integro-differential equation (1) by using four of the most recently developed methods, the first method is the Adomian decomposition Sumudu transform method (ADSTM) developed by Devedra Kumar 14 and used in 15,16 among many others, the second method is Homotpy perturbation method (HPM) developed by He 9, the third method is Variational iteration method (VIM) developed by He 10-13, the fourth method is Adomian decomposition Sumudu transform method with Pade approximant (ADSTM-PA), this method depends on the Pade approximant 17, sumudu transform 1 and decomposition method 2-8, the fourth methods, which accurately compute the solution in a series form or in an exact form.

 

2.          Analysis of numerical methods

 

2.1.            Basic Idea of the (ADSTM)

                         In this section, Adomian decomposition Sumudu transform method is applied to the following classes of nonlinear integro-differential equation (1).

The method depends of first applying the Sumudu transformation of both sides of Eq. (1);

.                                                          (3)

Using the forms of the Sumudu transform, we get;

.                                        (4)

Using the initial condition (2), we have;

.                                         (5)

If we apply the inverse operator to both sides of the equation (5), we obtain:

                                                        (6)

In the Adomian decomposition Sumudu transform method we assume the solution as an infinite series, given as follows;

       ,                                                                                      (7)

Where the terms  are to be recursively computed. Also, the nonlinear term  is decomposed as an infinite series of Adomian polynomials:

  ,                                                                        (8)

Where  are determined by the following recursive relation:

                          .                                                                         

Using (7) and (8), we rewrite (6) as;

                      .                                       (9)

Applying the linearity of the Sumudu transform, we have;

                                                          (10)

Now, we define the following iterative algorithm:

                                                                             (11)

 

2.2.            Basic Idea of The Pade Approximant

 

          Here we will investigate the construction of the Pade approximates for the functions studied. The main advantage of the Pade approximation gives a better approximation of the function than truncating its Taylor series.

The Pade approximation of a function, symbolized by m / n, is a rational function defined by;

                                                                    (12)

Where we considered , and numerator, denominator have no common factors.

In The (ADSTM-PA) we use the method of the Pade approximation as an after – treatment method to the solution obtained by the (ADSTM). This after – treatment method improves the accuracy of the proposed method.

 

2.3.            Basic Idea of the (HPM)

 

         To explain (HPM), we consider (1) as;

    ,                                                         (13)

With solution .Now, we can define homotopy   by;

 ,                                                         (14)

Where  is a functional operator with a solution , obtained easily. Now, we choose a convex homotopy by;

                           .                                                                (15)

And continuously trace an implicitly defined curve from a starting point  to a solution function .  Here the parameter  is monotonically increasing from zero to unit along – with the trivial problem  is continuously deformed to the original problem .

The (HPM) uses the homotopy parameter  as an expending parameter to obtain;

  ,                                                         (16)

When , Eq. (16) becomes the approximate solution of (13), i.e.

 .                                                                     (17)

Series (17) is convergent for most cases, and the rate of convergence depends on .

 

2.4.            Basic Idea of the (VIM)

 

         To clarify the basic ideas of (VIM), we consider Eq. (1) as correction functional as follows;

  .                (18)

Where  is general Lagrange multiplier which can be identified optimally via integrating by parts. The successive approximations  for the solution  will be readily obtained upon using the Lagrange multiplier and by using the selective function .  Consequently, the exact solution may be obtained by using:

                     .

3.      Application

 

        In this section, we demonstrate the analysis of all the numerical methods by applying the methods to the following two integro- differential equations. A comparison is also given in the forms of graphs and tables, presented here.

 

Example 1: consider the following integro- differential equation 18:

,                                                                         (19)

With the initial condition;

.                                                                            (20)

Solution: Taking the Sumudu transform of both sides of (19) gives;

.                                                 (21)

Using the initial condition (20), we have;

.                                                                (22)

If we apply the inverse operator to both sides of the equation (22), we obtain:

                                                                                    (23)

By the assumption (7) and (8), we rewrite (23) as;

                                                                          (24)

Where the nonlinear term   is decomposed in terms of the Adomian polynomials as suggested in (8). Few terms of the Adomian polynomials for  are given as follows:

                      

And so on. Following the Adomian decomposition Sumudu transform method, we define an iterative scheme;

                                                                                              (25)

Consequently, we obtain:

                                                        (26)

 

 

Similarly, we can also find other components. Finally, the solution takes the following form;

  .                                               (27)

 

Notes on (ADSTM):

     From the previous analysis, we can observe that:

1.      (ADSTM) can obtain a series solution, not converge, which must be truncated. The truncated series solution is an inaccurate solution in that region, which will greatly restrict the application area of the method.

2.      (ADSTM) needs some modification to overcome the Taylor series does not converge.

To overcome these disadvantages of ADSTM, the following ADSTM – PA method is suggested.

 

The m / n Pade approximant of the infinite series (27), with  and , which gives the following fraction approximation to the solution:

  .                                                                                          (28)                                            

 

Example 2: consider the following integro- differential equation 18:

,                                                                         (29)

Given the initial condition;

.                                                                            (30)

With the exact solution:

 .                                                                                    (31)

Solution: Proceeding as in Example 1, Eq. (29) becomes:

 

.                                                 (32)

Where the nonlinear term   is decomposed in terms of the Adomian polynomials as suggested in (8). We have a few terms of the Adomian polynomials of  which are given by:

 

               

And so on. Following the Adomain decomposition Sumudu transform method, we define an iterative scheme;

                                                                           (33)

Now, we obtain the following components:

                                                               (34)

Similarly, we can also find other components. the solution takes the following form;

  .                                                             (35)

 

The m / n Pade approximant of the infinite series (35), with  and , which gives the following fraction approximation to the solution:

  .                                                                            (36)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Error (HPM)

Error (ADST-PA)

Error(ADSTM)

Exact Sol.

Step Size

0.0000

             0.0000

0.0000

0.0000

0.0000

0.0000

             0.0000

0.0000

0.1253

0.1250

0.0000

0.0000

0.0000

0.2526

0.2500

0.0000

0.0000

0.0000

0.3840

0.3750

0.0001

0.0000

0.0001

0.5219

0.5000

0.0003

0.0000

0.0003

0.6691

0.6250

0.0010

0.0000

0.0010

0.8292

0.7500

0.0030

0.0000

0.0030

1.0069

0.8750

0.0081

0.0000

0.0081

1.2085

1.0000

0.0198

0.0000

0.0198

1.4431

1.1250

0.0452

0.0000

0.0452

1.7243

1.2500

0.0979

0.0000

0.0979

2.0737

1.3750

0.2051

0.0001

0.2051

2.5275

1.5000

 

 

Table 1: Comparison of (ADSTM), (HPM) and (ADST – PA method), for Example 2 

 

 

 

Fig 1:

             Combine between (ADSTM), (HPM) and (ADST-PA), for Example 2.

 

Error (VIM)

Error (ADST-PA)

Exact Sol.

Step Size

0.0000

             0.0000

0.0000

0.0000

0.0000

             0.0000

0.1253

0.1250

0.0000

0.0000

0.2526

0.2500

0.0000

0.0000

0.3840

0.3750

0.0000

0.0000

0.5219

0.5000

0.0000

0.0000

0.6691

0.6250

0.0001

0.0000

0.8292

0.7500

0.0002

0.0000

1.0069

0.8750

0.0009

0.0000

1.2085

1.0000

0.0029

0.0000

1.4431

1.1250

0.0087

0.0000

1.7243

1.2500

0.0242

0.0000

2.0737

1.3750

0.0644

0.0001

2.5275

1.5000

 

      Table 2: Comparison of (ADST – PA method) and (VIM), for Example 2

Numerical outcomes shown in Table 2, 3:  explain the significance of (ADST – PA method) over another numerical method.

 

 

.

 

        

 

    Fig 2:

                      Combine between (ADST-PA) and (VIM), for Example 2.          

 

 

 

Conclusion:

 

     In this paper, we have studied a few recent familiar numerical methods for solving integro-differential equations. The numerical studies showed that all the method gives highly accurate results for given equations. The (ADSTM), the (HPM) and the (VIM) are simple and easy. Despite this, they are not converging to a closed form. Since the method of the (ADSTM) is based on an approximation of the solution function in this study by the truncating of approximation the solution, this kind of approximation is an inaccurate solution, which will greatly restrict the application area of the method. To get the better of these demerits, we use the Pade approximations. This fact is showing by the second example given in the study.

 

 

References

 

1 G. K. Weatugala, Sumudu transforms – a new integral transform to solve differential equations and control engineering problems, Mathematic Engineering, 61(1993): 319-329.

2 G. Adomian, Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Comput. Math. Appl. (12) (1990) 9-12.

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4 G. Adomian, Nonlinear stochastic operator equation, Academic Press, San Diego, (1986).

5 G. Adomian, Solving frontier problems of physics, the decomposition method, Kluwer Boston, (1994).

6 G. Adomian, Solution of physical problems by decomposition, Computers and Mathematics with Application, 2(1994): 145-154.

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8 G. Adomian and R. Rach, Modified Adomian polynomials, Mathematical and Computer Modeling, 24(1996): 39-46.

9 He JH. Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999; 178:257–62.

10 J.H. He, A new approach to nonlinear partial differential equations, Comm. Nonlinear Sci. Numer. Simul. 2 (1997) 230–235.

11 J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Eng. 167(1998) 57-68.

12 J.H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Eng; 167 (1998) 69–73.

13 J.H. He, Variational iteration method—a kind of non-linear analytical technique: some examples, Internat. J. Nonlinear Mech. 34 (1999) 699-708.

14 D. Kumar, J. Singh and S. Rathore; Sumudu decomposition method for nonlinear equations, International Mathematical Forum, 7(2012):515-521.

15 S. Ahmed and T. Elzaki, the solution of nonlinear Volterra integro-differential   equations of second kind by combine Sumudu transforms and Adomian decomposition method, International Journal of Advanced and Innovative Research, 2(2013):90-93.

16 H. Eltayeb, A. Kilicman and S.Mesloub, Application of Sumudu decomposition method to solve nonlinear systems Volterra integro- differential equations, Hindawi Publishing Corporation Abstract and Applied Analysis, 2014: 1- 6.

17 A.M. Wazwaz, Partial Differential Equations and Solitary Wave’s Theory, Springer, Higher Education Press, 2009.

18 D. Bahugna, A. Ujlayan and D.N. Pandey, A comparative study of numerical methods for solving an integro-differential equation, Computers and Mathematics with Applications, 57(2009):1485-1493.

 

 

 

 

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