Abstract and Applied Analysis

Volume 2013 (2013), Article ID 278097, 6 pages

http://dx.doi.org/10.1155/2013/278097

Research Article

Optimal Homotopy Asymptotic Method for Solving the Linear Fredholm Integral Equations of the First Kind

Mohammad Almousa and Ahmad Ismail

School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Gelugor, Penang, Malaysia

Received 20 April 2013; Accepted 16 June 2013

Academic Editor: Santanu Saha Ray

Copyright © 2013 Mohammad Almousa and Ahmad Ismail. 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.

Abstract.

The aim of this study is to present the use of a semi analytical method called the optimal homotopy asymptotic method (OHAM) for solving the linear Fredholm integral equations of the first kind. Three examples are discussed to show the ability of the method to solve the linear Fredholm integral equations of the first kind. The results indicated that the method is very effective and simple.

1. Introduction

Integral equations of the first kind arise in several applications. These include applications in biology, chemistry, physics, and engineering. In recent years, much work has been carried out by researchers in mathematics and engineering in applying and analyzing novel numerical and semi analytical methods for obtaining solutions of integral equations of the first kind. Among these are the homotopy analysis method [1], operational Tau method [2], homotopy perturbation method [3], Adomian decomposition [3], quadrature rule [4], and automatic augmented Galerkin algorithms [5].

In this study, we develop the optimal homotopy asymptotic method (OHAM), which was proposed by Marinca et al. [6, 7], for solving the linear Fredholm integral equations of the first kind. This method is characterized by it is convergence criteria which are more flexible than other methods.

The general form of the linear Fredholm integral equations of the first kind is

where and are constant and the functions and are known.

It should be noted that OHAM has been applied to the nonlinear Fredholm integral equations of the second kind by [8].

2. Application of OHAM to the Linear Fredholm Integral Equations of the First Kind

In this section, we formulate the optimal homotopy asymptotic method (OHAM) for solving the linear Fredholm integral equations of the first kind following the procedure as outlined in [6, 7] and other papers. Let us consider a form of the linear Fredholm integral equation of the first kind:

Using OHAM, we can obtain a family of equations as follows:

where is an embedding parameter, is unknown function, and is an (nonzero) auxiliary function for and and given as where , , are auxiliary constants, and when and it holds that

respectively. For obtaining the approximate solution, we use Taylor’s series expansion about as follows:

If the series (5) convergence occurs when , one has

Substituting (5) in (3) and equating the coefficients of like powers of , we get as follows:

For finding the constants , we can get the result of the th-order approximations as follows:

If we substitute (8) into (1) we obtain the residual equation

If , then will be the exact solution. The least squares method can be used to determine . At first we consider the functional

By using Galerkin’s method we get the following system:

and then minimizing it to obtain the values of , we have

With these constants, the approximate solution is determined.

3. Numerical Examples and Discussion

In this section, three examples of the linear Fredholm integral equations of the first kind were solved to show the efficiency of the present method. Maple software with long format and double accuracy was used to carry out the computations.

Example 1. We consider the following equation [9]:

for which the exact solution is . Applying OHAM to the linear Fredholm integral equation of first kind yields

which satisfies

Now we use (7) to obtain a series of problems:

Hence the solutions are

By substituting , , and solutions in (6), we obtain

For the calculations of the constants and , the use of the technique mentioned in (8)–(12) yields

Substituting values in (18), the final solution becomes

This is the exact solution.

Table 1 shows some numerical results of these solutions calculated according to the present method.

Table 1: Numerical results of Example 1.

0
0
0
0

0.1
0.09983341665
0.09983341665
0

0.2
0.1986693308
0.1986693308
0

0.3
0.2955202067
0.2955202067
0

0.4
0.3894183423
0.3894183423
0

0.5
0.4794255386
0.4794255386
0

0.6
0.5646424734
0.5646424734
0

0.7
0.6442176872
0.6442176872
0

0.8
0.7173560909
0.7173560909
0

0.9
0.7833269096
0.7833269096
0

1.0
0.8414709848
0.8414709848
0

The exact solution, OHAM solution and absolute error of this example are shown in Figure 1.

(a) Results for Example 1

(b) Absolute error for Example 1

Figure 1

Example 2. We consider the following equation [10]:

for which the exact solution is . Applying OHAM to the linear Fredholm integral equation of first kind yields

which satisfies

Now we use (7) to obtain a series of problems:

Hence the solutions are

By substituting , , and solutions in (6), we obtain

For the calculations of the constants and , the use of the technique mentioned in (8)–(12) yields

Substituting values in (26), the final solution becomes

This is the exact solution.

Table 2 shows some numerical results of these solutions calculated according to the present method.

Table 2: Numerical results of Example 2.

0
0
0
0

0.1
0.005
0.005
0

0.2
0.02
0.02
0

0.3
0.045
0.045
0

0.4
0.08
0.08
0

0.5
0.125
0.125
0

0.6
0.18
0.18
0

0.7
0.245
0.245
0

0.8
0.32
0.32
0

0.9
0.405
0.405
0

1.0
0.5
0.5
0

The exact solution, OHAM solution and absolute error of this example are shown in Figure 2.

(a) Results for Example 2

(b) Absolute error for Example 2

Figure 2

Example 3. We consider the following equation [9]:

for which the exact solution is . Applying OHAM to the linear Fredholm integral equation of first kind yields

which satisfies

Now we use (7) to obtain a series of problems:

Hence the solutions are

By substituting , , and solutions in (6), we obtain

For the calculations of the constants and , the use of the technique mentioned in (8)–(12) yields

Substituting values in (34), the final solution becomes

This is the exact solution.

Table 3 shows some numerical results of these solutions calculated according to the present method.

Table 3: Numerical results of Example 3.

0
0
0
0

0.1
0.01
0.01
0

0.2
0.04
0.04
0

0.3
0.09
0.09
0

0.4
0.16
0.16
0

0.5
0.25
0.25
0

0.6
0.36
0.36
0

0.7
0.49
0.49
0

0.8
0.64
0.64
0

0.9
0.81
0.81
0

1.0
1.0
1.0
0

The exact solution, OHAM solution and absolute error of this example are shown in Figure 3.

(a) Results for Example 3

(b) Absolute error for Example 3

Figure 3

4. Conclusions

In this paper, we presented the application of the OHAM in solving the linear Fredholm integral equations of the first kind. This method was tested on three different examples. This method proved to be an accurate and efficient technique for finding approximate solutions for the linear Fredholm integral equations of the first kind.

References

A. Adawi, F. Awawdeh, and H. Jaradat, “A numerical method for solving linear integral equations,” International Journal of Contemporary Mathematical Sciences, vol. 4, no. 9–12, pp. 485–496, 2009.

M. H. AliAbadi and S. Shahmorad, “A matrix formulation of the tau method for Fredholm and Volterra linear integro-differential equations,” The Korean Journal of Computational & Applied Mathematics, vol. 9, no. 2, pp. 497–507, 2002.

S. Abbasbandy, “Numerical solutions of the integral equations: homotopy perturbation method and Adomian's decomposition method,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 493–500, 2006.

F. Mirzaee, “Numerical solution for Volterra integral equations of the first kind via quadrature rule,” Applied Mathematical Sciences, vol. 6, no. 17–20, pp. 969–974, 2012.

S. Abbasbandy and E. Babolian, “Automatic augmented Galerkin algorithms for linear first kind integral equations: non-singular and weak-singular kernels,” Bulletin of the Iranian Mathematical Society, vol. 21, no. 1, pp. 35–62, 1995.

V. Marinca and N. Herişanu, “Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer,” International Communications in Heat and Mass Transfer, vol. 35, no. 6, pp. 710–715, 2008.

N. Herişanu, V. Marinca, T. Dordea, and G. Madescu, “A new analytical approach to nonlinear vibration of an electrical machine,” Proceedings of the Romanian Academy. Series A, vol. 9, no. 3, pp. 229–236, 2008.

M. S. Hashmi, N. Khan, and S. Iqbal, “Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind,” Applied Mathematics and Computation, vol. 218, no. 22, pp. 10982–10989, 2012.

A. Sulaiman and I. Hassan, “Successive approximation method (S.A.M.) for solving integral equation of the first kind with symmetric kernel,” Journal of Education and Sciences, vol. 21, no. 4, pp. 149–159, 2008.

A. J. Mohammed and J. I. Mustafa, “Construction of a new technique in Aitken extrapolation method for solving Fredholm integral equation of the first kind with iterated kernel,” Journal of Education and Sciences, vol. 21, no. 2, pp. 143–149, 2008.