Vol.08 No.02(2017), Article ID:74419,6 pages
Numerical Solution of Nonlinear Mixed Integral Equation with a Generalized Cauchy Kernel
Fatheah Ahmed Hendi1, Manal Mohamed Al-Qarni2
1Department of Mathematics, Faculty of Science, King Abdul Aziz University, Jeddah, KSA
2Department of Mathematics, Faculty of Science, King Khalid University, Abha, KSA
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: January 25, 2017; Accepted: February 25, 2017; Published: February 28, 2017
In this article, we present approximate solution of the two-dimensional singular nonlinear mixed Volterra-Fredholm integral equations (V-FIE), which is deduced by using new strategy (combined Laplace homotopy perturbation method (LHPM)). Here we consider the V-FIE with Cauchy kernel. Solved examples illustrate that the proposed strategy is powerful, effective and very simple.
Singular Integral Equation, Linear and Nonlinear V-FIE, Homotopy Perturbation Method (HPM), Cauchy Kernel
The V-FIE arises from parabolic boundary value problems. Many authors have interested in solving the linear and nonlinear integral equation. The time collocation method was introduced by Pachpatta  and the projection method by Hacia  . Brunner  extended the Pachpatta’s  results to nonlinear Volterra-Hammerstein integral equations. In  treated Maleknejad and Hadizadeh V-FIE by using the Adomian decomposition method (ADM) presented in  . Wazwaz  introduced the modified ADM for solving the V-FIE.
We consider the nonlinear mixed V-FIE with a generalized singular kernel
The functions, and are given and called the kernel of Fredholm integral term, Volterra integral term and the free term respectively and denotes a (real or complex) parameter. Also, is the domain of integration with respect to position, and the time. While is the unknown function to be determined in the space. The existence and uniqueness results for Equation (1) were found in   .
Many authors have studied solutions of linear and nonlinear integral equations by utilizing different techniques, for example Abdou et al. in   considered the integral equation with singular kernel and used Toeplitz matrix method (TMM) and product Nystrom method (PNM) to obtain the solution. Abdou et al. in  discussed the solution of linear and nonlinear Hammerstein integral equations with continuous kernel and used two different methods (Adomian decomposition method and homotopy analysis method). In  , El-Kalla and Al-Bugami used ADM and degenerate kernel method for solving nonlinear V-FIE with continuous kernel.
With the quick advancement of nonlinear sciences, many analytical and numerical techniques have been produced and developed by various scientists, for example, the HPM introduced by He   . Many research works have been conducted recently in applying this method to a class of linear and nonlinear equations   . We extend the techniques to solve nonlinear mixed V-FIE with a generalized singular kernel.
In this article, we present new strategy which is the combined LHPM to obtain approximate solutions with high degree of accuracy for the nonlinear mixed V-FIE with a generalized Cauchy kernel.
2. The Homotopy Perturbation Method (HPM)
In this section, we will present the HPM. We consider a general integral equation
where is an integral operator. Define a convex homotopy by
where is a functional operator with solution. Then
and the process of changing from 0 to 1 is just that of changing from to. In topology, this is called deformation. and are called homotopies.
According to the HPM, we can use the embedding parameter as a “small parameter”, and assume that the solution of Equation (3) can be written as a power series in:
when, the approximate solution of Equation (2) is obtained with
The series Equation (6) is convergent for most cases; however, the rate of convergence depends upon the nonlinear operator L  .
3. The HPM Applied to Nonlinear Mixed V-FIE
To illustrate the HPM, for nonlinear mixed V-FIE let us consider the Equation (1)
By the HPM, we can expand into the form
and the approximate solution is
and in sum, according to  , He’s HPM considers the nonlinear term as
where are the so-called He’s polynomials  , which can be calculated by using the formula
Substituting (8) and (10) into (7) and equating the terms with identical powers of, we have
The components can be computing by using the recursive relations (12).
4. The Combined LHPM Applied to Nonlinear Mixed V-FIE 
We assume that the kernel of Equation (7) takes the form
Applying the Laplace transform to both sides of Equation (7), we represent the linear term from Equation (8) and the nonlinear term will be represented by the He’s polynomials from Equation (10) and equating the terms with identical powers of, we have:
Applying the inverse Laplace transform to the first part of Equation (13) gives, that will define. Utilizing will enable us to evaluate. The determination of and leads to the determination of that will allows us to determine, and so on. This in turn will lead to the complete determination of the components of, upon utilizing the second part of Equation (13). The series solution follows immediately after using Equation (9). The obtained series solution may converge to an exact solution if such a solution exists.
5. Numerical Examples
Example 5.1  :
Consider the linear mixed V-FIE with a generalized Cauchy kernel
we obtain Table 1.
Example 5.2  :
Consider the nonlinear mixed V-FIE with a generalized Cauchy kernel
we obtain Table 2.
Table 1. Results obtained for example 1 and error.
Table 2. Results obtained for example 2 and error.
In this article, we proposed LHPM and used it for solving nonlinear mixed V- FIE with a generalized singular kernel. As examples show, the displayed technique diminishes the computational difficulties of other methods. An interesting feature of this method is that the error is too small and all the calculations can be done straightforward. It can be concluded that LHPM is a very simple, powerful and effective method.
The authors would like to thank the king Abdulaziz city for science and technology.
Cite this paper
Hendi, F.A. and Al-Qarni, M.M. (2017) Numerical Solution of Nonlinear Mixed Integral Equation with a Generalized Cauchy Kernel. Applied Ma- thematics, 8, 209-214. https://doi.org/10.4236/am.2017.82017
- 1. Pachpatta, B.G. (1986) On Mixed Volterra-Fredholm Type Integral Equations. Indian Journal of Pure and Applied Mathematics, 17, 488-496.
- 2. Hacia, L. (1996) On Approximate Solution for Integral Equations of Mixed Type. ZAMM—Journal of Applied Mathematics and Mechanics, 76, 415-416.
- 3. Brunner, H. (1990) On the Numerical Solution of Nonlinear Volterra-Fredholm Integral Equation by Collocation Methods. SIAM Journal on Numerical Analysis, 27, 987-1000.
- 4. Maleknejad, K. and Hadizadeh, M. (1999) A New Computational Method for Volterra-Fredholm Integral Equations. Journal of Computational and Applied Mathematics, 37, 1-8.
- 5. Adomian, G. (1991) A Review of the Decomposition Method and Some Recent Results for Nonlinear Equation. Computers & Mathematics with Applications, 21, 101-127.
- 6. Wazwaz, A.M. (1999) A Reliable Modification of Adomian’s Decomposition Method. Applied Mathematics and Computation, 102, 77-86.
- 7. Abdou, M.A., El-Kalla, I.L. and Al-Bugami, A.M. (2011) Numerical Solution for Volterra-Fredholm Integral Equation with a Generalized Singular Kernel. Journal of Modern Methods in Numerical Mathematics, 2, 1-15.
- 8. Al-Bugami, A.M. (2013) Toeplitz Matrix Method and Volterra-Hammerstin Integral Equation with a Generalized Singular Kernel. Progress in Applied Mathematics, 6, 16-42.
- 9. Abdou, M.A., El-Kalla, I.L. and Al-Bugami, A.M. (2011) New Approach for Convergence of the Series Solution to a Class of Hammerstein Integral Equations. International Journal of Applied Mathematics and Computation, 3, 261-269.
- 10. El-Kalla, I.L. and Al-Bugami, A.M. (2012) Numerical Solution for Nonlinear Volterra-Fredholm Integral Equation with Applications in Torsion Problems. International Journal of Computational and Applied Mathematics, 7, 403-418.
- 11. He, J.H. (1999) Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262.
- 12. He, J.H. (2000) A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems. International Journal of Non-Linear Mechanics, 35, 37-43.
- 13. Abdulaziz, O., Hashim, I. and Chowdhury, M.S.H. (2008) Solving Variational Problems by Homotopy Perturbation Method. International Journal for Numerical Methods in Engineering, 75, 709-721.
- 14. Golbabai, A. and Keramati, B. (2009) Solution of Non-Linear Fredholm Integral Equations of the First Kind Using Modified Homotopy Perturbation Method. Chaos, Solitons and Fractals, 39, 2316-2321.
- 15. Ghorbani, A. (2009) Beyond Adomian Polynomials: He Polynomials. Chaos, Solitons and Fractals, 39, 1486-1492.
- 16. Hendi, F.A. (2011) Laplace Adomian Decomposition Method for Solving the Nonlinear Volterra Integral Equation with Weakly Kernels. Studies in Nonlinear Sciences, 2, 129-134.