Convergence Analysis for the Homotopy Perturbation Method for a Linear System of Mixed Volterra-Fredholm Integral Equations

In this paper, the homotopy perturbation method (HPM) is presented for treating a linear system of second-kind mixed Volterra-Fredholm integral equations. The method is based on constructing the series whose summation is the solution of the considered system. Convergence of constructed series is discussed and its proof is given; also, the error estimation is obtained. Algorithm is suggested and applied on several examples and the results are computed by using MATLAB (R2015a). To show the accuracy of the results and the effectiveness of the method, the approximate solutions of some examples are compared with the exact solution by computing the absolute errors.


Introduction
In this work, the linear system of second kind mixed Volterra-Fredholm integral equations (LSMVFIE2 nd ) will be considered which has the form:  (1). Karimi and Jozi solved the system of Fredholm integral equations by Taylor series-expansion method (2). Wazwaz (3) also approximated the exact solution of system of Fredholm integral equations by the Adomian decomposition method. Alturk (4) used weighted mean-value theorem for solving Fredholm integral and integro differential equations and their system. By contrast, in (5) the improvement of block-pulse functions has been discussed by Mirzaee to solve system of linear Volterra and Fredholm integral equations. In addition, Hesameddini and Shahbazi solved system of Volterra-Fredholm integral equations with hybrid Bernstein Block pulse functions (6). In (7), Radzuan et al. treated the system of Fredholm integral equations by halve-sweep kaudd successive over relaxation iterative. While, Hameed et al. treated the system of nonlinear Volterra integral equations by the Newton-Kantorovich method (8). Finally, in (9) the approximate solution of system of nonlinear integral equations is discussed. The Homotopy perturbation method (HPM) is a strong method in which the approximate solution is considered as a series solution which converges rapidly to the accurate approximation with a high accuracy. The method was proposed in1998 by He (10) and was further developed and improved for solving nonlinear problem in (11) and (12). The convergence of HPM for Fredholm integral equations and Volterra integral equations of the second kind is discussed in (13). It has also been used to find approximate and exact solution for Volterra-Fredholm integro differential equations in (14), and their systems in (15), while in (16) the method is used for system of Burgers equations. Presenting a convergence condition for HPM and the error estimation for the solution is our major purpose in this work.

Application of HPM on LSMVFIE2 nd
Recall system (1) and consider the i th equation of the system as To illustrate HPM, define the operator ℓ as follows:  If equation (7) has a radius of convergence that is ≮ 1 also, ∑ * ( ) converges absolutely, then the approximation of equation (1) is found: Substituting (7) in equation (6) gives ). = 1,2, … , . by equating the similar power terms of p gives a recurrence relations that leads to the approximate solution: for ≥ 1.
The above relations have been obtained by assuming that (7) is convergent. In the following Theorem, the conditions for such convergence will be discussed. If the following inequality The above assumptions imply the estimations below:
Which is a geometric series, with the common ratio = ( − ) 2 < 1, and it is convergent [by the assumption in equation (11)]. Consequently, series (8)  If it is impossible or difficult to calculate the series summation in (8), its partial sum can be accepted as an approximate solution of equation (1). In the limit → 1, terms of first + 1 in (8) produce -th order approximate solution as

=0
The level of error of the solution ̂( ) can be estimated by the following theorem: Theorem3.2. Estimated error of the th-order approximation will be determined as follows: Note: The above theorem can be considered as the rate of convergence of HPM, and its application can be considered in example 3.

The HPM Algorithm
To find an approximate solution of (LSMVFIE2 nd ) by HPM, perform the following steps: Step 1: Choose the integers n and m, and the bounds of integration a, and b.

Numerical Examples
In this part, several examples will be solved to show the implementation of the method and its strength.
Example1 . Consider the linear system of Volterra-Fredholm integral equations below:
While Table 2, shows a comparison between the computed least square errors of HPM and successive approximation method (SAM) which is discussed by many authors for different purposes. The results in the table show that our results are considerable accurate as it has less error than that of successive approximation method. The absolute errors for the exact solutions 1 ( ), 2 ( ) and the approximate solutions ̂1 ,10 ( ),̂2 ,10 ( ) are presented in Table 3, while Fig. 2 shows the convergence of the method for ̂1 , ( ) and ̂2 , ( ) for = 1, … ,5.  Where, the exact solution is not given.
First, it will be verified whether the inequality (11) satisfied or not.
In this example, we have  In Table 4, for = 1,2, different orders of approximation with the corresponding error estimations are presented for example 3.

Conclusion
In summary, the sufficient condition of convergence of the HPM for the LSMVFIE2 nd is formulated and proved; an estimation of error is also given. HPM is discussed for solving the system and two examples are presented for illustration. Good approximations are obtained while better results have been found by increasing the number of components of the partial sum ( ). Moreover, a comparison between exact solution and its approximation made to demonstrate application technique. It is worth mentioning that the technique can be used as a very accurate algorithm for solving LSMVFIE2 nd . These claims are supported by the results of the given numerical examples in Tables (1-4) and Figures (1-2). the permission for re-publication attached with the manuscript. -The author has signed an animal welfare statement. -Ethical Clearance: The project was approved by the local ethical committee in University of Salahaddin.