Solving Linear Volterra – Fredholm Integral Equation of the Second Type Using Linear Programming Method

In this paper, a new technique is offered for solving three types of linear integral equations of the 2 nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree (m − 1) and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those methods is produced. Finally, for more explanation, an algorithm is proposed and applied for testing examples to illustrate the effectiveness of the new technique.


Introduction:
Integral equations occur naturally in many fields of mechanics and mathematical physics. They also arise as representation formulas for the solutions of differential equations (1).
From the last few years, there has been interest to use the linear and nonlinear programming methods to find a numerical or approximate solution for integral equations. AL-Nasir in 1999 used the linear programming method to find numerical solution of Volterra integral equations of the 2 nd kind (2). Also, Saed in 1999 used linear programming method to find numerical solution of Fredholm integral equations of the 1 st kind (3). While Kalwi in 1999 used the same procedure to find numerical solution for Fredholm integral equations of the 2 nd kind (4). Shua"a in 2005 calculated numerical solution for Volterra integral equation using linear programming problem (5). Kamyad et al. in 2010 proposed a new approach for solving linear and nonlinear Volterra integral equations of the 1 st and 2 nd kinds first by defining a new problem in calculus of variations, which is equivalent to this kind of problem, then by using the optimal solution of the latest problem as an approximate one with a controllable error for the original solution is obtained (6). Nazemi and Farahi in 2011 considered a numerical method for nonlinear Fredholm integral equations of the second kind with the continuous kernel by converting the integral equation problem into an optimization problem (7). Skandari et al. in 2011 proposed a new approach for a class of optimal control problems to solve Volterra integral equations which is based on linear combination property of intervals (8). Erfanian and Mostahsan in 2011 considered two stages of approximation to find an approximate solution for a class of nonlinear Volterra integral equations by: first convert the integral equation to a moment problem, and then modify the new problem to two classes of optimization problems (nonconstraint optimization and optimal control problems) (9). Effati and Skandari in 2012 presented a new approach for linear Volterra integral equations based on optimal control theory and some optimal control problems corresponding Volterra integral equation are introduced which solved by discretization methods and linear programming approaches (10 (18). In this paper, LVFIE of the 2 nd kind of the following form are considered: where ( ) is the unknown function to be determined, ( ), 1 ( , ) 2 ( , ) are continuous known functions. Therefore an approximate solution depending on a polynomial of degree ( − 1) is proposed, which has the form: are arbitrary constants to be determined. Next using numerical integration includes (TR, SR, BR, and RI) to find an approximate solution for the integral parts in Eq.1, after that using LPP to find an approximate solution to this problem. This paper is well ordered as follows: section (2) includes basic definitions, section (3) contains transforms Eq.1 to the LPP, in section (4), the algorithm for solving LVFIE is proposed, section (5)  Note that, for Eq. 1 when 1 ( , ) is equal to zero Eq. 1 becomes FIE, furthermore when 2 ( , ) is equal to zero Eq. 1 becomes VIE. And it is called VFIE if the both integral appear at one time. In this paper Eq. 1 is taken as a general form to improve that the proposed method is effective for the three kinds of equations.

Calculate
( ) Using Numerical Integration Formulas: In this subsection, the numerical computation of the integral part in ( ) is shown using quadrature methods including (TR, SR, BR, and RI). For this purpose, the following notation is used: ∫ ( , ) −1 The Algorithm for Solving LVFIE of the 2 nd kind: The following steps are used to find an approximate solution of LVFIE of the 2 nd kind: Step 1. Select two positive integers and , (where represent the number of terms of series in Eq. 2, represent the number of subinterval for the closed interval [a,b]).
Step 7. Determine the approximate solution of Eq. 1 by substituting the values of , = 1,2, … , in Eq. 2. Note that in this paper MATLAB R2018a is used for implementation of the algorithm.

Numerical Test Examples:
In this section, some of the numerical test examples are given to illustrate the proposed method for solving the LVFIE of the 2 nd kind. In all the test examples ( ) is chosen in such a way that we know the exact solution. The exact solution is used only to show that the numerical solution obtained with our method is true. Then, in these test  Table 1 shows the absolute error obtained by using (TR, SR, BR, and RI) for ℎ = 0.1, = 10, = 10, where m is the degree of approximate polynomial g(x) appear in Eq. 2 and ‖ ‖ ∞ is the maximum absolute error, for ∈ [0,1].     Table 3 shows the absolute error obtained by using (TR, SR, BR, and RI) for ℎ = 0.1, = 10, = 10, and ∈ [0, 2 ].  Table 4 shows the maximum absolute error of test example 2 by using RI with 10 columns for , = 5,10, … ,30 and = 5,10, … ,20 and comparing with the minimum error in (21) using A wavelet based method. for which the exact solution is ( ) = x 5 . Table 5 shows the absolute error obtained by using (TR, SR, BR, and RI) for ℎ = 0.1, = 10, = 10, and ∈ [0,1].  Table 6 shows the maximum absolute error of test example 3 by using RI with 10 columns with , = 5,10, … ,30 and = 5,10, … ,20 and comparing with the minimum error in (22) using analytical techniques for a numerical solution.

Conclusions:
This paper presents a method of finding the solution of LVFIE of the 2 nd kind using the LPP. The polynomial of degree − 1 is used to convert the LVFIE of the 2 nd kind into LPP. TR, SR, BR and RI are proposed for computing the integral part and the comparison between those methods is made. The accuracy of the method has been shown by applying different test examples and comparing the results with the exact solution. The results for the LPP are improved using RI instead of the other methods to evaluate the integrals within linear programming method. Also, it is supposed that the best result can be obtained by increasing both the number of basic functions ( ) and the number of constraints (n) with keeping ( > ). For future work, we suggest using this method for solving 'Volterra-Fredholm integro-differential equations', by using suitable approximation for derivatives part.