Analytical Solution of System of Volterra Integral Equations Using OHAM

In this work, a reliable technique is used for the solution of a system of Volterra integral equations (VIEs), called optimal homotopy asymptotic method (OHAM). The proposed technique is successfully applied for the solution of diﬀerent problems, and comparison is made with the relaxed Monto Carlo method (RMCM) and hat basis function method (HBFM). The comparisons show that the present technique is more suitable and reliable for the solution of a system of VIEs. The presented technique uses auxiliary function containing auxiliary constants, which control the convergence. Moreover, OHAM does not require discretization like other numerical methods and is also free from small or large parameter.


Introduction
Differential and integral equations have their own importance and a lot of applications. ese equations have been used in many fields, such as control, economics, electrical engineering, medicine, and so on, and also the system of these equations arises in modeling of different phenomena in different fields of science and technology. e exact solutions of most of the nonlinear systems are difficult to obtain; therefore, the researchers used an alternative approach to find their approximate solutions, such as the residual power series method [1], homotopy analysis method [2], wavelet-Galerkin method [3], Adomian decomposition method [4], modified reproducing kernel method [5], new homotopy perturbation method [6], Chebyshev wavelet method [7], homotopy perturbation method [8], fixed-point method [9], hat basis functions [10], modified variational iterative method [11], relaxed Monte Carlo method [12], variational iterative method [13], collocation method [14], modified tanh-coth method [15], and generalized hypergeometric solutions [16].
In this work, OHAM is used [17,18] for the solution of a system of VIEs. It is motivated by the aspiration to acquire an exciting solution of a system of VIEs by using the proposed method. e novelty of the presented technique is its flexible convergence and it provides us with a convenient way to control and adjust the approximation series. Many researchers used the proposed technique for different types of problems in literature. Iqbal [23], three-dimensional Volterra integral equations [24], and fractional integro-differential equations [25]. We apply the method to obtain the approximate solution of a system of VIEs and test some numerical examples to show the effectiveness and accuracy of the method. is paper is organized as follows. Section 2 gives the basic idea of OHAM for a system of VIEs. e application of OHAM to the system of VIEs is given in Section 3, and conclusions of the paper are given in Section 4.

Basic Idea of OHAM
In this section, we discuss the formulation of OHAM for solving the system of Volterra integral equations of the second kind following the procedure outlined in [17,18]. Let us consider the system of VIEs of the form where (2) In equation (1), K(x, s) is the kernel, G(x) are given functions, and W(x) is the unknown solution of the system. Consider the ith equation of (1): First, we construct homotopy: where H(q) � m p�1 c p q p is an auxiliary function, c p are auxiliary constants, 0 ≤ q ≤ 1 is an embedding parameter, and H(0) � 0.
For q � 0, equation (4) becomes and for q � 1, When q approaches from zero to 1, then w i (x, 0) is continuously deformed to w i (x, 1).
For approximate solution of equation (3), using Taylor's series expansion about q, one can get Substituting equation (7) into equation (4) and comparing the by comparing the coefficient of the like powers of q, one can get a series of problems.
For k th order problem for w ik (x), i � 1, 2, . . . , n, it becomes At q � 1, equation (7) converges to the series solution: m th order approximation is We define the residual on substituting equation (11) into equation (3).
To find c p , p � 1, 2, 3, . . . , we use the least squares method as follows: Journal of Mathematics where limit of integration is the domain of the problem. Differentiating equation (13) with respect to the constant involved, we get One can easily obtain the values of c p from equation (14) for which the m th order approximation given in equation (11) is well determined.

System of VIEs
In this section, the consistency and reliability of OHAM are verified by some numerical problems, and the obtained results are compared with other methods in the form of tables; these tables clearly show the dominance of the proposed technique over these methods.
3.1. Problem 1. Consider the following system of VIEs [12]: Equation (15) has exact solutions w 1 (x) � x 2 + 1 and Applying the proposed algorithm discussed in Section 2, one can get different order solutions, as follows.
Zeroth-order solution: Journal of Mathematics Substituting equations (16)-(23) into w i � 3 k�0 w ik , for i � 1, 2, one can get the third-order OHAM solution. e approximate solution contains auxiliary constants; using the technique discussed in Section 2, one can get the following values of the auxiliary constants. Using these constants, the third-order OHAM solution becomes 3.2. Problem 2. Consider the following system of VIEs [10]: 3.3. Problem 3. Consider the following system of VIEs: (31) e system in equation (31) has exact solutions w 1 (x) � x and w 2 (x) � x 2 . Using the proposed algorithm, one can get the following auxiliary constants and third-order approximate solution. 0005926311682586243x)x)))))))), 0006263697932730246x)x))))))))) (34) Table 1: e exact solution and third-order OHAM solution for problem 1.

Journal of Mathematics
x  x    x    Tables 1 and 2 show the exact and third-order OHAM solutions for problems 1 and 2, respectively. ird-order OHAM solutions for problem 1 are compared with the relaxed Monte Carlo method (RMCM) with k � 8, h � 0.2 , and N � 100 in Table 3, while comparison of third-order OHAM solutions is made with hat basis functions (HBF) [10] with n � 64 in Table 4. e OHAM solution, exact solution, and absolute errors of OHAM for problem 3 are shown in Table 5. Figures 1-3 show the plot of OHAM and exact solutions for problems 1-3, respectively. e graphical representation and tables clearly show the reliability and consistency of the proposed method for the system of VIEs.

Conclusions
In this paper, the simple and easy algorithm of OHAM is successfully implemented to the system of VIEs. e obtained results confirmed the reliability of the proposed method. e accuracy of the result is a point of interest; moreover, if we increase the order of approximation, the results get closer to the exact solutions. e fast convergence and accuracy of the proposed technique are valid reasons for researcher to use the OHAM for different problems arising in various fields of science and technology.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.