Abstract
The Simpson’s 3/8 rule is used to solve the nonlinear Volterra integral equations system. Using this rule the system is converted to a nonlinear block system and then by solving this nonlinear system we find approximate solution of nonlinear Volterra integral equations system. One of the advantages of the proposed method is its simplicity in application. Further, we investigate the convergence of the proposed method and it is shown that its convergence is of order . Numerical examples are given to show abilities of the proposed method for solving linear as well as nonlinear systems. Our results show that the proposed method is simple and effective.
1. Introduction
We consider the system of second kind Volterra integral equations (VIE) given by where Numerical solution of Volterra integral equations system has been considered by many authors. See for example [1–5].
In recent years, application of HPM (Homotopy Perturbation Method) and ADM (Adomian Decomposition Method) in nonlinear problems has been undertaken by several scientists and engineers [6–8]. HPM [8] was proposed by He in for the first time and recently Yusufoğlu has proposed this method [9] for solving a system of Fredholm-Volterra type integral equations. Block by block method was suggested by Young [10] for the first time in connection with product integration techniques. On the other side, the engineers are facing certain challenges to deal with complexity and efficient mathematical modeling. Thus, there are several research have being carried out related to these problems. For example, more details on the modeling of complexity we refer to [11], direct operational method to solve a system of linear in-homogenous couple fractional differential equations, see [12] and for a class of fractional oscillatory system, see [13].
In this paper, we consider block by block method by using Simpson's rule for solving linear and nonlinear systems of Volterra integral equations. And we make a block by block method comparison between our method and HPM.
This paper is organized as follow: in Section 2, we present some background material on the use of this method. In Section 3, we prove the convergence result. Finally, numerical results are given in Section 4.
2. Starting Method
Consider a system of nonlinear Volterra integral equations in (1.1) and further, we suppose that the system equation (1.1) has a unique solution. However, the necessary and sufficient conditions for existence and uniqueness of the solution of (1.1) can be found in [14]. The idea behind the block by block methods is quite general but is most easily understood by considering a specific. Let us assume that in (1.1) and use the Simpson's rule as a numerical integration formulae. Let then approximating the integrals by Simpson's rule, we have where Further we also get In order to evaluate the integrals on the right hand sides, we introduce the points , and the corresponding values , and use the Simpson's rule with step size , then we gain where , , , have unknown values, which can be estimated by Lagrange interpolation points , , , . Therefore we obtain Thus, Substituting from (2.7) into (2.5) we obtain the following values for , Also we get to evaluate the integrals on the right-hand sides, we introduce points , and the corresponding values , and use the Simpson's rule with step size . Therefore we have where , , , have unknown values, similarly, that can be estimated by Lagrange interpolation points , , , . As a result we obtain and so Substituting (2.12) into (2.10), we obtain the following values for and : Equations (2.2), (2.8), and (2.13) are three pairs of simultaneous equations in terms of unknowns , , , , , and for the nonlinear system of VIE. The solutions of these equations may be found by the method of successive approximation or by a suitable software package such as Maple. For the linear system of VIE a direct method can be used for solving system of linear algebraic equations.
3. The General Scheme
Consider the system of VIE Let be a partition of with the step size , such that for . Then we can construct a block by block approach so that a system of simultaneous equations is obtained and thus a block of values of is also obtained simultaneously. We put for simplicity. Setting in (3.1) we get or equivalently Now, integration over can be accomplished by Simpson's rule and the integral over is computed by using a cubic interpolation. Hence and in a similar way the right-hand side is obtained for , where , .
By setting in (3.1) we get or equivalently Now, the integration over can be accomplished by Simpson's rule and the integral over is computed by using a cubic interpolation. Hence and a similar right-hand side obtains for , where , . In a similar manner we obtain Equations (3.4)–(3.9) form a system with six unknowns for . In fact, we have six simultaneous equations at each step.
4. Convergence Analysis
In this section we investigate the convergence of the proposed method. The following theorem shows that the order of convergence is at least four.
Theorem 4.1. The approximate method given by the systems Equations (3.4), (3.7), and (3.9), is convergent and its order of convergence is at least four.
Proof . We have using the Lipschitz condition it can be written as where , are the errors of integration rule. Without loss of generality, we assume that then let , hence then from Gronwall inequality, we have For functions and with at least fourth-order derivatives, we have and so and the proof is completed.
5. Numerical Results
In this section, some examples are given to certify the convergence and error bounds of the presented method. All results are computed using the well-known symbolic software Maple 12. Tables 1–3 show the obtained results for nonlinear examples and Tables 4–6 contain the results for linear examples. The results in Tables 1–6 show the absolute errors , , where is the exact solution evaluated at and is the corresponding approximate solution.
Example 5.1. Consider the following system [9]: with the exact solutions and . In Table 1 we compare the errors and obtained using the present method and method in [9].
Example 5.2. Consider the following system [7]: the exact solutions are and . The errors are given in Table 2.
Example 5.3. Consider the following system [15]: where the exact solutions are and .
Example 5.4. Consider the following system [7]: with the exact solutions and . Table 4 shows the errors.
Example 5.5. Consider the following Volterra system of integral equations [5]: the functions and are chosen such that the exact solutions are to be and . The errors are given in Table 5.
Example 5.6. Consider the following Volterra system of integral equations [5] the functions and are chosen such that the exact solutions are to be and . Table 6 compares errors.
6. Conclusion
Now let be error of the block by block using Simpson's rule where and are, respectively, a constant and order of the error, then we have . By computing the order from this formulae for the errors reported in Tables 1–6, we conclude that . This confirms the claim that was stated in introduction and was proved by the convergence analysis. For example in Table 2 for , where , so we get or in Table 4 for , where , so we get or in Table 6 for , , then .
Note that in the Table 1 we compare the order of convergence between the results which were obtained by Yusufoğlu [9] (by using HPM method) and block by block method. Further, the results of the Table 6 show that block by block method is efficient for the large values of , whereas HPM method isn't applicable. Also, the time of computation in block by block method is less than HPM method whenever programming of both method is done in MAPLE package.
Acknowledgments
This paper was prepared when the third author visits the Institute for Mathematical Research (INSPEM) thus the authors wish to thank the Institute for Mathematical Research (INSPEM), University Putra Malaysia. The authors would also like to express their sincere thanks and gratitude to the reviewer(s) for their valuable comments and suggestions.