Elsevier

Applied Mathematics and Computation

Volume 265, 15 August 2015, Pages 759-767
Applied Mathematics and Computation

Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations

https://doi.org/10.1016/j.amc.2015.05.125Get rights and content

Abstract

The main aim of this work is to give further studies for the multi-dimensional integral equations. In this work, we solve special types of the three-dimensional Volterra–Fredholm linear integral equations of the second kind via the modified block-pulse functions. Some theorems are included to show convergence and advantage of the method. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.

Introduction

The multi-dimensional integral equation is an integral equation in which the integration is carried out with respect to multiple variables. In this case, the unknown function depends on more than one independent variable. Moreover, the multi-dimensional integral equations can be found in many technologies, mechanics and biology. Many problems of fracture mechanics, aerodynamics, the theory of porous filtering, antenna problems in electromagnetic theory and others can be formulated as multi-dimensional integral equations of the first, second and third kind [1].

Solution methods for the multi-dimensional integral equations are very significant since they appear in the mathematical formulation. Because these equations are usually difficult to solve analytically, the aim in the present research is to develop an accurate as well as easy to implement analytic solution scheme to treat such equations. Numerous works have been focusing on the development of more advanced and efficient methods for solving Volterra–Fredholm integral equations. The literature on the numerical solution methods of such equations is fairly extensive [2], [3], [4], [5], [6], [7], [8], [9]. But the analysis of computational methods for two-dimensional integral equations seem to have been discussed in only a few papers [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].

Block-pulse functions have been studied and applied extensively as a basic set of functions for signal and function approximations. All these studies and applications show that block-pulse functions have definite advantages for solving problems involving integrals and derivatives due to their clearness in expressions and their simplicity in formulations.

This paper is divided into the following sections. The properties of modified three-dimensional block-pulse functions (M3D-BFs) are presented in the next section. Section 3 briefly reviews the description of the method based on M3D-BFs for solving the special types of the multi-dimensional Volterra–Fredholm integral equations. In Section 4, theorems are provided for convergence analysis. Numerical results are given in Section 5. Section 6 consists of a brief conclusion.

Section snippets

About M3D-BFs bases

Now we describe the fundamental idea of M3D-BFs. We define the (m+1)3-set of M3D-BFs over district D=[0,1)×[0,1)×[0,1) as follows ϕi1,i2,i3(x,y,z)={1(x,y,z)Di1,i2,i3,0otherwise,,i1,i2,i3=0(1)m,where Di1,i2,i3={(x,y,z)|xIi1,ɛ,yIi2,ɛ,zIi3,ɛ},Iα,ɛ={[0,hɛ)α=0,[αhɛ,(α+1)hɛ)α=1(1)(m1),[1ɛ,1)α=m,and h=1m which m is an arbitrary positive integer. Since each M3D-BF takes only one value in its subregion, the M3D-BFs can be expressed by the three modified one-dimensional block-pulse functions

Solving of three-dimensional integral equations via M3D-BFs

We will consider the numerical solution of a class of three-dimensional Volterra–Fredholm integral equations in the form f(x,y,z)=g(x,y,z)+λ10x0y0zk1(x,y,z,s,t,r)f(s,t,r)drdtds+λ2010101k2(x,y,z,s,t,r)f(s,t,r)drdtds,where x,y,z[0,1),λiR(i=1,2) and f(x, y, z) is an unknown function, g(x, y, z), k1(x, y, z, s, t, r) and k2(x, y, z, s, t, r) are analytical functions on D and D × D, respectively. The literature turn on no numerical solution methods of such models. Actually this is the first

Convergence analysis

In this section, we shall provide the error of the associated approximation. We show that the given approximation in Section 2, is convergent and its order of convergence is O(1km).

Theorem 1

Assume that fm,ɛ(x,y,z)=i1=0mi2=0mi3=0mfi1,i2,i3ϕi1,i2,i3(x,y,z),and fi1,i2,i3=1(Ii1,ɛ)(Ii2,ɛ)(Ii3,ɛ)010101f(x,y,z)ϕi1,i2,i3(x,y,z)dzdydx;i1,i2,i3=0(1)m.Then the criterion of this approximation is that the mean square error between f(x, y, z) and fm, ε(x, y, z) in the interval (x, y, z) ∈ D ϵ=010101(f(x,y

Numerical examples

Some different examples of three dimensional integral equations of the second kind are chosen to illustrate the proposed method. Also, the system of integral equations are solved to show applicability and flexibility of the method for solving the various problems. To compare the presented method with the other methods, the examples are selected from other papers. Comparison between results confirms the efficiency and simplicity of the presented method. All computations have been performed on a

Conclusion

In this article, we have studied a numerical scheme to solve three-dimensional Volterra–Fredholm integral equations. This method is based on the expansion of the solution as series of M3D-BFs. The problem has been reduced to a problem of solving linear system of algebraic equations. Furthermore, it is proved that M3D-BFs method is convergent and the order of convergence of this method is O(1km). The method is computationally attractive and its applications are demonstrated through illustrative

Acknowledgments

The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the paper.

References (23)

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