Numerical solution of two-dimensional stochastic Fredholm integral equations on hypercube domains via meshfree approach

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Abstract

In this article, radial basis functions (RBFs) and quadrature rules have been employed to estimate the solution of two-dimensional (2D) stochastic integral equations on hypercube areas. The main advantage of the suggested approach is that this algorithm can be easily implemented to estimate the solution of multidimensional stochastic integral equations defined on irregular domains. Also, it is established that the convergence order is proportional to hX,Dl, where hX,D denotes fill distance parameter. Finally, to reveal accuracy, efficiency and applicability of our scheme two test problems are included.

Introduction

We utilize the various kinds of stochastic models for mathematical modeling of real phenomena in nature [1], [2], [3]. Because of the randomness, solving the stochastic equations analytically are usually difficult and the numerical methods are applied to solve them. Since the beginning of 2010, the attention of many researchers was attracted to such equations and they tried to present some appropriate numerical methods to solve different types of stochastic equations. For instance, Maleknejad et al. used second order Runge–Kutta method to solve stochastic differential equations in [4]. Stochastic Itô–Volterra integral equations have been solved via collocation method and wavelet method in [5] and [6], respectively. Truncated Euler–Maruyama method was implemented by Mao in [7] to provide the approximate solution of stochastic differential equations. In [8], the solution of Stratonovich integral equations has been estimated via Bernoulli’s approximation. Furthermore, finite difference method [9], [10], Galerkin method [11], Bernstein approximation method [12], RBFs method [13], spectral collocation method [14] and operational matrices method [15] have been applied to solve different types of one dimensional stochastic integral equations. 2D stochastic integral equation is a knowledge tool for modeling different problems in engineering and science. Despite abundant application, there are a few published papers to produce the solution of such equations. Haar wavelet [16], block-pulse functions [17] and hat functions [18] have been used to solve 2D stochastic integral equations.

This paper is focused on using RBFs method [19], [20] for solving 2D stochastic integral equation which is formulated as f(u,v)=g(u,v)+0101k1(u,v,z,w)f(z,w)dwdz+0101k2(u,v,z,w)f(z,w)dB(w)dB(z),(u,v)D.

Section snippets

Preliminaries and notations

Traditional numerical methods require some sort of underlying computational mesh. Creating of these meshes become a rather difficult task in higher dimensional problems. This was the motivation for creating meshfree methods. Meshfree methods entered in the mathematics literature in 1980 for interpolation of an unknown function, but nowadays, it is used extensively for solving differential equations and integral equations. Scattered data fitting which is described in the following is one of the

Numerical scheme

The aim of current section is presenting an effective approach to solve stochastic integral equations (1). To start, we require positive definite RBFs Φ and n collocation points X={(u1,v1),(u2,v2),,(un,vn)} on the domain D. The solution of integral equation (1) can be approximated via RBFs as follows f(u,v)fn(u,v)=j=1ncjΨj(u,v),(u,v)D,where Ψj(u,v)=ψ((uuj)2+(vvj)2),j=1,2,3,,n.Substitute approximate function which is defined in Eq. (9) into Eq. (1). Thus Rn(u,v)=fn(u,v)g(u,v)0101k1(u,v

Error analysis

Define the integral operators k1,k2:C(D)C(D) as (k1f)(u,v)=0101k1(u,v,z,w)f(z,w)dwdz,(k2f)(u,v)=0101k2(u,v,z,w)f(z,w)dB(w)dB(z). The abstract form of integral equation (1) is as follows (Ik1k2)f=g.Assume that fn(u,v) denotes the numerical solution of Eq. (1) and consider projection operator Pn:C(D)Vn which has been defined in Eq. (3) and Vn=Span{Ψ1,Ψ2,,Ψn}C(D). So, the abstract form of Eq. (12) can be written as (IPnk1Pnk2)fn=Png.Define a sequence of numerical integration operators k

Test problems

This scheme is used to solve two numerical examples to reveal efficiency, accuracy and applicability of the proposed method. We measure the accuracy of this method by introducing definition of maximum error and RMS-errors as e=max(u,v)D{|f(u,v)fˆn(u,v)|},RMS-error=1ni=1n(f(ui,vi)fˆn(ui,vi))2, where f and fˆn are the exact and approximate solutions of Eq. (1), respectively.

Example 1

Consider the following stochastic integral equation f(u,v)=g(u,v)+0101uvzwf(z,w)dwdz+0101(u+v)f(z,w)dB(w)dB(z),(u,

Conclusion

In summary, to solve 2D linear stochastic Fredholm integral equations on the hypercube areas numerically, we introduce an efficient approach depending on meshfree method and quadrature rule. By using this scheme, solving considered stochastic integral equations are converted to the solving linear algebraic system. This algorithm is used as a powerful instrument to solve different kinds of multidimensional integral equations arising in various fields of science, i.e. there are no restriction on

Acknowledgments

The authors would like to express our very great appreciation to anonymous reviewers for their valuable comments and constructive suggestions which have helped to improve the quality and presentation of this paper.

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