On the numerical solution of stochastic quadratic integral equations via operational matrix method

Abstract

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h 2 ). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method. KEYWORDSBrownian motion process, error analysis, operational matrix, quadratic integral equations, stochastic integral equations | INTRODUCTIONThe various kinds of integral equations are important mathematical tools for describing knowledge models that appear in different areas of applied science. Because of extensive application of integral equations and not having the exact solutions in many cases, numerical solution of integral equations has attracted researcher's attention to develop numerical method for approximating solution of these equations. Among these methods, we refer to wavelet method, 1-4 homotopy perturbation method, 5-8 collocation method, 9,10 and meshless method. 11-13 A novel algorithm to get approximate solution of these equations is to express the solution as linear combination of orthogonal or nonorthogonal basis functions and polynomials such as block-pulse functions, 14,15 hat functions, 16,17 Bernoulli polynomials, 18 Legendre polynomials, 19 Bessel polynomials, 20 Chebyshev polynomials, 21 Fibonacci polynomials, 22 and orthonormal Bernstein polynomials. 23 Mathematical modeling of various phenomena in real world is leaded to a special kind of integral equations named quadratic integral equations. Quadratic integral equations always arise in many problems of mathematical physics and chemical such as theory of radiative transfer, the kinetic theory of gases, the theory of neutron transport, the queuing theory, and the traffic theory and many other applications. Existence solution and numerical method to solve these type of integral equations have been studied in previous papers. [24][25][26][27][28] Recently, with increasing computational power, it becomes feasible to utilize more accurate equations to model some problems that are often dependent on a noise source. So modeling these problems naturally require the use of

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