Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix

In this paper, we obtain stochastic operational matrix of block pulse functions on interval [0,1) to solve stochastic Volterra-Fredholm integral equations. By using block pulse functions and their stochastic operational matrix of integration, the stochastic Volterra-Fredholm integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution. We prove that the rate of convergence is O(h). Furthermore, the results show that the approximate solutions have a good degree of accuracy.

[1]  K. Maleknejad,et al.  Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions , 2011, Int. J. Syst. Sci..

[2]  Khosrow Maleknejad,et al.  A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation , 2011 .

[3]  Xicheng Zhang,et al.  Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations , 2008, 0812.0834.

[4]  Constantin Tudor,et al.  Approximation schemes for ito - volterra stochastic equations , 1995 .

[5]  Juan Carlos Cortés,et al.  Numerical solution of random differential equations: A mean square approach , 2007, Math. Comput. Model..

[6]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[7]  Xicheng Zhang,et al.  Euler schemes and large deviations for stochastic Volterra equations with singular kernels , 2008 .

[8]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[9]  M. Thoma,et al.  Block Pulse Functions and Their Applications in Control Systems , 1992 .

[10]  Khosrow Maleknejad,et al.  Numerical solution of stochastic differential equations by second order Runge-Kutta methods , 2011, Math. Comput. Model..

[11]  Svetlana Janković,et al.  One linear analytic approximation for stochastic integrodifferential equations , 2010 .

[12]  Bernt Øksendal,et al.  Stochastic differential equations (3rd ed.): an introduction with applications , 1992 .

[13]  Khosrow Maleknejad,et al.  Numerical solution of linear Fredholm integral equation by using hybrid Taylor and Block-Pulse functions , 2004, Appl. Math. Comput..

[14]  Khosrow Maleknejad,et al.  Modification of Block Pulse Functions and their application to solve numerically Volterra integral equation of the first kind , 2011 .

[15]  G. Rao.  Piecewise Constant Orthogonal Functions and Their Application to Systems and Control , 1983 .

[16]  Khosrow Maleknejad,et al.  Interpolation solution in generalized stochastic exponential population growth model , 2012 .

[17]  Khosrow Maleknejad,et al.  Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials , 2007, Appl. Math. Comput..