A computational wavelet method for numerical solution of stochastic Volterra-Fredholm integral equations

Document Type : Research Paper


Hormozgan University


A Legendre wavelet method is presented for numerical solutions of stochastic Volterra-Fredholm integral equations. The main characteristic of the proposed method is that it reduces stochastic Volterra-Fredholm integral equations into a linear system of equations. Convergence and error analysis of the Legendre wavelets basis are investigated. The efficiency and accuracy of the proposed method was demonstrated by some non-trivial examples and comparison with the block pulse functions method.


[1] P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, in: Applications of Mathematics, Springer-Verlag, Berlin, 1999.
[2] B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, fifth ed., springer-Verlag,  New York, 1998.
[3] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Re., 43(3)(2001), 525–546.
[4] K. Maleknejad, M. Khodabin, M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Modelling, 55(2012), 791–800.
[5] M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri, Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix, Comput. Math. Appl., Part A, 64(2012), 1903–1913.
[6] M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek, C. Cattani, A computational method for solving stochastic Ito-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, J. Comput. Phys., 270(2014), 402–415.
[7] J . C. Cortes, L.  Jodar, L. Villafuerte, Numerical solution of random differential equations: a mean square approach, Math. Comput. Modelling, 45 (7)(2007), 757–765.
[8] G. Strang, Wavelets and dilation equations, SIAM Rev., 31(1989), 613–627.
[9] A. Boggess, F. J. Narcowich, A first course in wavelets with Fourier analysis, John Wiley and Sons, 2001.
[10] M. Razzaghi, S. Yousefi, The Legendre wavelets operational matrix of integration, Int. J. Syst. Sci., 32(4)(2001), 495–502.
[11] M. Razzaghi, S. Yousefi, Legendre wavelets direct method for variational problems, Math. Comput. Simul., 53(3)(2000), 185–192.
[12] F. Mohammadi, M. M. Hosseini, and Syed Tauseef Mohyud-Din. Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution, Int. J. Syst. Sci.,42(4)(2011), 579–585.
[13] F. Mohammadi, M. M. Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations, J. Franklin Inst., 348(8)(2011), 1787–1796.
[14] Z. H. Jiang, W. Schaufelberger, Block Pulse Functions and Their Applications in Control Systems, Springer-Verlag, 1992.
[15] L. Nanshan, E. B. Lin, Legendre wavelet method for numerical solutions of partial differential equations, Numer.Methods Partial Differ. Equations, 26(1)(2010), 81–94.