# The Use of Modified Block Pulse Functions for Solving the Stochastic Volterra-Fredholm Integral Equations

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Keywords: Brownian Motion, Itô Integral, Stochastic Integration Operational Matrix, SV-FIEs, MBPFs.## Abstract

Abstract. A computational method based on modified block pulse functions is proposed for solving numerically stochastic VolterraFredholm integral equations. We obtain stochastic integration operational matrix of modified block pulse functions on interval [0,1). Amodified block pulse functions and their stochastic integration operational matrix can be reduced to a linear upper triangular system. Then, the problem under study is transformed to a system of linear algebraic equations which can be used to obtain an approximate solution of stochastic VolterraFredholm integral equations. Furthermore, the rate of convergence is (?) and error analysis of the proposed method are investigated. The results show that the approximate solutions have a good of efficiency and accuracy.

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## References

Numerical approach for solving stochastic VolterraFredholm integral equations by stochastic operational matrix. Computers and Mathematics with Applications 64:1903-1913. Klebaner, Fima C. 2005. Introduction to Stochastic Calculus with Applications 2th edition. Imperial College Press, New York. Kloeden, Peter E. and Eckhard Platen. 1992. Numerical Solution of Stochastic Differential Equations. Springer, Germany. Maleknejad, K. and B. Rahimi. 2011. Modification of Block Pulse Functions and their application to solve numerically Volterra integral equation of the first kind. Commun Nonlinear Sci Numer Simulat 16: 2469-2477. Maleknejad, K., M. Khodabin, and F. Hosseini Shekarabi. 2014. Modified Block Pulse Functions for Numerical Solution of Stochastic Volterra Integral Equations. Journal of Applied Mathematics 4: 1-10. Mao, Xuerong. 1997. Stochastic Differential Equations and Applications 2th edition. Horwood Publishing Limited, UK. Oksendal, Bernt. 1998. Stochastic Differential Equations: An Introduction with Applications 5th edition. Springer, USA.

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*Proceeding International Conference on Science and Engineering*,

*4*, 266–271. Retrieved from https://sunankalijaga.org/prosiding/index.php/icse/article/view/669