Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion

Abstract

We are involved in this study with hybrid functions consisting of Taylor polynomials and block-pulse functions and use them as basis functions to achieve the numerical solution of stochastic evolution equation with fractional Brownian motion (FBM). First, we compute stochastic operational matrix based on hybrid Taylor block-pulse functions (HTBPFs) that this operator able us obtain an applicable procedure for solving stochastic integral equations (SIEs) and stochastic differential equations (SDEs) such as stochastic evolution equation. Then, we use this operator and HTBPFs ordinary operational matrix to convert solving stochastic evolution equations with FBM into solving a system of easily solvable algebraic equations. Under some mild conditions we prove that our proposed method is convergent when M, N → ∞, where N and M are the order of block-pulse functions and Taylor polynomials, respectively. Finally, we utilize the mentioned method for solving two test problems that their exact solution is available. Comparison obtained approximate solution with the exact solution demonstrate that the values of absolute error can be ignored and thus obtained solution has a good degree of accuracy. Furthermore, the theoretical discussions and numerical examples confirm that by increasing the values of N and M, the approximate solution tends to the exact solution.