A numerical solution for a quasi solution of the time-fractional stochastic backward parabolic equation

Abstract

In the present study, we consider a time-fractional stochastic backward parabolic equation driven by standard Brownian motion. In this problem, the fractional derivative is considered in the Caputo sense. Using the minimization of a least-squares functional, stochastic variational formulation, Fréchet differentiability and utility theorems adopted directly from deterministic fractional backward equations, the existence and uniqueness theorems for a quasi solution of the proposed problem are proved. To approximate the quasi solution, a numerical technique based on 2D Chebyshev wavelets is applied. We employ the Levenberg–Marquardt regularization technique since the derived equivalent system of linear equations is ill-posed. Also, the convergence analysis for this numerical algorithm is investigated. Our results provide a new insight to find quasi solutions and apply adapted deterministic methods for some fractional stochastic backward equations. Moreover, a numerical example is provided to indicate the accuracy and efficiency of the Chebyshev wavelet method in solving the mentioned problem.