Discrete Chebyshev polynomials for the numerical solution of stochastic fractional two-dimensional Sobolev equation

Abstract

In this work, the stochastic fractional two-dimensional Sobolev equation is introduced and a collocation method is proposed to solve it. The discrete Chebyshev polynomials are used as a proper family of basis functions to establish this collocation method. Some operational matrices related to conventional and stochastic integrals, as well as fractional and ordinary derivatives of these polynomials, are extracted and successfully used in making the expressed method. More precisely, by approximating the problem solution via a finite expansion of the expressed polynomials (where the expansion coefficients are unknown) and substituting it into the stochastic fractional problem, as well as by employing the above obtained matrices, a system of linear algebraic equations is obtained. Finally, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are found by solving this system. The correctness of the established method is examined by solving some numerical examples.