Existence, uniqueness, and collocation solutions using the shifted Legendre spectral method for the Hilfer fractional stochastic integro-differential equations regarding stochastic Brownian motion

In this paper, the existence and uniqueness of a specific class of fractional stochastic integro-differential equations considering the stochastic Brownian motion equipped with an appropriate form of a random initial condition is introduced regarding the Hilfer fractional derivative. The proofs of the existence and uniqueness of the solution are presented utilizing sensible constraints upon the deterministic and stochastic coefficients, Schauder's fixed point theorem, and some stochastic theories. Moreover, to get approximations of the exact paths solving such equations we introduce a numerical technique based upon the time-dependent spectral collocation technique considering the shifted Legendre polynomials as a basis. The underlying concept of this technique involves transforming complex equations into a set of algebraic ones by selecting an appropriate set of collocation points within the specified domain where collocation is applied. Herein, the values of the stochastic Brownian motion are calculated using the Mathematica program. For approximating the integrals, the Gauss–Legendre integration scheme is implemented. In addition, we establish the convergence concerning the presented scheme with the error estimate in detail. For this purpose, we present the graphs of maximum errors under the log-log scale. The utilized procedure is leveraged to tackle a variety of stochastic examples encompassing various types to confirm the effectiveness of the obtained theoretical and numerical results. The acquired upshots expose the efficiency and applicability of the presented methodology in the fractional stochastic field.