Orthonormal discrete Legendre polynomials for stochastic distributed-order time-fractional fourth-order delay sub-diffusion equation

In this study, the stochastic distributed-order time-fractional version of the fourth-order delay sub-diffusion equation is defined by employing the Caputo fractional derivative. The orthonormal discrete Legendre polynomials, as a well-known family of discrete polynomials basis functions, are used to develop a numerical method to solve this equation. To employ these polynomials in constructing the expressed approach, the operational matrices of the classical integration, differentiation (ordinary, fractional and distributed-order fractional), and stochastic integration of these polynomials are extracted. The established method turns solving the introduced stochastic-fractional equation into solving a more simple linear algebraic system of equations. In fact, by representing the unknown solution in terms of the introduced polynomials and employing the extracted matrices, this system is obtained. The accuracy of the developed algorithm is numerically checked by solving two examples.