A new numerical approach for solving space–time fractional Schrödinger differential equations via fractional-order Chelyshkov functions

In this paper, a numerical method for solving space–time fractional Schrödinger equations is proposed. The method employs fractional-order Chelyshkov functions and their properties to derive the remainders associated with the main problem. The Riemann–Liouville fractional integral operator is applied to the basis functions, yielding exact results through the analytical representation of Chelyshkov polynomials. The real and imaginary parts of the functions involved in the problem are separated, transforming the Schrödinger equation into two equations. By approximating the fractional derivative of the unknown function and using a set of collocation points, the problem is reduced to a system of algebraic equations, the solution of which provides the numerical solution to the problem. Additionally, an error analysis is presented. Finally, numerical examples and their results demonstrate the efficiency and accuracy of the proposed scheme.