A numerical method based on the piecewise Jacobi functions for distributed-order fractional Schrodinger equation

Abstract

In this work, the distributed-order time fractional version of the Schrodinger problem is defined by replacing the first order derivative in the classical problem with this kind of fractional derivative. The Caputo fractional derivative is employed in defining the used distributed fractional derivative. The orthonormal piecewise Jacobi functions as a novel family of basis functions are defined. A new formulation for the Caputo fractional derivative of these functions is derived. A numerical method based upon these piecewise functions together with the classical Jacobi polynomials and the Gauss- Legendre quadrature rule is constructed to solve the introduced problem. This method converts the mentioned problem into an algebraic problem that can easily be solved. The accuracy of the method is examined numerically by solving some examples.