An Error Analysis Study for the Distributed-Order Time-Fraction Model Using an Efficient Numerical Method

In this paper, we propose a numerical scheme based on the shifted Legendre polynomials for solving the forced Korteweg–de Vries (fKdV) equation including a Caputo fractional operator of a distributed order. To obtain numerical solutions of these types of equations, we derive an operational matrix based on the shifted Legendre polynomials, and using this operational matrix, their equations change to a set of nonlinear algebraic systems. Then, by calculating these systems in the collocation points, we solve systems. Also, convergence and error are investigated in this paper. Finally, several numerical examples to show the applicability of our scheme are displayed.