Robust Approximation for Non-Linear Variable-Distributed Fractional Differential Equation with Non-Smooth Solutions

Abstract

This article introduces a spectral method aimed at estimating solutions for nonlinear Variable Distributed-Order Fractional Differential Equations (VDO-FDEs) with a non-smooth solution in one-dimensional and time-nonlinear VDOFDEs. Initially, we express the solution and its fractional derivatives using a series of Shifted Legendre Polynomials (SLPs). Subsequently, the expansion coefficients were derived by transforming the VDO-FDEs in addition to the conditions related to the algebraic system. We illustrate effectiveness and feasibility through various numerical tests.