Shifted Monic Ultraspherical Approximation for solving some of Fractional Orders Differential Equations

TL;DR

This paper introduces a new spectral Galerkin method using Shifted Monic Ultraspherical Polynomials to efficiently solve linear and nonlinear fractional differential equations with Caputo derivatives.

Contribution

It develops a novel formula for derivatives of SMUPs of any degree and fractional order, enabling a direct spectral method for fractional differential equations.

Findings

Derivation of a new formula for derivatives of SMUPs of any degree and fractional order

Development of a spectral Galerkin method for solving fractional differential equations

Demonstration of the method's effectiveness for linear and nonlinear FDEs

Abstract

The purpose of this paper is to show and explain a new formula that indicates with finality the derivatives of Shifted Monic Ultraspherical polynomials (SMUPs) of any degree and for any fractional-order using the shifted Monic Ultraspherical polynomials themselves. We also create a direct method solution for the linear or nonlinear multi-order fractional differential equations (FDEs) with constant coefficients involving a spectral Galerkin method. The spatial approximation with its fractional order derivatives (described in the Caputo sense) are built using shifted Monic Ultraspherical polynomials.