Petrov-Galerkin and Spectral Collocation Methods for distributed Order Differential Equations

TL;DR

This paper introduces two spectrally-accurate numerical schemes, a Petrov-Galerkin spectral method and a spectral collocation method, for solving distributed order fractional differential equations, with stability and error analyses.

Contribution

The paper develops novel spectral methods based on fractional Sturm-Liouville eigenfunctions for distributed order fractional differential equations.

Findings

Both methods demonstrate high accuracy and spectral convergence.

Numerical tests confirm the stability and efficiency of the schemes.

The approaches effectively handle the complexity of distributed order operators.

Abstract

Distributed order fractional operators offer a rigorous tool for mathematical modelling of multi-physics phenomena, where the differential orders are distributed over a range of values rather than being just a fixed integer/fraction as it is in standard/fractional ODEs/PDEs. We develop two spectrally-accurate schemes, namely a Petrov-Galerkin spectral method and a spectral collocation method for distributed order fractional differential equations. These schemes are developed based on the fractional Sturm-Liouville eigen-problems (FSLPs). In the Petrov-Galerkin method, we employ fractional (non-polynomial) basis functions, called \textit{Jacobi poly-fractonomials}, which are the eigenfunctions of the FSLP of first kind, while, we employ another space of test functions as the span of poly-fractonomial eigenfunctions of the FSLP of second kind. We define the underlying \textit{distributed Sobolev space} and the associated norms, where we carry out the corresponding discrete stability and error analyses of the proposed scheme. In the collocation scheme, we employ fractional (non-polynomial) Lagrange interpolants satisfying the Kronecker delta property at the collocation points. Subsequently, we obtain the corresponding distributed differentiation matrices to be employed in the discretization of the strong problem. We perform systematic numerical tests to demonstrate the efficiency and conditioning of each method.