Multi-domain Spectral Collocation Method for Variable-Order Nonlinear Fractional Differential Equations

TL;DR

This paper introduces a multi-domain spectral collocation method for solving nonlinear and variable-order fractional PDEs efficiently, overcoming limitations of traditional spectral methods for such complex equations.

Contribution

The paper develops a novel multi-domain spectral collocation approach with variable-order differentiation and interface penalty techniques for nonlinear fractional PDEs.

Findings

Effective for fractional Helmholtz equations with variable order.

Improved stability and convergence with penalty parameter tuning.

Accurate solutions for fractional Burgers equation compared to single domain methods.

Abstract

Spectral and spectral element methods using Galerkin type formulations are efficient for solving linear fractional PDEs (FPDEs) of constant order but are not efficient in solving nonlinear FPDEs and cannot handle FPDEs with variable-order. In this paper, we present a multi-domain spectral collocation method that addresses these limitations. We consider FPDEs in the Riemann-Liouville sense, and employ Jacobi Lagrangian interpolants to represent the solution in each element. We provide variable-order differentiation formulas, which can be computed efficiently for the multi-domain discretization taking into account the nonlocal interactions. We enforce the interface continuity conditions by matching the solution values at the element boundaries via the Lagrangian interpolants, and in addition we minimize the jump in (integer) fluxes using a penalty method. We analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme. We demonstrate the effectiveness of the new method for the fractional Helmholtz equation of constant and variable-order using $h-p$ refinement for different values of the penalty parameter. We also solve the fractional Burgers equation with constant and variable-order and compare with solutions obtained with a single domain spectral collocation method.