A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations

TL;DR

This paper introduces a fast, accurate contour method for solving time-fractional PDEs, especially applied to fractional viscoelastic beam equations, with advantages in efficiency, parallelization, and error control.

Contribution

The paper develops a novel contour-based numerical method for time-fractional PDEs that improves efficiency and error management, and applies it to fractional viscoelastic beam equations.

Findings

Method achieves exponential convergence with optimal complexity.

Efficiently models energy evolution and surface deformation in viscoelastic materials.

Approach is adaptable to various time-fractional PDEs and parameters.

Abstract

We develop a rapid and accurate contour method for the solution of time-fractional PDEs. The method inverts the Laplace transform via an optimised stable quadrature rule, suitable for infinite-dimensional operators, whose error decreases like $\exp(-cN/\log(N))$ for $N$ quadrature points. The method is parallisable, avoids having to resolve singularities of the solution as $t\downarrow 0$, and avoids the large memory consumption that can be a challenge for time-stepping methods applied to time-fractional PDEs. The ODEs resulting from quadrature are solved using adaptive sparse spectral methods that converge exponentially with optimal linear complexity. These solutions of ODEs are reused for different times. We provide a complete analysis of our approach for fractional beam equations used to model small-amplitude vibration of viscoelastic materials with a fractional Kelvin-Voigt stress-strain relationship. We calculate the system's energy evolution over time and the surface deformation in cases of both constant and non-constant viscoelastic parameters. An infinite-dimensional ``solve-then-discretise'' approach considerably simplifies the analysis, which studies the generalisation of the numerical range of a quasi-linearisation of a suitable operator pencil. This allows us to build an efficient algorithm with explicit error control. The approach can be readily adapted to other time-fractional PDEs and is not constrained to fractional parameters in the range $0<\nu<1$.