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

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 down arrow 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. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

We propose a rapid and accurate contour method for solving time-fractional PDEs. The method utilizes an optimized stable quadrature rule to invert the Laplace transform, making it suitable for infinite-dimensional operators. It is parallelizable, avoids resolving singularities of the solution as t approaches 0, and overcomes the memory consumption challenge of time-stepping methods for time-fractional PDEs. The resulting ODEs from quadrature are solved using adaptive sparse spectral methods with exponential convergence. Our approach simplifies the analysis of fractional beam equations and allows for efficient algorithm with explicit error control.