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

• Published in: Journal of computational physics Vol. 454; p. 110995
• Main Authors: Colbrook, Matthew J., Ayton, Lorna J.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.04.2022; Elsevier Science Ltd
• Subjects: Algorithms; Computational physics; Contour methods; Contours; Error control; Fractional derivative; Laplace transforms; Parameters; Quadratures; Spectral methods; Stress-strain relationships; Viscoelastic beam structures; Viscoelastic materials; Viscoelasticity; Contour methods; Spectral methods; Error control; Fractional derivative; Viscoelastic beam structures
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2022.110995

•Contour integral methods allow rapid and accurate solution of time-fractional PDEs.•Infinite-dimensional approach simplifies analysis and provides error control.•Parallelizable method avoids typical challenges of large memory and singularities.•Use of spectral methods gives exponential convergence and linear complexity.•Application to vibration of thin viscoelastic beams with variable parameters. 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↓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<ν<1.