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