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

•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.