Exponentially convergent method for time-fractional evolution equation
Abstract
An exponentially convergent numerical method for solving a differential equation with a right-hand fractional Riemann-Liouville time-derivative and an unbounded operator coefficient in Banach space is proposed and analysed for a homogeneous/inhomogeneous equation of the Hardy-Tichmarsh type. We employ a solution representation by the Danford-Cauchy integral on hyperbola that envelopes spectrum of the operator coefficient with a subsequent application of an exponentially convergent quadrature. To do that, parameters of the hyperbola are chosen so that the integration function has an analytical extension into a strip around the real axis and then apply the Sinc-quadrature. We show the exponential accuracy and illustrate the results by a numerical example confirming the {\it a priori} estimate. Existence conditions for the solution of the inhomogeneous equation are established.
Cite
@article{arxiv.2412.17521,
title = {Exponentially convergent method for time-fractional evolution equation},
author = {V. Vasylyk and V. L. Makarov},
journal= {arXiv preprint arXiv:2412.17521},
year = {2024}
}