Exponentially convergent method for time-fractional evolution equation

TL;DR

This paper introduces an exponentially convergent numerical method for solving time-fractional evolution equations with unbounded operator coefficients, utilizing complex analysis and quadrature techniques for high accuracy.

Contribution

The paper develops a novel exponentially convergent numerical approach for fractional evolution equations with unbounded operators, combining complex integral representations and Sinc-quadrature.

Findings

Method achieves exponential accuracy in numerical solutions.

Numerical example confirms theoretical error estimates.

Existence conditions for solutions are established.

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.