Exponentially convergent method for time-fractional evolution equation

TL;DR

The paper develops an exponentially convergent framework for time-fractional evolution equations with a right Riemann--Liouville derivative and an unbounded operator $A$ in a Banach space, of Hardy--Titchmarsh type. It represents the solution via the Danford--Cauchy integral on a spectral hyperbola enveloping the spectrum of $A$ and discretizes with a Sinc-quadrature to achieve exponential accuracy, supported by a priori error estimates. Existence conditions for the inhomogeneous case are established, and a numerical example confirms the theory in the homogeneous setting, with extensions to the inhomogeneous problem providing a parallel, highly efficient computational approach. The methods offer near-optimal complexity and are amenable to parallel computation, enabling scalable simulation of fractional dynamics with unbounded operator coefficients in applied contexts.

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.

Theorems & Definitions (7)

• Theorem 1.1
• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Theorem 2.1
• Theorem 2.2