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Licensed Unlicensed Requires Authentication Published by De Gruyter September 2, 2017

The Numerical Computation of the Time Fractional Schrödinger Equation on an Unbounded Domain

  • Dan Li , Jiwei Zhang EMAIL logo and Zhimin Zhang

Abstract

A fast and accurate numerical scheme is presented for the computation of the time fractional Schrödinger equation on an unbounded domain. The main idea consists of two parts. First, we use artificial boundary methods to equivalently reformulate the unbounded problem into an initial-boundary value (IBV) problem. Second, we present two numerical schemes for the IBV problem: a direct scheme and a fast scheme. The direct scheme stands for the direct discretization of the Caputo fractional derivative by using the L1-formula. The fast scheme means that the sum-of-exponentials approximation is used to speed up the evaluation of the Caputo fractional derivative. The resulting fast algorithm significantly reduces the storage requirement and the overall computational cost compared to the direct scheme. Furthermore, the corresponding stability analysis and error estimates of two schemes are established, and numerical examples are given to verify the performance of our approach.

MSC 2010: 65M12; 65M06; 65M15

Award Identifier / Grant number: 91430216

Award Identifier / Grant number: U1530401

Award Identifier / Grant number: 11471031

Award Identifier / Grant number: DMS-1419040

Funding statement: The research of the second author was supported in part by the National Natural Science Foundation of China under grants 91430216 and U1530401. The research of the third author was supported in part by the National Natural Science Foundation of China under grants 11471031, 91430216, U1530401, and by the US National Science Foundation under grant DMS-1419040.

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Received: 2016-11-11
Revised: 2017-8-19
Accepted: 2017-8-19
Published Online: 2017-9-2
Published in Print: 2018-1-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

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