Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem

Chaobao Huang; Xiaohui Liu; Xiangyun Meng; Martin Stynes

doi:10.1515/cmam-2019-0042

Published by De Gruyter January 15, 2020

Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem

Chaobao Huang , Xiaohui Liu , Xiangyun Meng and Martin Stynes

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2019-0042

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Abstract

An initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is ( 0 , 1 ) d for some d ∈ { 1 , 2 , 3 } . This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where d = 1 and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.

Keywords: Caputo Fractional Derivative; Initial-Boundary Value Problem; Multiterm Fractional; Weak Singularity; Graded Mesh; L1 Scheme

MSC 2010: 65M60; 65M12; 35R11

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11801332

Award Identifier / Grant number: 11971259

Award Identifier / Grant number: NSAF-U1930402

Funding statement: The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grant NSAF U1930402. The research of Chaobao Huang is supported in part by the National Natural Science Foundation of China (Grant Nos. 11801332 and 11971259).

Acknowledgements

We thank both reviewers for their careful reading of the paper, and in particular we thank one reviewer for suggesting to us a significant simplification in the stability analysis.

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Received: 2019-03-02

Revised: 2019-10-29

Accepted: 2019-12-05

Published Online: 2020-01-15

Published in Print: 2020-10-01

Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem

Abstract

Acknowledgements

References

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