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The solution of the anomalous diffusion equation by a Finite Element Method based on the Caputo derivative

Correa, R.M.; Carrer, J.A.M.; Solheid, B.S.; Trevelyan, J.

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Authors

R.M. Correa

J.A.M. Carrer

B.S. Solheid



Abstract

A Finite Element Method formulation is developed for the solution of the anomalous diffusion equation. This equation belongs to the branch of mathematics called fractional calculus; it is governed by a partial differential equation in which a fractional time derivative, whose order ranges in the interval (0,1), replaces the first order time derivative of the classical diffusion equation. In this work, the Caputo integro-differential operator is employed to represent the fractional time derivative. After assuming a linear time variation for the variable of interest, say u, in the intervals in which the overall time is discretized, the integral in the Caputo operator is computed analytically. To demonstrate the usefulness of the proposed formulation, some examples are analysed, showing a good agreement between the FEM results the analytical solutions, even for small orders of the time derivative.

Citation

Correa, R., Carrer, J., Solheid, B., & Trevelyan, J. (2022). The solution of the anomalous diffusion equation by a Finite Element Method based on the Caputo derivative. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 44(6), Article 250. https://doi.org/10.1007/s40430-022-03544-5

Journal Article Type Article
Acceptance Date Apr 19, 2022
Online Publication Date May 27, 2022
Publication Date 2022-06
Deposit Date May 11, 2022
Publicly Available Date May 27, 2023
Journal Journal of the Brazilian Society of Mechanical Sciences and Engineering
Print ISSN 1678-5878
Electronic ISSN 1806-3691
Publisher Springer Berlin Heidelberg
Peer Reviewed Peer Reviewed
Volume 44
Issue 6
Article Number 250
DOI https://doi.org/10.1007/s40430-022-03544-5
Public URL https://durham-repository.worktribe.com/output/1206385

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