Stability analysis of a second-order difference scheme for the time-fractional mixed sub-diffusion and diffusion-wave equation

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model wh...

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Bibliographic Details
Published inFractional calculus & applied analysis Vol. 27; no. 1; pp. 102 - 123
Main Authors Alikhanov, Anatoly A., Asl, Mohammad Shahbazi, Huang, Chengming
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.02.2024
Subjects
Abstract Harmonic Analysis
Analysis
Functional Analysis
Integral Transforms
Mathematics
Mathematics and Statistics
Operational Calculus
Original Paper
Stability and Convergence analysis
76Rxx
Mixed sub-diffusion and diffusion-wave equation
Caputo derivative
65M06
35R11
35L05
L2 formula
Riemann-Liouville integral
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Summary:This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0, 1). By providing an a priori estimate of the exact solution, we have established the continuous dependence on the initial data and uniqueness of the solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2-type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method’s potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings.
ISSN:1311-0454
1314-2224
DOI:10.1007/s13540-023-00229-1