An Order Reduction Method for the Nonlinear Caputo-Hadamard Fractional Diffusion-Wave Model

Abstract

In this paper, the numerical solutions of the nonlinear Hadamard fractional diffusion-wave model with the initial singularity are investigated. Firstly, the model is transformed into coupled equations by virtue of a symmetric fractional-order reduction method. Then the $L_{\log,2-1_{\sigma} }$ formula on nonuniform grids is applied to approach to the time fractional derivative. In addition, the discrete fractional Grönwall inequality is used to analyze the optimal convergence of the constructed numerical scheme by the energy method. The accuracy of the theoretical analysis will be demonstrated by means of a numerical experiment at the end.

Keywords

Caputo-Hadamard fractional differential equations (FDEs) / Symmetric fractional-order reduction method / Nonuniform mesh / Stability and convergence / 65M06 / 65M12 / 35R11

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Funding

National Natural Science Foundation of China(11701103)

Natural Science Foundation of Guangdong Province(2022A1515012147)

RIGHTS & PERMISSIONS

Shanghai University