Networks and Heterogeneous Media

2023, Issue 1: 291-309. doi: 10.3934/nhm.2023011
Research article Special Issues

Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations

• Received: 21 September 2022 Revised: 13 November 2022 Accepted: 05 December 2022 Published: 21 December 2022
• This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.

Citation: Zhe Pu, Maohua Ran, Hong Luo. Effective difference methods for solving the variable coefficient fourth-order fractional sub-diffusion equations[J]. Networks and Heterogeneous Media, 2023, 18(1): 291-309. doi: 10.3934/nhm.2023011

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• This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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