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Volume 16, Issue 5
Finite Element Scheme with H2N2 Interpolation for Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation

Huiqin Zhang, Yanping Chen, Jianwei Zhou & Yang Wang

Adv. Appl. Math. Mech., 16 (2024), pp. 1197-1222.

Published online: 2024-07

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  • Abstract

In this paper, two numerical schemes for the multi-term fractional mixed diffusion and diffusion-wave equation (of order $α,$ with $0<α<2$) are developed to solve the initial value problem. Firstly, we study a direct numerical scheme that uses quadratic Charles Hermite and Newton (H2N2) interpolation polynomials approximations in the temporal direction and finite element discretization in the spatial direction. We prove the stability of the direct numerical scheme by the energy method and obtain a priori error estimate of the scheme with an accuracy of order $3−α.$ In order to improve computational efficiency, a new fast numerical scheme based on H2N2 interpolation and an efficient sum-of-exponentials approximation for the kernels is proposed. Numerical examples confirm the error estimation results and the validity of the fast scheme.

  • AMS Subject Headings

65M10, 78A48

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COPYRIGHT: © Global Science Press

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@Article{AAMM-16-1197, author = {Zhang , HuiqinChen , YanpingZhou , Jianwei and Wang , Yang}, title = {Finite Element Scheme with H2N2 Interpolation for Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2024}, volume = {16}, number = {5}, pages = {1197--1222}, abstract = {

In this paper, two numerical schemes for the multi-term fractional mixed diffusion and diffusion-wave equation (of order $α,$ with $0<α<2$) are developed to solve the initial value problem. Firstly, we study a direct numerical scheme that uses quadratic Charles Hermite and Newton (H2N2) interpolation polynomials approximations in the temporal direction and finite element discretization in the spatial direction. We prove the stability of the direct numerical scheme by the energy method and obtain a priori error estimate of the scheme with an accuracy of order $3−α.$ In order to improve computational efficiency, a new fast numerical scheme based on H2N2 interpolation and an efficient sum-of-exponentials approximation for the kernels is proposed. Numerical examples confirm the error estimation results and the validity of the fast scheme.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2023-0117}, url = {http://global-sci.org/intro/article_detail/aamm/23291.html} }
TY - JOUR T1 - Finite Element Scheme with H2N2 Interpolation for Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation AU - Zhang , Huiqin AU - Chen , Yanping AU - Zhou , Jianwei AU - Wang , Yang JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1197 EP - 1222 PY - 2024 DA - 2024/07 SN - 16 DO - http://doi.org/10.4208/aamm.OA-2023-0117 UR - https://global-sci.org/intro/article_detail/aamm/23291.html KW - The multi-term fractional mixed diffusion and diffusion-wave equation, finite element method, H2N2 interpolation, fast algorithm, stability and convergence. AB -

In this paper, two numerical schemes for the multi-term fractional mixed diffusion and diffusion-wave equation (of order $α,$ with $0<α<2$) are developed to solve the initial value problem. Firstly, we study a direct numerical scheme that uses quadratic Charles Hermite and Newton (H2N2) interpolation polynomials approximations in the temporal direction and finite element discretization in the spatial direction. We prove the stability of the direct numerical scheme by the energy method and obtain a priori error estimate of the scheme with an accuracy of order $3−α.$ In order to improve computational efficiency, a new fast numerical scheme based on H2N2 interpolation and an efficient sum-of-exponentials approximation for the kernels is proposed. Numerical examples confirm the error estimation results and the validity of the fast scheme.

Huiqin Zhang, Yanping Chen, Jianwei Zhou & Yang Wang. (2024). Finite Element Scheme with H2N2 Interpolation for Multi-Term Time-Fractional Mixed Sub-Diffusion and Diffusion-Wave Equation. Advances in Applied Mathematics and Mechanics. 16 (5). 1197-1222. doi:10.4208/aamm.OA-2023-0117
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