A Generalized BDF2-<i>θ</i> Finite Element Method for Nonlinear Distributed-Order Time-Fractional Hyperbolic Wave Equations

HOU Yaxin; LIU Yang; LI Hong

doi:10.21656/1000-0887.450013

Volume 46 Issue 1

Jan. 2025

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Article Navigation > Applied Mathematics and Mechanics > 2025 > 46(1): 114-128

HOU Yaxin, LIU Yang, LI Hong. A Generalized BDF2-θ Finite Element Method for Nonlinear Distributed-Order Time-Fractional Hyperbolic Wave Equations[J]. Applied Mathematics and Mechanics, 2025, 46(1): 114-128. doi: 10.21656/1000-0887.450013

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A Generalized BDF2-θ Finite Element Method for Nonlinear Distributed-Order Time-Fractional Hyperbolic Wave Equations

doi: 10.21656/1000-0887.450013

HOU Yaxin^1
,,
LIU Yang^{2
,
,},
LI Hong^2
,

1.
Department of Mathematics, College of Science, Inner Mongolia University of Technology, Hohhot 010051, P.R.China
2.
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, P.R.China

Received Date: 2024-01-17
Rev Recd Date: 2024-03-25
Publish Date: 2025-01-01

Abstract

Abstract

A finite element (FE) method based on the generalized backward differentiation θ formula (generalized BDF2-θ) was presented to solve nonlinear distributed-order time-fractional hyperbolic wave equations. The temporal direction was approximated with the generalized BDF2-θ to get the FE fully discrete scheme. The proposed model with high-order temporal derivatives was transformed into a coupled system including 2 lower-order equations. The stability of the proposed FE scheme and the optimal error estimates for 2 functions u and p were discussed. Several numerical examples indicate the feasibility and efficiency of the schemes.
- nonlinear distributed-order time-fractional hyperbolic wave equation,
- FE method,
- generalized BDF2-θ,
- stability,
- error estimate,
- numerical simulation

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References

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