scholarly journals A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations

2014 ◽  
Vol 38 (15-16) ◽  
pp. 3755-3762 ◽  
Author(s):  
Qianqian Yang ◽  
Ian Turner ◽  
Timothy Moroney ◽  
Fawang Liu
Keyword(s):  
Finite Volume ◽  
Dimensional Space ◽  
Reaction Diffusion ◽  
Lanczos Method ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Two Dimensional ◽  
Volume Scheme ◽  
Fractional Reaction

scholarly journals A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations

2021 ◽  
Vol 7 (2) ◽  
pp. 1941-1970
Author(s):  
Zhichao Fang ◽  
◽  
Ruixia Du ◽  
Hong Li ◽  
Yang Liu
Keyword(s):  
Finite Volume ◽  
Matrix Theory ◽  
Reaction Diffusion ◽  
Volume Element ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Optimal Error Estimates ◽  
Fine Grid ◽  
Finite Volume Element ◽  
Fractional Reaction

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

scholarly journals A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 600 ◽  
Author(s):  
Jie Zhao ◽  
Hong Li ◽  
Zhichao Fang ◽  
Yang Liu
Keyword(s):  
Finite Volume ◽  
A Priori ◽  
Reaction Diffusion ◽  
Auxiliary Variable ◽  
Volume Element ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Unknown Variable ◽  
Finite Volume Element ◽  
Fractional Reaction

In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.

scholarly journals Curved fronts of bistable reaction–diffusion equations with nonlinear convection

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Hui-Ling Niu ◽  
Jiayin Liu
Keyword(s):  
Dimensional Space ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Asymptotically Stable ◽  
Two Dimensional ◽  
Nonlinear Convection ◽  
Globally Asymptotically Stable

Abstract This paper is concerned with traveling curved fronts of bistable reaction–diffusion equations with nonlinear convection in a two-dimensional space. By constructing super- and subsolutions, we establish the existence of traveling curved fronts. Furthermore, we show that the traveling curved front is globally asymptotically stable.

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