Abstract
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.
Highlights
Fractional partial differential equations (FPDEs) have received more and more attention from scholars in the fields of science and engineering because of their various practical applications [1,2,3,4,5,6].theoretical research and some problems of the solutions of FPDEs have been closely studied by many scholars [7,8,9,10,11]
In order to formulate the mixed finite volume element (MFVE) approximate scheme, we introduce an auxiliary variable σ ( X, t) = −∇u( X, t), and rewrite the problem (1) as
We introduce the generalized MFVE projection : [0, T ] → Lh × Hh, satisfies
Summary
Fractional partial differential equations (FPDEs) have received more and more attention from scholars in the fields of science and engineering because of their various practical applications [1,2,3,4,5,6]. Yang et al [27] constructed an FV scheme with pretreatment Lanczos method to solve two-dimensional space fractional reaction-diffusion equations. We will develop a mixed finite volume element (MFVE) method to solve the time-fractional reaction-diffusion equations as follows α. Our aim is to construct an MFVE scheme [36,37,38,39,40,41] by combining the MFE method [42,43,44,45] with the FVE method [46,47,48,49,50,51] to treat the time Caputo fractional reaction-diffusion equation. The mark C will denote a generic positive constant which is independent of the spatial mesh parameter h and time discretization parameter τ
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