Computers and Mathematics with Applications Numerical Approximation of Nonlinear Fractional Differential Equations with Subdiffusion and Superdiffusion
Computers and Mathematics with Applications Numerical Approximation of Nonlinear Fractional Differential Equations with Subdiffusion and Superdiffusion
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通讯作者：
Changpin Li;Zhengang Zhao;Y. Chen
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作者：
Changpin Li;Zhengang Zhao;Y. Chen
关键词：时空分数阶扩散方程次扩散超扩散Riemann-Liouville导数卡普托导数差分方法有限元方法a b S t r a c t本文研究了时空分数阶(为简单起见)的非线性次扩散和超扩散方程，它可以将物质通量矢量与一般意义上的浓度梯度联系起来，描述了反常扩散、分数布朗运动等现象。分析了半离散和全离散的数值逼近，其中分别分析了1+β∈[1，2]阶空间Riemann-Liouville分数阶导数的Galerkin有限元方法和α∈(0，1)阶时间Caputo导数的有限差分格式(次扩散)和(1，2)阶(超扩散)。给出了弱解的存在唯一性、数值稳定性和误差估计的结果。数值算例验证了理论分析的正确性。在我们的模拟过程中，观察到了一个有趣的粒子扩散现象，即平均而言，0<；α<；1的扩散速度比α=1的扩散速度慢，而1<；α<；2的扩散速度比α=1的快。对于空间扩散，我们也有类似的观察。
Keywords: Time–space fractional diffusion equation Subdiffusion Superdiffusion Riemann–Liouville derivative Caputo derivative Difference method Finite element method a b s t r a c t In this paper, we study the time–space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann–Liouville fractional derivative with order 1 + β ∈ [1, 2] and the finite difference scheme for the time Caputo derivative with order α ∈ (0, 1) (for subdiffusion) and (1, 2) (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for 0 < α < 1 is slower than that for α = 1, but the diffusion velocity for 1 < α < 2 is faster than that for α = 1. For the spatial diffusion, we have a similar observation.