Efficient difference method for time-space fractional diffusion equation with Robin fractional derivative boundary condition

Abstract

In this paper, a numerical method is proposed to solve the time-space fractional diffusion equation with Robin fractional derivative boundary condition. Under the weak regularity assumptions of solution, we present a numerical scheme based on the L1 method for time discretization on graded mesh and the Grünwald-Letnikov formula for spatial discretization on uniform mesh. And a fast scheme for the considered problem is constructed based on the exponential summation approximation of the kernel function t^−α. Meanwhile, a detailed analysis of stability and convergence is given. Then, the extrapolation method is applied to the space direction to make it reach the second-order accuracy. Finally, numerical experiments show that the proposed method is effective.

Appendix A: The proof of the inequality (46)

Proof

It is proved by induction. When n = 1, it follows from (45) that

$$ ||e^{1}||_{\infty}\leq\frac{1}{\sum\limits_{l=1}^{m}\xi_{1,1}^{l}}|\bar{R}_{i_{0}}^{1}|. $$

(67)

So, the inequality (46) holds for n = 1. Suppose that it holds for j ≤ n − 1. Next, we prove it is also holds for j = n. By (45), we have

$$ \small \begin{array}{ll} ||e^{n}||_{\infty}&\leq\frac{1}{\sum\limits_{l=1}^{m}\xi_{n,n}^{l}}\left[\sum\limits_{l=1}^{m}\frac{a_{l}}{{\varGamma}(1-\alpha_{l})}\sum\limits_{k=1}^{n-1}(c_{n,k+1}^{l}-c_{n,k}^{l})||e^{k}||_{\infty} +\sum\limits_{l=1}^{m}\xi_{n,1}^{l}\frac{|\bar{R}_{i_{0}}^{n}|}{\sum\limits_{l=1}^{m}\xi_{n,1}^{l}}\right]\\ &\leq\frac{1}{\sum\limits_{l=1}^{m}\xi_{n,n}^{l}}\left[\sum\limits_{l=1}^{m}\frac{a_{l}}{{\varGamma}(1-\alpha_{l})}\sum\limits_{k=1}^{n-1}(c_{n,k+1}^{l}-c_{n,k}^{l})\max\limits_{1\leq a\leq k}\frac{|\bar{R}_{i_{0}}^{a}|}{\sum\limits_{l=1}^{m}\xi_{a,1}^{l}}+\sum\limits_{l=1}^{m}\xi_{n,1}^{l}\frac{|\bar{R}_{i_{0}}^{n}|}{\sum\limits_{l=1}^{m}\xi_{n,1}^{l}}\right]\\ &\leq\frac{1}{\sum\limits_{l=1}^{m}\xi_{n,n}^{l}}\left[\sum\limits_{l=1}^{m}\frac{a_{l}}{{\varGamma}(1-\alpha_{l})}\sum\limits_{k=1}^{n-1}(c_{n,k+1}^{l}-c_{n,k}^{l})\max\limits_{1\leq a\leq n}\frac{|\bar{R}_{i_{0}}^{a}|}{\sum\limits_{l=1}^{m}\xi_{a,1}^{l}}+\sum\limits_{l=1}^{m}\xi_{n,1}^{l}\frac{|\bar{R}_{i_{0}}^{n}|}{\sum\limits_{l=1}^{m}\xi_{n,1}^{l}}\right]. \end{array} $$

(68)

Thus, it is available that

$$ \begin{array}{ll} ||e^{n}||_{\infty}&\leq\frac{1}{\sum\limits_{l=1}^{m}\xi_{n,n}^{l}}\left[\sum\limits_{l=1}^{m}\frac{a_{l}}{{\varGamma}(1-\alpha_{l})}\sum\limits_{k=1}^{n-1}(c_{n,k+1}^{l}-c_{n,k}^{l})+\sum\limits_{l=1}^{m}\xi_{n,1}^{l}\right] \underset{1\leq k\leq n}{\max}\frac{|\bar{R}_{i_{0}}^{k}|}{\sum\limits_{l=1}^{m}\xi_{k,1}^{l}}\\ &\leq\frac{1}{\sum\limits_{l=1}^{m}\xi_{n,n}^{l}}\left[\sum\limits_{l=1}^{m}\frac{a_{l}}{{\varGamma}(1-\alpha_{l})}\left( \sum\limits_{k=1}^{n-1}(c_{n,k+1}^{l}-c_{n,k}^{l})+c_{n,1}^{l}\right)\right]\underset{1\leq k\leq n}{\max}\frac{|\bar{R}_{i_{0}}^{k}|}{\sum\limits_{l=1}^{m}\xi_{k,1}^{l}}\\ &\leq\underset{1\leq k\leq n}{\max}\frac{|\bar{R}_{i_{0}}^{k}|}{\sum\limits_{l=1}^{m}\xi_{k,1}^{l}}. \end{array} $$

(69)

This completes the proof. □