Abstract
The paper is devoted to the construction of high-precision unconditionally stable finite difference methods for solving time-space fractional diffusion equation with the Caputo fractional derivative (of order β, with β ∈ (0, 1)) in time and the Rimann-Liouville fractional derivatives (of order α, with α ∈ (1, 2]) in space. Two kinds of difference schemes with the approximation orders O(τ2−β + h3) and O(τ2 + h3) respectively are constructed. The stability and convergence are analyzed in detail. The obtained results are illustrated numerically by some examples, and a comparative study of several high-order schemes is also carried out.
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Acknowledgements
Authors thank the editor and referees for their constructive comments and useful suggestions which improved greatly the quality of our paper.
Funding
The work was partially supported by the CAPES and CNPq, Brazil; the NSF of China (Nos. 1701196, 11701197); the China Postdoctoral Sustentation Fund; the Natural Science Foundation of Fujian Province (No. 2016J05007); the Natural Science Foundation of Xinjiang Province (No. 2016D01C07) and the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (ZQN-YX502), China.
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Appendix
Appendix
In this appendix, we present the proof of h(α; x) ≤ 0. Thus, we list the following lemma.
Lemma A.1
For anyx ∈ [0,π] andα ∈ (1, 2), we haveh(α; x) decreases withrespect toα,that is
Proof
Taking the partial derivative of h(α, x) with respect to α, we have
where
with \(A=\frac {\alpha (x-\pi )}{2}-x\).
Next, we estimate Eq. 28 via the following two steps.
- I: :
-
We first consider L1(α; x) ≤ 0. It is easy to obtain
$$ L_{1}(\alpha,x)=\frac{6\alpha-13}{12}\cos(A)\cos(x)+\frac{1}{2}\sin(A)\sin(x)+\frac{-6\alpha+ 13}{12}\cos(A). $$(2)For any x ∈ [π/2,π] and α ∈ (1, 2), it is obviously L1(α; x) ≤ 0.
For any x ∈ [0,π/2) and α ∈ (1, 2), since
$$\frac{\partial}{\partial x}\left[\frac{6\alpha-13}{12}\cos A\cos x+\frac{1}{2}\sin A\sin x\right]\leq 0,$$we have
$$\frac{6\alpha-13}{12}\cos A\cos x+\frac{1}{2}\sin A\sin x\leq \frac{6\alpha-13}{12}\cos \left( -\frac{\alpha\pi}{2}\right), $$and combining with
$$\frac{-6\alpha+ 13}{12}\cos A\leq\frac{-6\alpha+ 13}{12}\cos\left( \frac{-\alpha\pi}{2}\right),$$so we obtain L1(α; x) ≤ 0.
- II: :
-
We now consider L2(α; x) ≤ 0. It is easy to obtain
$$ L_{2}(\alpha; x)=\frac{3\alpha^{2}-7\alpha}{12}\sin A\cos x+\frac{2-\alpha}{2}\sin\left( \frac{\alpha(x-\pi)}{2}\right)+\frac{-3\alpha^{2}+ 13\alpha}{12}\sin(A). $$(3)For any x ∈ [π/2,π] and α ∈ (1, 2), it is obviously L2(α; x) ≤ 0.
For any x ∈ [0,π/2) and α ∈ (1, 2), we have
$$\begin{array}{@{}rcl@{}} L_{2}(\alpha; x)&=&\frac{3\alpha^{2}-7\alpha}{12}\sin A\cos x+\frac{2-\alpha}{2}\sin(A+x)+\frac{-3\alpha^{2}+ 13\alpha}{12}\sin(A)\\ &=&\frac{3\alpha^{2}-13\alpha}{12}\sin A(\cos x-1)+\sin A\cos x+\frac{2-\alpha}{2}\cos A\cos x\\ &\leq& 0. \end{array} $$
As a result of (I) and (II), we obtain the expected result.□
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Zhai, S., Weng, Z., Feng, X. et al. Investigations on several high-order ADI methods for time-space fractional diffusion equation. Numer Algor 82, 69–106 (2019). https://doi.org/10.1007/s11075-018-0594-z
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DOI: https://doi.org/10.1007/s11075-018-0594-z