Abstract
In the present work, orthogonal spline collocation (OSC) method with convergence order O(τ3−α + hr+ 1) is proposed for the two-dimensional (2D) fourth-order fractional reaction-diffusion equation, where τ, h, r, and α are the time-step size, space size, polynomial degree of space, and the order of the time-fractional derivative (0 < α < 1), respectively. The method is based on applying a high-order finite difference method (FDM) to approximate the time Caputo fractional derivative and employing OSC method to approximate the spatial fourth-order derivative. Using the argument developed recently by Lv and Xu (SIAM J. Sci. Comput. 38, A2699–A2724, 2016) and mathematical induction method, the optimal error estimates of proposed fully discrete OSC method are proved in detail. Then, the theoretical analysis is validated by a number of numerical experiments. To the best of our knowledge, this is the first proof on the error estimates of high-order numerical method with convergence order O(τ3−α + hr+ 1) for the 2D fourth-order fractional equation.
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Acknowledgements
The authors thank the anonymous reviewers for their constructive comments and suggestions and Professor Graeme Fairweather for stimulating discussions and for his constant encouragement and support.
Funding
The work is supported by the National Natural Science Foundation of China (11701168, 11601144, 11626096), Hunan Provincial Natural Science Foundation of China (2018JJ3108, 2018JJ3109, 2018JJ4062), Scientific Research Fund of Hunan Provincial Education Department (16K026,YB2016B033), China Postdoctoral Science Foundation (2016M600964), Science Challenge Project (TZ2016002).
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Appendices
Appendix A
Proof
Since the first step equation is easier to treat,we will only consider the case for the time stepsn ≥ 2.By using the reformulations (3.14), (3.18) and (3.19), (3.20) can be rewritten as
where (⋅,⋅) isthe usual L2-innerproduct, ∥⋅∥ iscorresponding norm.
By choosing \(\chi = 2\overline {u}^{n}\),\(\psi = 2\alpha _{0}\beta _{0}^{-1}{\Delta }\overline {u}^{n}\) in (A.63) and (A.64), respectively, we subtract the resulting equations to obtain
In virtue of the identity
and
Substituting the above two equation to (A.65), using Lemma 4 and Schwartz inequality, wehave
From (1) and (3) ofLemma 4, we attain
Next we will prove the following estimates (A.67) by the mathematical induction.
It is easy to check that (A.67) hold for the casen = 2. Assuming the estimate(A.67) is true for n = 3, 4,⋯ , i − 1, we want to prove that it also true for n = i,and deduce from (A.66)
Thus the estimate (A.67) is proven.
From Lemma 4(1) and (4), we have
Through a recombination of the terms in (A.67) and (A.68), we obtain
Finally, since \(\overline {u}^{n}=u^{n}-pu^{n-1}\), wenow turn to estimate ∥un∥2. Applying the triangle inequality and (A.69) yields
using arguments similar to (A.70), we have
Combining the above estimate with (A.69) gives (3.21). The proof is completed.□
Appendix B
Proof
Firstly, by using the definition \({\Delta }\overline {\xi }^{n}={\Delta }\xi ^{n}-p{\Delta }\xi ^{n-1}\), we have
then,
Also,
Substituting (B.71) into (B.72) to get (4.48), the proof is completed. □
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Zhang, H., Yang, X. & Xu, D. A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation. Numer Algor 80, 849–877 (2019). https://doi.org/10.1007/s11075-018-0509-z
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DOI: https://doi.org/10.1007/s11075-018-0509-z