A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation

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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.

Keywords

Fourth-order fractional equation Orthogonal spline collocation Finite difference method Error estimate 

Mathematics Subject Classification (2010)

65M12 65M06 65M70 35S10 

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Notes

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.

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Authors and Affiliations

  1. 1.School of ScienceHunan University of TechnologyZhuzhouChina
  2. 2.National Key Laboratory of Science and Technology on Computational PhysicsInstitute of Applied Physics and Computational MathematicsBeijingChina
  3. 3.Department of MathematicsHunan Normal UniversityChangshaChina

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