Discrete-time orthogonal spline collocation method for a modified anomalous diffusion equation

Abstract

The modified anomalous sub-diffusion equation (MASDE) describes processes that become less anomalous as time progresses. In this paper, based on Crank-Nicolson and weighted and shifted Grünwald difference (WSGD) formula, we develop orthogonal spline collocation (OSC) method with second-order temporal accuracy and fourth-order spatial accuracy for MASDE. The stability and convergence of our numerical schemes are analysed in detail. Also, our numerical results are consistent with our theoretical analysis.

Keywords:

• Modified anomalous fractional equation
• weighted and shifted Grünwald difference operator
• orthogonal spline collocation
• stability
• convergence

Mathematics Subject Classification (2000):

• 65M70
• 65M12
• 65M15
• 35R11

Acknowledgements

We thank the anonymous referees for their valuable comments and suggestions which helped us to improve the manuscript a lot.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by National Natural Science Foundation of China (11701168, 11601144), Hunan Provincial Natural Science Foundation of China (2018JJ3108, 2018JJ3109, 2018JJ4062), Scientific Research Fund of Hunan Provincial Education Department (18B304, YB2016B033), and China Postdoctoral Science Foundation (2018M631403).