Fourth-order methods for space fractional reaction–diffusion equations with non-smooth data

ABSTRACT

We propose two fourth-order methods in time for one-dimensional space fractional reaction–diffusion equations. The methods are based on fourth-order Exponential Time Differencing Runge–Kutta method. Padé approximations of matrix exponential functions are used to construct an L-stable and an A-stable method. Partial fraction splitting technique is applied to construct more reliable and computationally efficient versions of the methods. Solution profiles as well as convergence rates in time are presented for fractional enzyme kinetics equation and fractional Fisher equation. The L-stable method performs well in the presence of non-smooth mismatched initial-boundary data while the A-stable method is more economical for smooth matched initial-boundary data.

KEYWORDS:

• Fractional partial differential equations
• anomalous diffusion
• Riesz fractional derivative
• exponential time differencing
• Padé approximation

2010 AMS SUBJECT CLASSIFICATION:

• 35R11
• 65N12
• 65N06
• 35B65
• 35B30

Acknowledgements

The authors are grateful to the referees for their constructive comments and valuable suggestions which have improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

M. Yousuf  http://orcid.org/0000-0003-3893-4776.