Second-order predictor-corrector schemes for nonlinear distributed-order space-fractional differential equations with non-smooth initial data

ABSTRACT

We propose second-order linearly implicit predictor-corrector schemes for diffusion and reaction-diffusion equations of distributed-order. For diffusion equations of distributed order, we propose an analytical solution based on the spectral representation of the fractional Laplacian. Numerically, we approximate the integral term of the equation by the midpoint quadrature rule to obtain a multi-term space-fractional differential equation. The matrix transfer technique is used for spatial discretization of the resulting differential equation and methods based on Padé approximations to the exponential function are used in time. In particular, we discuss the (0,2)- and (1,1)-Padé approximations to the exponential function. The method based on the (1,1)-Padé approximation to the exponential function are seen to produce oscillations for some time steps and we propose a constraint on the choice of the time step to avoid these unwanted oscillations. Stability and convergence of the schemes are discussed. Numerical experiments are performed to support our theoretical observations.

KEYWORDS:

• Predictor-corrector methods
• distributed-order space-fractional differential equations
• nonlinear PDEs
• matrix transfer technique
• non-smooth initial data
• fractional Laplacian

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12
• 35R11

Acknowledgments

The author is grateful to the anonymous referees whose suggestions and comments greatly improved the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author.