A Runge–Kutta Gegenbauer spectral method for nonlinear fractional differential equations with Riesz fractional derivatives

ABSTRACT

A Runge–Kutta Gegenbauer spectral method is proposed to solve an initial-boundary value problem for a nonlinear two-dimensional fractional differential equation with variable coefficients. The solution to the problem at each time step is approximated by a bivariate polynomial based on shifted Gegenbauer polynomials and then the Runge–Kutta method of order 3 is applied to the problem. The convergence rate of the derived method is analysed. Numerical results are presented to verify the effectiveness of the method.

KEYWORDS:

• Fractional differential equation
• Riemann–Liouville fractional derivative
• Riesz fractional derivative
• Runge–Kutta method
• shifted Gegenbauer polynomial

2010 AMS SUBJECT CLASSIFICATIONS:

• 35R11
• 26A33
• 65R10
• 78M22

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions which improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partly supported by National Natural Science Foundation of China under contract No. 11771265.