Numerical Solution for Fractional Partial Differential Equations Using Crank-Nicolson Method with Shifted Grünwald Estimate

Abstract

In this paper, the novel hybrid finite difference type Crank-Nicolson scheme with the aid of shifted Grünwald estimate is proposed to solve fractional partial differential equations. Consistency of the proposed method is confirmed using fractional Taylor’s expansion. Error analysis and properties of the scheme are proved. It is proved that the truncation error for this scheme is of the order of the fractional. Stability and convergence of the proposed method is proved. The exact solution is obtained via two-steps Adomian decomposition method. Companions are made between this proposed scheme and the closed analytical form solution. Numerical results are given.

Keywords

• Crank-Nicolson method
• fractional partial differential equations
• fractional Taylor's series
• Riemann-Liouville fractional derivative
• shifted Grünwald Estimate
• two-step Adomian decomposition method