On a numerical investigation of the time fractional Fokker– Planck equation via local discontinuous Galerkin method

ABSTRACT

This paper presents two numerical solutions of time fractional Fokker– Planck equations (TFFPE) based on the local discontinuous Galerkin method (LDGM). Two time-discretization schemes for the fractional order part of TFFPE are investigated. The first discretization utilizes the fractional finite difference scheme (FFDS) and in the second scheme the fractional derivative is replaced by the Volterra integral equation which it computed by the trapezoidal quadrature scheme (TQS). Then the LDGM has been applied for space-discretization in both schemes. Additionally, the stability and convergence analysis of the proposed methods have been discussed. Finally some test problems have been investigated to confirm the validity and convergence of two proposed methods. The results show that FFDS and TQS are $2-\alpha$ and second-order accurate in time variable, respectively.

KEYWORDS:

• Time fractional Fokker–Planck equation
• discontinuous Galerkin method
• local discontinuous Galerkin method
• Caputo derivative
• Mittag–Leffler function
• stability and convergence analysis

2010 AMS SUBJECT CLASSIFICATIONS:

• 45D05
• 45G05
• 41A30

Disclosure statement

No potential conflict of interest was reported by the authors.