Finite difference scheme for the time-fractional Fokker–Planck equation with time- and space-dependent forcing

Abstract

This paper presents the finite difference scheme for the time-fractional Fokker–Planck equation with the external force and source term which depends on space and time. We prove the stability and convergence of the numerical scheme by the energy method. The spatial error is $O\left(h^2\right)$ in the discrete $L_2$-norm, and $O\left(\tau \right)$ for the time discretization with uniform stepsize τ when $\alpha \in (\frac{1}{2},1)$. The numerical examples verify these theoretical results.

KEYWORDS:

• Fokker–Planck equation
• fractional
• finite-difference scheme
• time- and space-dependent forcing
• stability
• convergence

2020 AMS SUBJECT CLASSIFICATIONS:

• 65M12
• 65M15
• 65M60
• 35Q84
• 45K05

Acknowledgments

The authors would like to thank the Editor and the referees for their valuable comments and helpful suggestions which improved the paper greatly.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by Natural Science Foundation of Shandong Province [grant number ZR2014AM013], National Natural Science Foundation Committee funded by NSAF Joint Fund of China Institute of Engineering Physics [grant number U1430101], National Natural Science Foundation of China [grant numbers 11171189 and 11471194].