A backward euler orthogonal spline collocation method for the time-fractional Fokker-Planck equation

2014 ◽  
Vol 31 (5) ◽  
pp. 1534-1550 ◽  
Author(s):  
Graeme Fairweather ◽  
Haixiang Zhang ◽  
Xuehua Yang ◽  
Da Xu
Keyword(s):  
Collocation Method ◽  
Planck Equation ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
Orthogonal Spline Collocation ◽  
Fokker Planck Equation ◽  
Backward Euler ◽  
Fokker Planck

A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing

2019 ◽  
Vol 40 (2) ◽  
pp. 1217-1240 ◽  
Author(s):  
Can Huang ◽  
Kim Ngan Le ◽  
Martin Stynes
Keyword(s):  
Numerical Method ◽  
Blow Up ◽  
Planck Equation ◽  
Stability And Convergence ◽  
Time Stability ◽  
Backward Euler Scheme ◽  
Fokker Planck Equation ◽  
Backward Euler ◽  
Numer Anal ◽  
Fokker Planck

Abstract First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763–1784) to solve the time-fractional Fokker–Planck equation on a domain $\varOmega \times [0,T]$ with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the $L^2(\varOmega )$ norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative $\alpha $ approaches the classical value of $1$. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker–Planck problem, we present a new $L^2(\varOmega )$ convergence proof that avoids a flaw in the analysis of Le et al.’s paper for the semidiscrete (backward Euler scheme in time) method.

scholarly journals Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation

2020 ◽  
Vol 18 (1) ◽  
pp. 67-86
Author(s):  
Xiaoyong Xu ◽  
Fengying Zhou
Keyword(s):  
Collocation Method ◽  
Wave Equations ◽  
Stability And Convergence ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
Orthogonal Spline Collocation ◽  
Fractional Wave Equations ◽  
Discrete Orthogonal ◽  
The Stability ◽  
Crank Nicolson

Abstract In this paper, a discrete orthogonal spline collocation method combining with a second-order Crank-Nicolson weighted and shifted Grünwald integral (WSGI) operator is proposed for solving time-fractional wave equations based on its equivalent partial integro-differential equations. The stability and convergence of the schemes have been strictly proved. Several numerical examples in one variable and in two space variables are given to demonstrate the theoretical analysis.

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