Third order difference schemes (without using points outside of the domain) for one sided space tempered fractional partial differential equations

Abstract

Power-law probability density function (PDF) plays a key role in both subdiffusion and Lévy flights. However, sometimes because of the finite of the lifespan of the particles or the boundedness of the physical space, tempered power-law PDF seems to be a more physical choice and then the tempered fractional operators appear; in fact, the tempered fractional operators can also characterize the transitions among subdiffusion, normal diffusion, and Lévy flights. This paper focuses on the quasi-compact schemes for space tempered fractional diffusion equations, being much different from the ones for pure fractional derivatives. By using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Some numerical simulations are performed to testify the effectiveness and numerical accuracy of the obtained schemes.

Fast solution algorithms for exponentially tempered fractional diffusion equations

“…One can generalize the proposed fast algorithms for 2D TFDEs with different numbers of spatial nodes in ‐direction and ‐direction. In addition, based on the Toeplitz structure, the proposed fast numerical solution strategies can be employed for TFDEs discretized by the high order scheme in . However, more parameters are involved in those schemes which affect the properties of the coefficient matrices so that further study are required on the theoretical analysis for the efficiency of the proposed methods.…”

Algebraic construction of a third order difference approximations for fractional derivatives and applications

“…Hao et al [6] derived a fourth order approximation from the first order Grünwald approximation (4) by considering a convex combination of three of its shifted forms with an appropriate P x and called it a quasi-compact approximation. YanYan Yu et al [23] used this technique to derive third order schemes for tempered fractional diffusion equations. In those papers, the approximation schemes were derived directly from the first order Grünwald approximation (1 − z) α .…”

The implicit midpoint method for Riesz tempered fractional diffusion equation with a nonlinear source term

“…In recent years, differential equations with tempered fractional derivatives have widely been used for modeling many special phenomena, such as geophysics [9][10][11] and finance [12,13] and so on. It has attracted many authors' attention in constructing the numerical algorithm for tempered fractional partial differential equation (see e.g., [1,2,[14][15][16][17][18][19][20][21][22][23][24]). Li and Deng proposed the tempered weighted and shifted Grünwald-Letnikov formula with second-order accuracy for Riemann-Liouville tempered fractional derivative in [24], and its approximation is applied in the numerical simulation of the tempered fractional Black-Scholes equation for European double barrier option by Zhang et al [14].…”

“…Based on this approximation, Qu and Liang [15] constructed a Crank-Nicolson scheme for a class of variable-coefficient tempered fractional diffusion equation, and disscussed the stability and convergence. Yu et al [16] proposed a third-order difference scheme for one side Riemann-Liouville tempered fractional diffusion equation and given the stability and convergence analysis. Yu et al [19] constructed a fourth-order quasi-compact difference operator for Riemann-Liouville tempered fractional derivative and tested its effectiveness by numerical experiment.…”