Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian

Abstract

Tempered fractional Laplacian is the generator of the tempered isotropic Lévy process [

Numerical Algorithms of the Two-dimensional Feynman–Kac Equation for Reaction and Diffusion Processes

“…Proof The proof of the first inequality of (A.2) can be found in [28]. For the second one, by simple calculation, we obtain…”

“…This section provides a finite difference discretization for the two-dimensional tempered fractional Laplacian on a bounded domain Ω = (−l, l)×(−l, l) with extended homogeneous Dirichlet boundary conditions: G(x, y) ≡ 0 for (x, y) ∈ Ω c , which is based on our previous work [28] and modifies the regularity requirement according to [32]. Afterwards, we give the error analysis of the space semi-discrete scheme.…”

“…For the stochastic process x 0 (t) described by a continuous time random walk (CTRW) with power law waiting time and jump length distributions, Barkai and his collaborators derive the governing equation of the distribution of the corresponding functional in Fourier space [9,30]. Because of the finite lifespan and the bounded physical space, sometimes the power law waiting time and jump length distributions of the CTRW need to be tempered [25]; then the tempered fractional Feynman-Kac equations are derived in [31], and the numerical methods for the equations are discussed in [12,13,16,28,32,33]. For x 0 (t) characterized by Langevin pictures, the readers can refer to [7,8] for the derivation of Feynman-Kac equations.…”

“…Theorem 4.1 and Theorem 4.3 in Sec. 4 show that the convergence order only depends on the regularity of source term f instead of the exact solution G. The second problem is to discretize (1.4); so far, the discretizations of (1.4) are mainly for the one-dimensional case [32,33]; our previous work [28] provides a finite difference scheme for the two-dimensional tempered fractional Laplacian (∆ + γ) β 2 . Here, we modify the discretization according to [32], so that the regularity requirement can be relaxed from the whole space to Ω , and the optimal convergence rate is achieved.…”

Maximum Principle for Variable-Order Fractional Conformable Differential Equation with a Generalized Tempered Fractional Laplace Operator

“…In 2018, Deng, Li, Tian and Zhang [17] gave the mathematic definition of the tempered fractional Laplace operator. In 2018, Sun, Nie and Deng [18] advanced the finite difference discretization for the tempered fractional Laplace operator by the weighted trapezoidal rule and bilinear interpolation. On this basis, Zhang et al [19] proposed a new type of generalized tempered fractional p-Laplace operator in 2020.…”