In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Grünwald difference (tempered-WSGD) operators for the tempered fractional diffusion equation. We also show stability and convergence analysis for the fully discrete scheme based a Crank--Nicolson scheme in time. A third-order scheme for the tempered Black--Scholes equation is also proposed and tested numerically. Some numerical experiments are carried out to confirm accuracy and effectiveness of these proposed methods.
This paper discusses the fractional diffusion equation forced by a tempered fractional Gaussian noise. The fractional diffusion equation governs the probability density function of the subordinated killed Brownian motion. The tempered fractional Gaussian noise plays the role of fluctuating external source with the property of localization. We first establish the regularity of the infinite dimensional stochastic integration of the tempered fractional Brownian motion and then build the regularity of the mild solution of the fractional stochastic diffusion equation. The spectral Galerkin method i…
We consider the numerical solution of time-dependent space tempered fractional diffusion equations. The use of Crank-Nicolson in time and of second-order accurate tempered weighted and shifted Grünwald difference in space leads to dense (multilevel) Toeplitz-like linear systems. By exploiting the related structure, we design an ad-hoc multigrid solver and multigrid-based preconditioners, all with weighted Jacobi as smoother. A new smoothing analysis is provided, which refines state-of-the-art results expanding the set of the suitable Jacobi weights. Furthermore, we prove that if a multigrid me…
The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high order fully discrete local discontinuous Galerkin (LDG) method based on the generalized alternating numerical fluxes for the tempered fractional diffusion equation. From a practical point of view, the generalized alternating numerical flux which is different from the purely alternating numerical flux has a broader range of applications. We first design an effici…
Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University（兰州大学数学与统计学院，甘肃省应用数学与复杂系统重点实验室）
Laguerre Functions and Their Applications to Tempered Fractional Differential Equaitons on Infinite Intervals
Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing and School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）；Department of Mathematics, Purdue University（普渡大学数学系）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学学院）
In the continuous time random walk model, the time-fractional operator usually expresses an infinite waiting time probability density. Different from that usual setting, this work considers the tempered time-fractional operator, which reflects a finite waiting time probability density. Firstly, we analyse the solution of a tempered benchmark problem, which shows a weak singularity near the initial time. The L1 scheme on graded mesh and the WSGL formula with correction terms are adapted to deal with the non-smooth solution, in which we compare these two methods systematically in terms of the co…
In this paper, we propose an accurate finite difference method to discretize the $d$-dimensional (for $d\ge 1$) tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various applications. Compared to other existing methods, our method has higher accuracy and simpler implementation. Our numerical method has an accuracy of $O(h^ε)$, for $u \in C^{0, α+ε} (\barΩ)$ if $α< 1$ (or $u \in C^{1, α-1+ε} (\barΩ)$ if $α\ge 1$) with $ε> 0$, suggesting the minimum consistency conditions. The accuracy can be improved to $O(h^2)$, for $u \in …
To model wave propagation in inhomogeneous media with frequency-dependent power-law attenuation, it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time. The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative white noise and fractional Gaussian noise, because of the potential fluctuations of the external sources. The purpose of this work is to discuss the Galerkin finite element approximation for the semilinear stochastic…
In this paper, we study the numerical schemes for the two-dimensional Fokker-Planck equation governing the probability density function of the tempered fractional Brownian motion. The main challenges of the numerical schemes come from the singularity in the time direction. When $0<H<0.5$, a change of variables $\partial \left(t^{2H}\right)=2Ht^{2H-1}\partial t$ avoids the singularity of numerical computation at $t=0$, which naturally results in nonuniform time discretization and greatly improves the computational efficiency. For $0.5<H<1$, the time span dependent numerical scheme and nonunifor…
Modeling transient currents in time-of-flight experiments with tempered time-fractional diffusion equations
Instituto Superior Técnico（里斯本高等技术学院）；University of Trás-os-Montes e Alto Douro, UTAD（特拉斯-奥-蒙特斯和阿尔托杜罗大学）；Instituto de Telecomunicações, Lisboa（里斯本电信研究所）
In this paper, we are concerned with the numerical solution for the two-dimensional time fractional Fokker-Planck equation with tempered fractional derivative of order $α$. Although some of its variants are considered in many recent numerical analysis papers, there are still some significant differences. Here we first provide the regularity estimates of the solution. And then a modified $L$1 scheme inspired by the middle rectangle quadrature formula on graded meshes is employed to compensate for the singularity of the solution at $t\rightarrow 0^{+}$, while the five-point difference scheme is …
An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are proved under several suitable assumptions, and numerical examples show that the convergence orders of these two schemes are $1$ in both time and space. Secondly, a nonlinear all-at-once system is derived based on the nonlinear implicit scheme, which may s…
Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks
National University of Singapore（新加坡国立大学）；Worcester Polytechnic Institute（伍斯特理工学院）；Brown University（布朗大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
本文将 MC-fPINN 扩展为 MC-tfPINN，并用一维 Gaussian 求积替代 Monte Carlo 采样，在高达 100,000 维的分数阶与缓增分数阶 PDE 正、反问题中提升了精度和收敛速度。
Numerical algorithms of the two-dimensional Feynman-Kac equation for reaction and diffusion processes
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University（兰州大学数学与统计学院，甘肃省应用数学与复杂系统重点实验室）
Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the corresponding model problems, theoretical analysis such as the regularity result, or the relationship between the left-side and right-side fractional operators are seldom mentioned. In stead of considering the fractional derivative spaces, this paper starts from discussing the image spaces of Riemann-Liouville fractional integrals of $L_p(Ω)$ functions, since the fractional derivative operators that often used are all pseudo-diff…
In this paper, we propose a new choice of poles to define reliable rational Krylov methods. These methods are used for approximating function of positive definite matrices. In particular, the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations. The results of the numerical experiments we have carried out on some fractional models confirm that the proposed approach is promising.
A new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order (so-called fractional derivatives) and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation. The suggested method is the development of Podlubny's matrix approach (Fractional Calculus and Applied Analysis, vol. 3, no. 4, 2000, 359--386). Four examples of numerical solution of fractional diffusion equation with various combinations of time/space fractional derivatives (integer/integer, fractional/integer, i…
