A compact finite difference scheme for the fractional sub-diffusion equations

In this paper, a compact finite difference scheme for the fractional sub-diffusion equations is derived. After a transformation of the original problem, the L1 discretization is applied for the time-fractional part and fourth-order accuracy compact approximation for the second-order space derivative. The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme in maximum norm are proved using the energy method, where a new inner product is introduced for the theoretical analysis. The technique is quite novel and different from previous analytical methods. Finally, a numerical example is provided to show the effectiveness and accuracy of the method.     

A box-type scheme for fractional sub-diffusion equation with Neumann boundary conditions

Combining order reduction approach and L1 discretization, a box-type scheme is presented for solving a class of fractional sub-diffusion equation with Neumann boundary conditions. A new inner product and corresponding norm with a Sobolev embedding inequality are introduced. A novel technique is applied in the proof of both stability and convergence. The global convergence order in maximum norm is O(τ^2−α + h²). The accuracy and efficiency of the scheme are checked by two numerical tests.     

Finite difference scheme for the time-space fractional diffusion equations

In this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes.     

Alternative boundary conditions for solving optical diffusion equations by a finite difference time domain analysis in three-dimensional scattering medium

Alternative boundary conditions for solving optical diffusion equations in three-dimensional (3-D) scattering medium by a finite difference time domain (FDTD) analysis formulated by the author are proposed. The previous boundary conditions were defined only by fluence rate, which, although essential, is only one factor needed to solve approximated diffusion equations for fluence rate. In this paper, alternative boundary conditions defined both by fluence rate and radiant flux have been proposed for use in the FDTD analysis, which is derived from the two coupled differential equations for fluence rate and radiant flux. It has been become clear that these boundary conditions are almost equivalent to the previous boundary conditions in the FDTD analysis for sufficiently fine grid spacing. For the analysis with coarser grid spacing, the proposed boundary conditions suppress analytical errors, especially in intensity of time-resolved reflectance and transmittance.     

Poisson finite difference boundary equations for the magnetic potentials

Magnetostatic solutions may be obtained by using Laplace's equation in finite difference form. The boundary equations are of primary importance and even these have been put forward as special forms of Laplace's equation. They can be derived in what appears to be a simpler and more flexible manner by assuming that Poisson's equation applies at the interface for both scalar and vector potentials.     

A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations

An implicit finite difference method with non-uniform timesteps for solving fractional diffusion and diffusion-wave equations in the Caputo form is presented. The non-uniformity of the timesteps allows one to adapt their size to the behaviour of the solution, which leads to large reductions in the computational time required to obtain the numerical solution without loss of accuracy. The stability of the method has been proved recently for the case of diffusion equations; for diffusion-wave equations its stability, although not proven, has been checked through extensive numerical calculations.     

一种色散介质FDTD通用吸收边界

由单轴各向异性介质所满足的场方程出发,并根据相位匹配原理以得到介质中的色散关系和无反射条件.结合频域到时域的转换关系（即用/t代替jω）和移位算子时域有限差分（SO-FDTD）方法提出了一种适用于各向同性常见色散介质模型,包括德拜模型、洛伦兹模型、德鲁德模型等的通用UPML吸收边界.数值结果表明了该算法的通用性和高效性. 关键词： 时域有限差分 吸收边界 移位算子 色散介质     

A smart nonstandard finite difference scheme for second order nonlinear boundary value problems

A new kind of finite difference scheme is presented for special second order nonlinear two point boundary value problems. An artificial parameter is introduced in the scheme. Symbolic computation is proposed for the construction of the scheme. Local truncation error of the method is discussed. Numerical examples are illustrated. Numerical results show that the method is very effective.     

On multi-term fractional differential equations with multi-point boundary conditions

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples. The paper concludes with the study of multi-term fractional integro-differential equations supplemented with multi-point boundary conditions. Our results are new and contribute significantly to the existing literature on the topic.     

On fractional Langevin differential equations with anti-periodic boundary conditions

In this paper, we provide existence criteria for the solutions of p-Laplacian fractional Langevin differential equations with anti-periodic boundary conditions. The Caputo fractional as well as Caputo q-fractional operators are used to address the derivatives. The main results are verified by the help of Leray–Schaefer’s fixed point theorem. We construct an example to illustrate the feasibility of the main theorems. Our results are new and provide extensions to some known theorems in the literature.     

Existence of solutions for sequential fractional differential equations with four-point nonlocal fractional integral boundary conditions

This paper investigates the existence of solutions for a nonlinear boundary value problem of sequential fractional differential equations with four-point nonlocal Riemann-Liouville type fractional integral boundary conditions. We apply Banach’s contraction principle and Krasnoselskii’s fixed point theorem to establish the existence of results. Some illustrative examples are also presented.     

A Fourier method for the fractional diffusion equation describing sub-diffusion

In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.     

A Fourier method for the fractional diffusion equation describing sub-diffusion

In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.     

Efficient absorbing boundary conditions for Biot's equations in time-harmonic finite element applications

Absorbing boundary conditions for two phase media previously presented by Zerfa and Loret [Earthquake Eng. Struct. Dyn. 33, 89-110 (2004)] have been improved by considering additionally absorbing waves with auxiliary angles of incidence. These angles are defined at each point on the boundaries, so one can easily implement tensor impedances as analogous to those defined by Krenk and Kirkegaard for isotropic, nonporous media [J. Sound Vib. 247, 875-896 (2001)]. The boundary conditions have been tested and validated in two-dimensional frequency domain simulations.     

A symmetry-preserving difference scheme for high dimensional nonlinear evolution equations

In this paper, a procedure for constructing discrete models of the high dimensional nonlinear evolution equanons is presented. In order to construct the difference model, with the aid of the potential system of the original equation and compatibility condition, the difference equations which preserve all Lie point symmetries can be obtained. As an example, invariant difference models of the （2＋1）-dimensional Burgers equation are presented.     

室内声学时域有限差分模拟中的边界条件

给出了时域有限差分法用于室内声学问题模拟中的边界条件,结合声波方程的基本差分格式,模拟并分析了高斯脉冲在4m×4m房间中的波动过程和脉冲响应;模拟了一9m×6m×4m房间的简正频率,并与经典理论计算值进行了对比;模拟了一12m×5m×4m水平地面房间中的坐席吸声低谷效应,并与Joe LoVtri的模拟结果进行了对比;模拟并实际测量了一10.6m×5.8m×3.4m房间在几个受声点的脉冲响应和早期衰变时间EDT,将模拟结果与实际测量结果进行了对比分析,计算程序是用Metlab语言编写的。模拟与经典理论、相关研究、实际测量几方面的对比分析,验证了本边界条件的可靠性。     

Boundary conditions for time dependent problems with an artificial boundary

In this paper different procedures for treating artificial boundaries are analyzed, and it is shown that many commonly used methods give bad results. A new procedure is developed for the case where the asymptotic behaviour of the coefficient matrices is known.     

Compact finite difference method for the fractional diffusion equation

High-order compact finite difference scheme for solving one-dimensional fractional diffusion equation is considered in this paper. After approximating the second-order derivative with respect to space by the compact finite difference, we use the Grünwald–Letnikov discretization of the Riemann–Liouville derivative to obtain a fully discrete implicit scheme. We analyze the local truncation error and discuss the stability using the Fourier method, then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis. Numerical results are provided to verify the accuracy and efficiency of the proposed algorithm.