Finite Difference/Collocation Method for Two-Dimensional Sub-Diffusion Equation with Generalized Time Fractional Derivative

In this paper, we propose a finite difference/collocation method for two-dimensional time fractional diffusion equation with generalized fractional operator. The main purpose of this paper is to design a high order numerical scheme for the new generalized time fractional diffusion equation. First, a finite difference approximation formula is derived for the generalized time fractional derivative, which is verified with order $2-\alpha$ $(0<\alpha<1)$. Then, collocation method is introduced for the two-dimensional space approximation. Unconditional stability of the scheme is proved. To make the method more efficient, the alternating direction implicit method is introduced to reduce the computational cost. At last, numerical experiments are carried out to verify the effectiveness of the scheme.     

分数阶延迟微分方程的样条配置方法

首次利用三次样条配置方法采用直接法求解了一类非线性分数阶延迟微分方程初值问题,并给出了方法的局部截断误差和若干数值算例.数值结果表明方法求解分数阶延迟微分方程初值问题是非常有效的,结果对于未来研究分数阶延迟微分方程的数值方法具有重要的意义.     

The Global Behavior of Finite Difference-Spatial Spectral Collocation Methods for a Partial Integro-Differential Equation with a Weakly Singular Kernel

The $z$-transform is introduced to analyze a full discretization method for a partial integro-differential equation (PIDE) with a weakly singular kernel. In this method, spectral collocation is used for the spatial discretization, and, for the time stepping, the finite difference method combined with the convolution quadrature rule is considered. The global stability and convergence properties of complete discretization are derived and numerical experiments are reported.     

基于径向基函数配点法的梁板弯曲问题分析

采用径向基函数配点法分析考虑剪切效应的梁板弯曲问题,该方法利用径向基函数作为近似函数,基于配点法离散方程,通过最小二乘法求解。径向基函数配点法在离散和计算过程中不需要任何形式的网格划分,是一种真正的无网格法;径向基函数可以用一元函数来描述多元函数,存在明显的储存和运算简单的特点;而基于配点法求解不需要积分,提高了计算效率。分析考虑剪切效应的薄梁板问题时,传统的有限元法或无网格法求解均会存在剪切锁闭问题,而径向基函数在全域内存在无限连续性,能够准确地满足Kirchhoff约束条件,因此径向基函数配点法能够消除剪切锁闭现象,而且不会出现应力波动。该方法的优势在于,其不仅易于离散、精度高,而且具有指数收敛率,计算效率高。数值算例验证了上述结论和该方法的稳定性。     

一种新的求解对流扩散边值问题的无网格方法

基于配点法和楔形基函数,提出了一种新的求解对流扩散边值问题的无网格方法。通过一维和二维的问题验证了该数值方法的可行性;并根据数值算例和分析,可以看到该数值方法能达到满意的收敛效果。该数值方法的隐格式形式能够有效地消除对流占优问题的数值振荡现象,是一种真正的无网格方法。     

The Collocation Method and the Splitting Extrapolation for the First Kind of Boundary Integral Equations on Polygonal Regions

In this paper, the collocation methods are used to solve the boundary integral equations of the first kind on the polygon. By means of Sidi's periodic transformation and domain decomposition, the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers $h_{i}^{3}(i=1,...,d)$, which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations. Numerical experiments are carried out to show that the methods are very efficient.     

Collocation Methods for a Class of Volterra Integral Functional Equations with Multiple Proportional Delays

In this paper, we apply the collocation methods to a class of Volterra integral functional equations with multiple proportional delays (VIFEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solutions for this type of equations, and then analyze the convergence orders of the collocation solutions and give corresponding error estimates. The numerical results verify our theoretical analysis.     

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto(GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective,reliable and does not require any restrictive assumptions for nonlinear terms.     

Strong-form framework for solving boundary value problems with geometric nonlinearity

In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite element methods (FEMs) or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches computationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method (RBCM) motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the proposed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the incremental-iterative process.     

Numerical simulation of flocculation and transport of suspended particles: Application to metered-dose inhalers

This study investigates the dynamics of flocculation and transport of solid particles suspended in a liquid propellant. Polydisperse particles with lognormal size distribution are considered. Collision of particles is presumed to be controlled by upward velocity differential and Brownian motion. These mechanisms are enhanced by the van der Waals force. Flocculation of the particles is described using the continuous form of the Smoluchowski equation. Upward transport of the particles is specified via a convection term. The general dynamics of the system is governed by a nonlinear transient partial integro-differential equation which is solved numerically. The technique employed is based on discretizing the size distribution function using orthogonal collocation on finite elements. This is combined with a finite difference discretization of the physical domain, and an explicit Runge–Kutta–Fehlberg time marching scheme. The numerical analysis is validated by comparing with a closed form analytical solution. The simulation results represent the particle size distribution as a function of time and position. The method allows prediction of the effects of the initial conditions and physical properties of the suspension on its dynamic behavior and phase separation.