Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function

This paper presents a numerical method for one-dimensional Burgers’ equation by the Hopf–Cole transformation and a reproducing kernel function, abbreviated as RKF. The numerical solution is given as explicit integral expressions with the RKF at each time step, so that the computation is fully parallel. The stability and error estimates are derived. Numerical results for some test problems are presented and compared with the exact solutions. Some numerical results are also compared with the results obtained by other methods. The present method is easily implemented and effective.     

Numerical stability of a discrete model in the dynamics of ferromagnetic bodies

A discrete penalty method for the numerical study of the evolution equations for soft and undeformed magnetoelastic solids is proposed. Some results concerning the stability and the boundedness of the numerical solution are established. Since the aim of the article is also to show the numerical development of singularities (these are expected, due to the similarities with the evolution equations for liquid crystals), some numerical results on a specific test problem are reported and discussed. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 544–557, 1999     

Stability of linear multistep methods for nonlinear neutral delay differential equations in Banach space

This paper is devoted to investigating the nonlinear stability properties of linear multistep methods for the solution to neutral delay differential equations in Banach space. Two approaches to numerically treating the “neutral term” are considered, which allow us to prove several results on numerical stability of linear multistep methods. These results provide some criteria for choosing the step size such that the numerical method is stable. Some examples of application and a numerical experiment, which further confirms the main results, are given.     

A computational meshless method for the generalized Burger’s–Huxley equation

A numerical solution of the generalized Burger’s–Huxley equation, based on collocation method using Radial basis functions (RBFs), called Kansa’s approach is presented. The numerical results are compared with the exact solution, Adomian decomposition method (ADM) and Variational iteration method (VIM). Highly accurate and efficient results are obtained by RBFs method. Excellent agreement with the exact solution is observed while better (or same) accuracy is obtained than other numerical schemes cited in this work.     

线性中立型随机延迟微分方程Euler方法的均方稳定性

本文讨论Euler方法用于求解线性中立型随机延迟微分方程初值问题时数值解的稳定性,利用了一种不同于以往文献中的证明技巧,给出了Euler方法均方稳定的一个充分条件.文末的数值试验证实了本文所获理论结果的正确性.     

非线性延迟积分微分方程连续Runge-Kutta方法的稳定性分析

本文主要研究了一般形式的延迟积分微分方程,将连续Runge-Kutt,a方法用于求解该类问题,并讨论了方法的稳定性,证明了(k,l)-代数稳定的Runge-Kutta方法当0k1时对应的连续Runge-Kutta方法是渐近稳定的.最后我们通过数值试验验证了方法的有效性及所获结论的正确性.     

Numerical method with high order accuracy for solving a anomalous subdiffusion equation

In this paper, a numerical method with second order temporal accuracy and fourth order spatial accuracy is developed to solve a anomalous subdiffusion equation; by Fourier analysis, the convergence, stability and solvability of the numerical method are analyzed; the theoretical results are strongly supported by the numerical experiment.     

Numerical solution of the three‐dimensional parabolic equation with an integral condition

Developement of numerical methods for obtaining approximate solutions to the three dimensional diffusion equation with an integral condition will be carried out. The numerical techniques discussed are based on the fully explicit (1,7) finite difference technique and the fully implicit (7,1) finite difference method and the (7,7) Crank‐Nicolson type finite difference formula. The new developed methods are tested on a problem. Truncation error analysis and numerical examples are used to illustrate the accuracy of the new algorithms. The results of numerical testing show that the numerical methods based on the finite difference techniques discussed in the present article produce good results. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 193–202, 2002; DOI 10.1002/num.1040     

Numerical stability of one-leg methods for neutral delay differential equations

This paper is concerned with the analytic and numerical stability of a class of nonlinear neutral delay differential equations. A sufficient condition for the stability of the problems itself is given. The numerical stability results are obtained for A-stable one-leg methods when they are applied to above mentioned problems. Numerical examples are given to confirm our theoretical results.     

Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems

In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decomposed into a sum of regular and singular components. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.

     

On the compact difference scheme for the two-dimensional coupled nonlinear Schrödinger equations

A high-order finite difference method for the two-dimensional coupled nonlinear Schrödinger equations is considered. The proposed scheme is proved to preserve the total mass and energy in a discrete sense and the solvability of the scheme is shown by using a fixed point theorem. By converting the scheme in the point-wise form into a matrix–vector form, we use the standard energy method to establish the optimal error estimate of the proposed scheme in the discrete L²-norm. The convergence order is proved to be of a fourth-order in space and a second-order in time, respectively. Finally, some numerical examples are given in order to confirm our theoretical results for the numerical method. The numerical results are compared with exact solutions and other existing method. The comparison between our numerical results and those of Sun and Wangreveals that our method improves the accuracy of space and time directions.     

Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burgers–Huxley equation

In this paper, a numerical solution of the generalized Burgers–Huxley equation is presented. This is the application of spectral collocation method. To reduce roundoff error in this method we use Darvishi’s preconditionings. The numerical results obtained by this method have been compared with the exact solution. It can be seen that they are in a good agreement with each other, because errors are very small and figures of exact and numerical solutions are very similar.     

A lattice Boltzmann model for the eddy–stream equations in two-dimensional incompressible flows

A lattice Boltzmann model for two-dimensional incompressible flows with eddy–stream equations is proposed. By using two kinds of distribution functions and employing several higher-order moments of equilibrium distribution functions, the eddy equation and stream function equation with the second-order truncation error are obtained. In the numerical examples, we compared the numerical results of this scheme with those obtained by other classical method. The numerical results agree well with the classical ones.     

A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative

In this article, we consider Stokes’ first problem for a heated generalized second grade fluid with fractional derivative (SFP-HGSGF). Implicit and explicit numerical approximation schemes for the SFP-HGSGF are presented. The stability and convergence of the numerical schemes are discussed using a Fourier method. In addition, the solvability of the implicit numerical approximation scheme is also analyzed. A Richardson extrapolation technique for improving the order of convergence of the implicit scheme is proposed. Finally, a numerical test is given. The numerical results demonstrate the good performance of our theoretical analysis.     

Oscillation analysis of θ‐methods for the Nicholson's blowflies model

This paper is concerned with the oscillation of numerical solution for the Nicholson's blowflies model. Using two kinds of θ‐methods, namely, the linear θ‐method and the one‐leg θ‐method, several conditions under which the numerical solution oscillates are derived. Moreover, it is shown that every non‐oscillatory numerical solution tends to equilibrium point of the original continuous‐time model. Finally, numerical experiments are provided to illustrate the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.     

Quartic B-spline Galerkin approach to the numerical solution of the KdVB equation

The nonlinear Korteweg-de Vries–Burgers’ equation is solved numerically by method of Galerkin using quartic B-splines as both shape and weight functions over the finite intervals. Five test problems are studied to demonstrate the accuracy and efficiency of the proposed method. A comparison of numerical results of both algorithm and some published articles is done in computational section. The numerical results are found in good agreement with exact solutions.     

Numerical treatment of retarded differential–algebraic equations by collocation methods

This paper investigates retarded differential–algebraic equations of index zero to two with state-dependent delay. The theory needed to understand the numerical approach and analyze the numerical treatment by collocation methods is developed. Different strategies for tracking the jump discontinuities are considered and numerical examples are presented to support the convergence results. This revised version was published online in August 2006 with corrections to the Cover Date.     

EfficientP-stable methods for periodic initial value problems

Efficient families ofP-stable formulae are developed for the numerical integration of periodic initial value problems where the required solution has an unknown period. Formulae of orders 4 and 6 requiring respectively 2 and 4 function evaluations per step are derived and some numerical results are given.     

带有不连续系数的线性输运方程差分格式的收敛性

提出了一个基于三角形网格的显式差分格式逼近带有不连续系数的线性输运方程. 通过对数值解的有界性、TVD(total variation decreasing)和空间、时间方向的平移估计, 利用Kolmogorov紧性原理证明了数值解在L¹_loc模下收敛于初值问题的唯一弱解.从而得到了初值问题解的存在唯一性和关于初值的稳定性. 数值算例表明本文提出的格式计算方便而且比 Lax-Friedrichs格式更有效.       

种群动力系统的数值解的振动性分析

本文主要研究下面动力系统的非线性延迟微分方程x't(t)+αVmx(t)p(t-τ)/βp+xp(t-τ)=λ,t≥0 数值解的振动性.这是由Mackey和Glass[1]提出来的关于动力系统疾病的方程.本文得到了数值方法振动的条件.同时对非振动的数值解的性质也做了研究,为了验证得到的结果,给出了数值算例.