用线方法求解带混合边值条件的抛物方程

研制了分别用显式Euler法、隐式Euler法、Crank-Nicolson格式(梯形方法)求解带第一、第二及混合边值条件的抛物问题的应用软件,通过求解若干抛物问题对该软件作了测试,获得了预期的数值结果,讨论了时间和空间步长的变化对格式计算结果的影响,得到了三种方法的稳定性、收敛精度和计算量.     

Numerical differentiation with noisy signal

The aim of this paper is to present a new numerical method, which ables one to filter and compute numerical derivatives of a function whose values are known in some points from experimental measurements, inducing noisy data. We use a piecewise cubic spline interpolation to generate a function whose Fourier coefficients give an approximation of the numerical derivatives we are looking for. Error and stability analysis of this numerical algorithm are provided. Numerical results are presented for data smoothing and for the first and second derivatives computed from noisy data. They show that this method gives good numerical results. Comparison with other methods is done.     

单物种人口模型指数隐式Euler方法的振动性

本文研究了用以描述单物种人口模型的延迟Logistic方程的数值振动性.对方程应用隐式Euler方法进行求解,针对离散格式定义了指数隐式Euler方法,证明了该方法的收敛阶为1.根据线性振动性理论获得了数值解振动的充分条件.进而还对非振动数值解的性质作了讨论.最后用数值算例对理论结果进行了验证.     

Recovering the local volatility in Black–Scholes model by numerical differentiation

In this article, a numerical method for recovering the local volatility in Black–Scholes model is proposed based on the Dupire formula in which the numerical derivatives are used. By Tikhonov regularization, a new numerical differentiation method in two-dimensional (2-D) case is presented. The convergent analysis and numerical examples are also given. It shows that our method is efficient and stable.     

Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems

In this letter, a new numerical method is proposed for solving second order linear singularly perturbed boundary value problems with left layers. Firstly a piecewise reproducing kernel method is proposed for second order linear singularly perturbed initial value problems. By combining the method and the shooting method, an effective numerical method is then proposed for solving second order linear singularly perturbed boundary value problems. Two numerical examples are used to show the effectiveness of the present method.     

A boundary preserving numerical algorithm for the Wright-Fisher model with mutation

The Wright-Fisher model is an It? stochastic differential equation that was originally introduced to model genetic drift within finite populations and has recently been used as an approximation to ion channel dynamics within cardiac and neuronal cells. While analytic solutions to this equation remain within the interval [0,1], current numerical methods are unable to preserve such boundaries in the approximation. We present a new numerical method that guarantees approximations to a form of Wright-Fisher model, which includes mutation, remain within [0,1] for all time with probability one. Strong convergence of the method is proved and numerical experiments suggest that this new scheme converges with strong order 1/2. Extending this method to a multidimensional case, numerical tests suggest that the algorithm still converges strongly with order 1/2. Finally, numerical solutions obtained using this new method are compared to those obtained using the Euler-Maruyama method where the Wiener increment is resampled to ensure solutions remain within [0,1].     

A meshless method for Burgers’ equation using MQ-RBF and high-order temporal approximation

In this paper, a new numerical method is proposed to solve one-dimensional Burgers’ equation using multiquadric (MQ) radial basis function (RBF) for spatial approximation and a second-order compact finite difference scheme for temporal approximation. The numerical results obtained by this way for different Reynolds number have been compared with the existing numerical schemes to show the accuracy and efficiency of the approach. To show the superiority of this meshless method, numerical experiments with non-uniform MQ interpolation node distribution are also performed.     

Implicit numerical approximation scheme for the fractional Fokker-Planck equation

In this paper, we consider an anomalous subdiffusion process, governed by fractional Fokker-Planck equation. An effective numerical method for approximating Fokker-Planck equation in a bounded domain is presented. The stability and convergence of the numerical method are analyzed. Some numerical examples are presented to show the application of the present technique. The numerical results exhibit the good performance of our theoretical analysis.     

非线性积分方程迭代配置法的渐近展开及其外推

非线性积分方程迭代配置法的渐近展开及其外推韩国强（华南理工大学计算机工程与科学系）ＡＳＹＭＰＴＯＴＩＣＥＲＲＯＲＥＸＭＮＳＩＯＮＳＡＮＤＥＸＴＲＡＰＯＬＡＴＩＯＮＦＯＲＴＨＥＩＴＥＲＡＴＥＤＣＯＬＬＯＣＡＴＩＯＮＭＥＴＨＯＤＳＯＦＮＯＮＬＩＮＥＡＲＩ...     

Solution of dual fuzzy polynomial equations by modified Adomian decomposition method

In this paper, we present some efficient numerical algorithm for solving dual fuzzy polynomial equations based on Newton’s method. The modified Adomian decomposition method is applied to construct the numerical algorithms. Some numerical illustrations are given to show the efficiency of the algorithms.     

Stability analysis for a fully discrete spectral scheme for Boussinesq systems

In this paper we perform a stability analysis of a fully discrete numerical method for the solution of a family of Boussinesq systems, consisting of a Fourier collocation spectral method for the spatial discretization and a explicit fourth order Runge–Kutta (RK4) scheme for time integration. Our goal is to determine the influence of the parameters, associated to this family of systems, on the efficiency and accuracy of the numerical method. This analysis allows us to identify which regions in the parameter space are most appropriate for obtaining an efficient and accurate numerical solution. We show several numerical examples in order to validate the accuracy, stability and applicability of our MATLAB implementation of the numerical method.     

Convergence analysis for numerical solution of Fredholm integral equation by Sinc approximation

In this study one of the new techniques is used to solve numerical problems involving integral equations known as Sinc-collocation method. This method has been shown to be a powerful numerical tool for finding accurate solutions. So, in this article, some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some algebraic equations. Then by a theorem we show error in the approximation of the solution decays at an exponential rate. Finally, numerical examples are included to demonstrate the validity and applicability of the technique.     

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.     

Booster Method for Singularly-Perturbed One-Dimensional Convection-Diffusion Neumann Problems

An improved numerical method for singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations subject to Neumann-type boundary conditions is proposed. In this method, an asymptotic approximation is incorporated into a finite-difference scheme to improve the numerical solution. Uniform error estimates are derived when implemented in known difference schemes. Numerical results are presented in support of the proposed method.     

Theoretical investigation on error analysis of Sinc approximation for mixed Volterra-Fredholm integral equation

In this study, we propose one of the new techniques used in solving numerical problems involving integral equations known as the Sinc-collocation method. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. So, in this article, a mixed Volterra-Fredholm integral equation which has been appeared in many science an engineering phenomena is discredited by using some properties of the Sinc-collocation method and Sinc quadrature rule to reduce integral equation to some algebraic equations. Then exponential convergence rate of this numerical technique is discussed by preparing a theorem. Finally, some numerical examples are included to demonstrate the validity and applicability of the convergence theorem and numerical scheme.     

Numerical Method for Singularly Perturbed Third Order Ordinary Differential Equations of Convection-Diffusion Type

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of convection-diffusion type of third order Ordinary Differential Equations (ODEs) in which the SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. In order to get a numerical solution for the derivative of the solution, the domain is divided into two regions namely inner region and outer region. The shooting method is applied to the inner region while standard finite difference scheme (FD) is applied for the outer region. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing.     

A semi‐discrete defect correction finite element method for unsteady incompressible magnetohydrodynamics equations

In this report, we give a semi‐discrete defect correction finite element method for the unsteady incompressible magnetohydrodynamics equations. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear magnetohydrodynamics equations is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect‐correction technique. Then, we give the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. In order to show the effect of our method, some numerical results are shown. Copyright © 2017 John Wiley & Sons, Ltd.     

A coupled system of singularly perturbed parabolic reaction-diffusion equations

In this paper systems with an arbitrary number of singularly perturbed parabolic reaction-diffusion equations are examined. A numerical method is constructed for these systems which involves an appropriate layer-adapted piecewise-uniform mesh. The numerical approximations generated from this method are shown to be uniformly convergent with respect to the singular perturbation parameters. Numerical experiments supporting the theoretical results are given.     

Contributions to the development of differential systems exactly solved by multistep finite-difference schemes

The motivation underlying this contribution has been to complete some of the topics concerning exact schemes for numerically solving ordinary differential equations. A procedure for obtaining differential systems exactly solved by a given finite-difference method is described. Examples illustrating the application of the procedure for obtaining first-order, second-order and systems of differential equations exactly solved by different numerical methods are given. Among the numerical methods considered there are the Trapezoidal Rule, the two-step Adams-Bashforth method and the Numerov method. Some numerical examples are presented to provide evidence that the procedure works properly.