混合广义Jacobi和Chebyshev谱配置法求解时间分数阶对流扩散方程

研究时间Caputo分数阶对流扩散方程的高效高阶数值方法.对于给定的时间分数阶偏微分方程,在时间和空间方向分别采用基于移位广义Jacobi函数为基底和移位Chebyshev多项式运算矩阵的谱配置法进行数值求解.这样得到的数值解可以很好地逼近一类在时间方向非光滑的方程解.最后利用一些数值例子来说明该数值方法的有效性和准确性.     

A numerical method for some nonlinear differential equation models in biology

In this paper, the authors present a full discretization of nonlinear generalisations of the Fischer and Burgers equations with the zero flux on the boundary. Efficiency of the method is derived via a numerical comparison between their numerical solution and the exact solution.     

A starting method for the numerical solution of Volterra's integral equation of the second kind

The acquisition of starting values is one of the chief difficulties encountered in computing a numerical solution of Volterra's integral equation of the second kind by a multi-step method. The object of this note is to present a procedure which is derived from certain quadrature formulas and which provides these starting values, to provide a sufficient condition for the approximate solution to be unique, to bound the approximate solution and the error, and to give a numerical example.     

Numerical Solution of Singularly Perturbed Boundary Value Problems Based on Optimal Control Strategy

In this paper, we develop a numerical technique for singularly perturbed boundary value problems using B-spline functions and least square method. The approximate solution derived in this article is convergent to the exact solution and can be applied both to linear and nonlinear models. The numerical examples and computational results illustrate and guarantee a higher accuracy for this technique.     

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.     

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

A numerical method for singularly-perturbed self-adjoint boundary-value problems for second-order ordinary differential equations subject to Neumann boundary conditions is proposed. In this method (booster method), an asymptotic approximation is incorporated into a finite-difference scheme to improve the numerical solution. Uniform error estimates are derived for this method when implemented in known difference schemes. Numerical examples are presented to illustrate the present method.     

On a stefan problem involving a spherical region: application of a Bergman-type expansion

A Bergman-type series expansion method is used in the analysis of a spherical reaction problem. The small-time solution so derived is employed as the starting point for the numerical solution, and the time for complete reaction is calculated.     

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.     

Mean-square stability of stochastic age-dependent delay population systems with jumps

In this paper, we present the compensated stochastic θ method for stochastic age-dependent delay population systems (SADDPSs) with Poisson jumps. The definition of mean-square stability of the numerical solution is given and a sufficient condition for mean-square stability of the numerical solution is derived. It is shown that the compensated stochastic θ method inherits stability property of the numerical solutions. Finally, the theoretical results are also confirmed by a numerical experiment.     

On series solution for unsteady boundary layer equations in a special third grade fluid

Analytic solution for the time-dependent boundary layer flow over a moving porous surface is derived by using homotopy analysis method (HAM). A special third grade fluid model has been used in the problem formulation. The obtained HAM solution is also compared with the numerical solution and a reasonable agreement is noted.     

On the solution of a mixed nonlinear integral equation

In this paper, we consider a mixed nonlinear integral equation of the second kind in position and time. The existence of a unique solution of this equation is discussed and proved. A numerical method is used to obtain a system of Harmmerstein integral equations of the second kind in position. Then the modified Toeplitz matrix method, as a numerical method, is used to obtain a nonlinear algebraic system. Many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived. The rate of convergence of the total error is discussed. Finally, numerical examples when the kernel of position takes a logarithmic and Carleman forms, are presented and the error estimate, in each case, is calculated.     

Modelling dispersion in laminar and turbulent flows in an open channel based on centre manifolds using 1D-IRBFN method

Centre manifold method is an accurate approach for analytically constructing an advection–diffusion equation (and even more accurate equations involving higher-order derivatives) for the depth-averaged concentration of substances in channels. This paper presents a direct numerical verification of this method with examples of the dispersion in laminar and turbulent flows in an open channel with a smooth bottom. The one-dimensional integrated radial basis function network (1D-IRBFN) method is used as a numerical approach to obtain a numerical solution for the original two-dimensional (2-D) advection–diffusion equation. The 2-D solution is depth-averaged and compared with the solution of the 1-D equation derived using the centre manifolds. The numerical results show that the 2-D and 1-D solutions are in good agreement both for the laminar flow and turbulent flow. The maximum depth-averaged concentrations for the 1-D and 2-D models gradually converge to each other, with their velocities becoming practically equal. The obtained numerical results also demonstrate that the longitudinal diffusion can be neglected compared to the advection.     

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.     

可压缩可混溶驱动问题的有限元配置法

多孔介质中可压缩可混溶驱动问题是非线性抛物系统,压力方程和饱和度方程用有限元配置方法来求解,证明了配置解的存在唯一性,最后得到了最优阶的误差估计．     

Integral equations and the interior Dirichlet potential problem

The two standard approaches for reformulating the interior Dirichlet potential problem as a boundary integral equation of the second kind are discussed. The integral equation derived from the representation of the solution as a double layer is shown to be more general than the one derived from Green's theorem. The boundary integral equation of the latter method, however, has definite analytical and numerical value. From it a new integral equation is derived whose solution can be represented as a convergent Neumann series and it is shown that the Green's function of the first kind can be obtained from it. An example is supplied to illustrate the method.     

Verification of Functional A Posteriori Error Estimates for Obstacle Problem in 1D

We verify functional a posteriori error estimates of numerical solutions of a obstacle problem proposed by S. Repin. The simplification into 1D allows for the construction of a nonlinear benchmark for which an exact solution can be derived and also an exact quadrature can be applied. Quality of a numerical solution obtained by the finite element method is compared with the exact solution to demonstrate the sharpness of the studied error estimated. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类广义Lyapunov矩阵方程的正定解

研究了双线性系统中的一类广义Lyapunov矩阵方程的正定解.基于混合单调算子不动点定理,给出新的存在正定解的充分条件,构造了求其正定解的不动点迭代方法,并给出了迭代误差估计公式.数值实验表明新方法是可行的.     

A Generalized Chebyshev Method for Non-linear Parabolic Equations in One Space Variable

The derivation and implementation of a generalized Chebyshevmethod is described for the numerical solution of non-linearparabolic equations in one space dimension. The solution isobtained by using the method of lines and is approximated inthe space variable by piecewise Chebyshev polynomial expansions.These expansions are normally few in number and of high order.It is shown that the method can be derived from a perturbedform of the original equation. A numerical example is givento illustrate its performance compared with the finite elementand finite difference method. A comparison of various Chebyshev methods is made by applyingthem to two-point eigenproblems. It is shown by analysis andnumerical examples that the approach used to derive the generalizedChebyshev method is comparable, in terms of the accuracy obtained,with existing Chebyshev methods.     

混合型分段连续微分方程的数值稳定性

This paper deals with the stability analysis of the Euler-Maclaurin method for differential equations with piecewise constant arguments of mixed type. The expression of analytical solution is derived and the stability regions of the analytical solution are given. The necessary and sufficient conditions under which the numerical solution is asymptotically stable are discussed. The conditions under which the analytical stability region is contained in the numerical stability region are obtained and some numerical examples are given.     

The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement

A numerical technique is presented for the solution of a parabolic partial differential equation with a time-dependent coefficient subject to an extra measurement. The method is derived by expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem can be reduced to a set of algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous works and also it is efficient to use.