Issues in the numerical solution of evolutionary delay differential equations

Delay differential equations are of sufficient importance in modelling real-life phenomena to merit the attention of numerical analysts. In this paper, we discuss key features of delay differential equations (DDEs) and consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide an introduction to the existing literature and numerical codes, and in particular we indicate the approaches adopted by the authors. We also indicate some of the unresolved issues in the numerical solution of DDEs. Communicated by J.C. Mason     

Issues in the numerical solution of evolutionary delay differential equations

Delay differential equations are of sufficient importance in modelling real-life phenomena to merit the attention of numerical analysts. In this paper, we discuss key features of delay differential equations (DDEs) and consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide an introduction to the existing literature and numerical codes, and in particular we indicate the approaches adopted by the authors. We also indicate some of the unresolved issues in the numerical solution of DDEs. Communicated by J.C. Mason     

Nonlinear instability in advection-diffusion numerical models

The phenomenon of nonlinear instability, which affects all propagation equations, is simply presented and explained; its origin lies in the numerical dispersion of fundamental solutions by the discrete schemes. Methods to avoid it or to minimize its effects on solutions are proposed and discussed.     

Local Linearization method for the numerical solution of stochastic differential equations

The Local Linearization (LL) approach for the numerical solution of stochastic differential equations (SDEs) is extended to general scalar SDEs, as well as to non-autonomous multidimensional SDEs with additive noise. In case of autonomous SDEs, the derivation of the method introduced gives theoretical support to one of the previously proposed variants of the LL approach. Some numerical examples are given to demonstrate the practical performance of the method.     

基于迭代正则化高斯-牛顿法的非线性Urysohn积分方程数值解

非线性Urysohn积分方程在许多领域中都有广泛的应用,但由于该方程具有不适定性的特点,数据的微小扰动可能导致解的巨大变化,给数值求解带来很大困难.为了获得稳定的、准确的数值解,本文利用迭代正则化高斯-牛顿法对此方程进行求解,给出了利用Sigmoid-型函数确定迭代正则化参数的方法.对一类重力测定问题进行了数值模拟,将得到的数值解和相应的精确解作比较.结果表明,本文提出的方法在求解非线性Urysohn积分方程时是可行的也是有效的.     

On the solution of a fractional diffusion integrodifferential equation with Rothe time discretization

This article deals with a fractional diffusion equation of the second-order differential Volterra operator and fractional integral condition. Existence and uniqueness of a weak solution in an appropriate sense as well as some regularity results are obtained by the use of Rothe’s method. Finally, an example is given to demonstrate the effectiveness of the proposed approach.     

The numerical solution of fractional differential equations: Speed versus accuracy

This paper is concerned with the development of efficient algorithms for the approximate solution of fractional differential equations of the form D y(t)=f(t,y(t)), R ⁺–N.()We briefly review standard numerical techniques for the solution of () and we consider how the computational cost may be reduced by taking into account the structure of the calculations to be undertaken. We analyse the fixed memory principle and present an alternative nested mesh variant that gives a good approximation to the true solution at reasonable computational cost. We conclude with some numerical examples.     

The probability approach to numerical solution of nonlinear parabolic equations

A number of new layer methods for solving semilinear parabolic equations and reaction‐diffusion systems is derived by using probabilistic representations of their solutions. These methods exploit the ideas of weak sense numerical integration of stochastic differential equations. In spite of the probabilistic nature these methods are nevertheless deterministic. A convergence theorem is proved. Some numerical tests are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 490–522, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10020     

A numerical solution of the dirichlet problem on some special doubly connected regions

The aim of this paper is to give a convergence proof of a numerical method for the Dirichlet problem on doubly connected plane regions using the method of reflection across the exterior boundary curve (which is analytic) combined with integral equations extended over the interior boundary curve (which may be irregular with infinitely many angular points).     

A unified framework for the numerical solution of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: N.karcanias{at}city.ac.uk^**** Email: ixp{at}le.ac.uk Algebraic quadratic equations are special cases of a singlegeneralized algebraic quadratic matrix equation (GQME). Thispaper focuses on the numerical solution of the GQME using probability-1homotopy methods. A synoptic review of these methods and theirapplication to algebraic matrix equations is provided as background.A large variety of analysis and design problems in systems andcontrol are reported as special cases of the presented frameworkand some of them are illustrated via numerical examples fromthe literature.     

The planar radiosity equation and its numerical solution

This article gives properties of the planar radiosity equationand methods for its numerical solution. Regularity propertiesof the radiosity solution are examined, including both the effectsof corners and the effects of the visibility function. Theseare taken into account in the design of collocation methodswith piecewise polynomial approximating functions. Numericalexamples conclude the paper.     

Numerical study of the transverse stability of the Peregrine solution

We generalize a previously published numerical approach for the one-dimensional (1D) nonlinear Schrödinger (NLS) equation based on a multidomain spectral method on the whole real line in two ways: first, a fully explicit fourth-order method for the time integration, based on a splitting scheme and an implicit Runge-Kutta method for the linear part, is presented. Second, the 1D code is combined with a Fourier spectral method in the transverse variable both for elliptic and hyperbolic NLS equations. As an example we study the transverse stability of the Peregrine solution, an exact solution to the 1D NLS equation and thus a y-independent solution to the 2D NLS. It is shown that the Peregine solution is unstable agains all standard perturbations, and that some perturbations can even lead to a blow-up for the elliptic NLS equation.     

A new numerical approach for the solution of nonlinear Fredholm integral equations system of second kind by using Bernstein collocation method

This paper presents an efficient numerical method for finding solutions of the nonlinear Fredholm integral equations system of second kind based on Bernstein polynomials basis. The numerical results obtained by the present method have been compared with those obtained by B‐spline wavelet method. This proposed method reduces the system of integral equations to a system of algebraic equations that can be solved easily any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

非线性二维Volterra积分方程的一个高阶数值格式

对非线性二维Volterra积分方程构造了一个高阶数值格式.block-byblock方法对积分方程来说是一个非常常见的方法,借助经典block-by-block方法的思想,构造了一个所谓的修正block-by-block方法.该方法的优点在于除u(x_1,y),u(x_2,y),u(x,y_1)和u(x,y_2)外,其余的未知量不需要耦合求解,且保存了block-by-block方法好的收敛性.并对此格式的收敛性进行了严格的分析,证明了数值解逼近精确解的阶数是4阶。     

On the numerical solution of hyperbolic PDEs with variable space operator

The first and second order of accuracy in time and second order of accuracy in the space variables difference schemes for the numerical solution of the initial‐boundary value problem for the multidimensional hyperbolic equation with dependent coefficients are considered. Stability estimates for the solution of these difference schemes and for the first and second order difference derivatives are obtained. Numerical methods are proposed for solving the one‐dimensional hyperbolic partial differential equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

Improved outer inverses for the numerical solution of Euler–Lagrange equations

In this paper, improved outer inverses are used for solving the overdetermined differential‐algebraic equations that arise in the simulation of mechanical systems. The method is based on Newton‐type iterations using outer inverses. Copyright © 2004 John Wiley & Sons, Ltd.     

Estimates on the weak solution of semilinear hyperbolic systems

The Cauchy problem for a semilinear hyperbolic system of the type

浅水长波近似方程组的拟小波解

以浅水长波近似方程组为例,提出了拟小波方法求解(1 1)维非线性偏微分方程组数值解,该方程用拟小波离散格式离散空间导数,得到关于时间的常微分方程组,用四阶Runge-K utta方法离散时间导数,并将其拟小波解与解析解进行比较和验证.