Similarity Solutions of a Non-linear Diffusion Equation

In several physical contexts the equations for the dispersionof a buoyant contaminant can be approximated by the Erdogan-Chatwin(1967) equation {dot}c = {dot}_y{[D_o + ({dot}_yc)²D₂]{dot}_yc}. Here it is shown that in the limit of strong non-linearity (i.e.D_o = 0) there are similarity solutions for a concentration jumpand for a finite discharge. A stability analysis for the latterproblem involves a new family of orthogonal polynomials Y_n(z)where (1 – z⁴)Y – 6z³Y + n(n + 5)z² Y_n = 0 and the degree n is restricted to the values 0, 1, 4, 5, 8,9,.... A numerical solution of the Erdogan-Chatwin equationis given which describes the transition between the non-linearand linear (Gaussian) similarity solutions.     

Asymptotic Solution of a Non-linear Second Order Hyperbolic Differential Equation with Large Time Delay

The solution is given for a certain hyperbolic partial differentialequation with small non-linearity and large time delay includedinto both unperturbed and perturbed parts of the equation.     

A Ray Method for the Asymptotic Solution of the Diffusion Equation

A general method is presented for finding asymptotic solutionsof problems posed for linear partial differential equationscontaining lower order (dispersive) terms. Powers of a largeparameter appear in these equations multiplying the lower orderterms. The expansion procedure is a "ray method", i.e. all ofthe functions that appear in the expansion satisfy ordinarydifferential equations along certain curves called rays. Special attention is paid to the equation of heat conduction.In this case, 1/ is related to the thermal conductivity of themedium. For this equation several problems are considered inwhich the parameter enters the data or the inhomogeneous (source)term in various ways.     

The Small-time Evolution of a Non-linear Diffusion Equation

The small-time evolution of the equation . t 0, – -$$\infty $$ < y < $$\infty $$ is consideredfor initial conditions appropriate in physical terms to botha concentration jump and a finite discharge. It is shown thatthe strong non-linear similarity solution is not uniformly validfor all y but is singular along an unknown moving boundary inthe (y, t) plane which has to be determined as part of the solution.The nature of this singularity and its resolution is investigatedby the introduction of a transition region between the strongnon-linear and linear regimes.     

On the Asymptotic Behaviour of a Certain Non-linear Difference Equation

Arising from the study of the convergence properties of a rationalapproximation method for determining a zero of the functionf(x)is a certain non-linear difference equation. This equation hasthe form v_n+1 = g_p–1(v_n)/g_p(v_n), Where g_p(v_n) is a polynomialin v_n whose coefficients depend on a parameter p, the orderof the zero of f The asymptotic behaviour of the differenceequation is studied and it is shown that if there is a limitorder of convergence it is always linear for multiple zeros.     

一个非线性方程的渐近激波解

该文是利用简捷的方法得到了高精度的非线性问题渐近激波解。     

具源项的奇异扩散方程解的存在性及渐近性

讨论了一类具源项的奇异扩散方程的第二初边值问题,证明了经典解的存在性以及时间充分大时解的渐近性质.     

Initial Motion of the Free Boundary for a Non-linear Diffusion Equation

The free boundary between v > 0 and v = 0 for the porousmedium equation v₁= |v|² +nv²v can remain stationary for somepositive waiting-time and then start moving. It is of interestto know the way in which any part of the boundary first moves.It is already known that if the waiting time is given by purelylocal considerations then the boundary speed is continuous,i.e. the initial speed is zero. For cases where the boundary moves before the time found bysimply considering v(x, 0) close to the boundary, a perturbationanalysis indicates that it starts moving with positive speed.Two-dimensional problems show the possible formation of verticesin the boundary. At these points the normal velocity jumps fromzero to some positive value.     

Asymptotic Solutions of a Model Diffusion-Reaction Equation

In this paper we study the large-time solution of the nonlineardiffusion reaction equation , m>1, p>0 subject to u(0, t)= and initial data with finitesupport. If we regard the steady state as the leading term inan asymptotic expansion of the solution as t then we show thatthis expansion is non-uniform in x. The nature of the non-uniformity,located at the moving interface, is shown to depend cruciallyon the location in (p, m) parameter space. For p<1<m weconstruct a uniformly valid solution using strained coordinates.In the remaining regions uniform zeroth-order composite solutionsare constructed via matched expansions.     

关于非线性狄立克雷问题的一致有效渐近解

本文研究最高阶导数项带小参数的二阶拟线性椭圆型方程的狄立克雷问题.在退化方程不存在奇点的情形下,当参数ε是充分小时,证明了解的存在性和唯一性,并在整个区域导出解的一致有效渐近近似式.     

Asymptotic Behavior of Solutions to the Diffusion Equation

We study asymptotic behavior of solutions to an initial value problem posed for heat equation. For which, we construct an approximate solution to the initial value problem in terms of derivatives of Gaussian by incorporating the moments of initial function. Spatial shifts are introduced into the leading order term as well as penultimate term of the approximation. This paper is continuation to the work of Yanagisawa [14].     

数值求解对流扩散方程有限分析方法的稳定性与收敛性

本文利用椭圆型偏微分方程所满足的最大最小值原理研究有限分析方法,证明了数值求解对流扩散方程有限分析方法的稳定性与收敛性,顺便指出了前人理论中的错误.     

微分方程指数形式渐近解

本文对常微分方程的指数形式渐近解作了进一步讨论,首先给出了二阶方程一致有效指数形式渐近解的正交条件;接着讨论了匹配渐近法中的指数形式渐近解;最后举例说明。     

The Asymptotic Behavior for Numerical Solution of a Volterra Equation

Long-time asymptotic stability and convergence properties for the numerical solution of a Volterra equation of parabolic type are studied.The methods are based on the first-second order backward difference methods.The memory term is approximated by the comvolution quadrature and the interpolant quadrature.Discretization of the spatial partial differential operators by the finite element method is also considered.     

一类非线性积分微分反应扩散方程的渐近解

研究了一类非线性积分———微分反应扩散方程.在适当的条件下,利用微分不等式理论,讨论了退化解具有两个交叉解的初值问题解的渐近性态.     

Boundary Integral Equation Solution of Non-linear Plane Potential Problems

This paper presents the boundary integral equation (BIE) formulation,and numerical solution procedure for two-dimensional problemsgoverned by Laplac's equation and subject to non-linear boundaryconditions. The introduction of non-linear terms constitutesa fundamental extension of the BIE method, as previous applicationshave been restricted entirely to linear problems. Furthermore,non-linearities necessitate the use of iterative solution techniqueswhich present the conceptual disadvantage that a solution isnot guaranteed. However, such difficulties were not encounteredwith the Newton—Raphson method employed in this study.The various features of the BIE technique are illustrated bythe application to a physical problem which is of significancein heat exchanger design.     

Asymptotic Analysis of a Perturbed Periodic Solution for the KdV Equation

We consider the solution of the Korteweg–de Vries (KdV) equation with periodic initial value where C , A , k , μ, and β are constants. The solution is shown to be uniformly bounded for all small ɛ, and a formal expansion is constructed for the solution via the method of multiple scales. By using the energy method, we show that for any given number   T > 0  , the difference between the true solution v ( x , t ; ɛ) and the N th partial sum of the asymptotic series is bounded by  ɛ ^{N +1}  multiplied by a constant depending on T and N , for all  −∞ < x < ∞, 0 ≤ t ≤ T /ɛ  , and  0 ≤ɛ≤ɛ₀  .     

具扩散的生物控制模型的渐进性

讨论了具有空间扩散的生物控制系统在齐次Neumann边界条件下解的整体性态.通过线性化方法给出了局部渐近稳定条件,通过构造Lyapunov函数给出了全局稳定性的证明.     

Boundary Integral Solution of the Time-Fractional Diffusion Equation

Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.      

一类可靠性模型解的渐近稳定性

研究一个可靠机器，一个不可靠机器和一个缓冲库构成的系统。通过研究相应算子的谱特征得到该系统解的渐近稳定性。