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1.
讨论了一类具有超抛物型方程的反应扩散问题.首先,证明了比较定理.其次,构造了形式渐近解.然后,利用微分不等式方法,研究了问题解的存在、唯一性和渐近性态.最后得到了原问题解的渐近展开式. 相似文献
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《Quaestiones Mathematicae》2013,36(2):229-248
Several investigations have been made on singularly perturbed integral equations. This paper aims at presenting an algorithm for the construction of asymptotic solutions and then provide a proof for asymptotic correctness to singularly perturbed systems of Volterra integro-differential equations. 相似文献
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Walter Renger 《Mathematische Nachrichten》2001,228(1):163-187
Given an operator H1 for which a limiting absorption principle holds, we study operators H2 which are produced by perturbing H1 in the sense that the difference between some powers of the resolvents is compact.We show that (except for possibly a discrete set of eigenvalues) a limiting absorption principle holds for H2. We apply this theory to study potential and domain perturbations of Feller operators. While our theory mostly reproduces known results in the case of potential perturbations, for domain perturbations we get results which appear to be new. 相似文献
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I. I. Klevchuk 《Ukrainian Mathematical Journal》2002,54(4):693-699
We obtain a representation of the integral manifold of a system of singularly perturbed differential-difference equations with periodic right-hand side. We show that, under certain conditions imposed on the right-hand side, the Poincaré map for the perturbed system has a transversal homoclinic point. 相似文献
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Samoilenko A. M. Kaplun Yu. I. Samoilenko V. H. 《Ukrainian Mathematical Journal》2002,54(8):1309-1323
We propose and justify an algorithm for the construction of asymptotic solutions of singularly perturbed differential equations with impulse action. 相似文献
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We analyze Runge-Kutta discretizations applied to singularly perturbed gradient systems. It is shown in which sense the discrete dynamics preserve the geometric properties and the longtime behavior of the underlying ordinary differential equation. If the continuous system has an attractive invariant manifold then numerical trajectories started in some neighbourhood (the size of which is independent of the step-size and the stiffness parameter) approach an equilibrium in a nearby manifold. The proof combines invariant manifold techniques developed by Nipp and Stoffer for singularly perturbed systems with some recent results of the second author on the global behavior of discretized gradient systems. The results support the favorable behavior of ODE methods for stiff minimization problems. 相似文献
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讨论了一类超抛物型方程的非线性奇摄动问题.利用比较定理,研究了问题解的存在性及其渐近性态. 相似文献
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The singularly perturbed problem for the semilinear elliptic equations is considered. Under appropriate conditions, by using the comparison theorem, the existence and asymptotic behavior of solution for the boundary value problems are studied. 相似文献
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An averaged system to approximate the slow dynamics of a two timescale nonlinear stochastic control system is introduced.
Validity of the approximation is established. Special cases are considered to illustrate the general theory. 相似文献
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一类非线性奇摄动方程的激波问题 总被引:3,自引:1,他引:3
利用奇摄动理论和匹配原理,讨论了一类非线性奇摄动方程的激波问题.首先,构造了原问题的外部解和内层解.其次,研究了当激波在区间的边界附近和内部的激波解.最后,得出了与边界条件相对应的激波位置及解的表达式. 相似文献
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奇摄动时滞反应扩散方程 总被引:1,自引:0,他引:1
研究了一类带有小延迟的微分-差分反应扩散方程初值边值问题.在适当的条件下,利用伸长变量法,构造了问题的形式渐近解.再用微分不等式理论证明了解的一致有效性. 相似文献
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本文讨论了半线性椭圆型方程非局部奇摄动问题 .在适当条件下 ,利用不动点原理 ,对边值问题解的存在性和渐近性态作了研究 . 相似文献
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Differential Equations - We consider a boundary value problem for a singularly perturbed system of two second-order ordinary differential equations with distinct powers of the small parameter... 相似文献
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Devendra Kumar 《高等学校计算数学学报(英文版)》2015,8(4):496-514
This paper deals with a more general class of singularly perturbed boundary value problem for a differential-difference equations with small shifts. In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter $ε$ and the shifts depend on the small parameter $ε$ has been
considered. The fitted-mesh technique is employed to generate a piecewise-uniform
mesh, condensed in the neighborhood of the boundary layer. The cubic B-spline
basis functions with fitted-mesh are considered in the procedure which yield a tridiagonal system which can be solved efficiently by using any well-known algorithm.
The stability and parameter-uniform convergence analysis of the proposed method
have been discussed. The method has been shown to have almost second-order
parameter-uniform convergence. The effect of small parameters on the boundary
layer has also been discussed. To demonstrate the performance of the proposed
scheme, several numerical experiments have been carried out. 相似文献
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Aditya Kaushik Anil K. Vashishth Vijayant Kumar 《Numerical Functional Analysis & Optimization》2019,40(13):1550-1571
A weakly coupled convection dominated system of m-equations is analyzed. A higher order accurate asymptotic-numerical method is presented. The solutions of convection dominated problem are known to exhibit multi-scale character. There exist narrow region across the boundary of the domain where the solution exhibit steep gradient. This region is termed as boundary layer region and the solution of problem is said to have a boundary layer. Outside of this region, the solution of system behaves smoothly. To capture this multi-scale nature given system is factorized into two explicit systems. The degenerate system of initial value problems (IVPs), obtained by setting ??=?0, corresponds to the smooth solution, which lies outside of boundary layers. For solution inside boundary layers, a system of boundary value problems (BVPs) is obtained using stretching transformation. Regardless of this simple factorization, solutions of these systems preserve the key features of the given coupled system. Runge–Kutta method is used to solve the degenerate system of IVPs, whereas the system of BVPs is solved analytically. Stability and consistency of the proposed method is established. A uniform convergence of higher order is obtained. Possible extension to differential difference equations are also brought to attention. A comparative study of the present method with some state of art existing numerical schemes is carried out by means of several test problems. The results so obtained demonstrate the effectiveness and potential of present approach. 相似文献
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奇摄动问题有很强的自然科学的背景,它在生态环境、大气物理、海洋科学、催化反应、激波和量子物理中都有很广泛的应用.本文研究了一类带有小延迟的微分--差分反应扩散方程初值问题.在适当的条件下,利用奇摄动伸长变量法,构造了问题的形式渐近解.再用微分不等式理论证明了解的一致有效性. 相似文献