An Initial Value Technique for Singularly Perturbed Convection–Diffusion Problems with a Negative Shift

In this paper, a numerical method named as Initial Value Technique (IVT) is suggested to solve the singularly perturbed boundary value problem for the second order ordinary differential equations of convection–diffusion type with a delay (negative shift). In this technique, the original problem of solving the second order equation is reduced to solving two first order differential equations, one of which is singularly perturbed without delay and other one is regular with a delay term. The singularly perturbed problem is solved by the second order hybrid finite difference scheme, whereas the delay problem is solved by the fourth order Runge–Kutta method with Hermite interpolation. An error estimate is derived by using the supremum norm. Numerical results are provided to illustrate the theoretical results.     

A Nonlinear Singularly Perturbed Problem for Reaction Diffusion Equations with Boundary Perturbation

A nonlinear singularly perturbed problems for reaction diffusion equation with boundary perturbation is considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations is studied using the theory of differential inequalities.     

Estimates in Hölder Classes for the Solution of an Inhomogeneous Dirichlet Problem for a Singularly Perturbed Homogeneous Convection–Diffusion Equation

Computational Mathematics and Mathematical Physics - An inhomogeneous Dirichlet boundary value problem for a singularly perturbed homogeneous convection–diffusion equation with constant...     

Solution of Contrast Structure Type for a Parabolic Reaction–Diffusion Problem in a Medium with Discontinuous Characteristics

We consider a reaction–diffusion-type equation in a two-dimensional domain containing the interface between media with distinct characteristics along which the reactive term has a discontinuity of the first kind. We assume that the interface between the media, as well as the functions describing the reactions, periodically varies in time. We study the existence of a stable periodic solution of a problem with an internal layer. To prove the existence, stability, and local uniqueness of the solution, we use the asymptotic method of differential inequalities, which we generalized to a new class of problems with discontinuous nonlinearities.     

A Class of Nonlinear Singularly Perturbed Problems for Reaction Diffusion Equations with Boundary Perturbation

A class of nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems is studied using the theory of differential inequalities.     

Uniform Global Convergence of a Hybrid Scheme for Singularly Perturbed Reaction–Diffusion Systems

We consider a system of coupled singularly perturbed reaction–diffusion two-point boundary-value problems. A hybrid difference scheme on a piecewise-uniform Shishkin mesh is constructed for solving this system, which generates better approximations to the exact solution than the classical central difference scheme. Moreover, we prove that the method is third order uniformly convergent in the maximum norm when the singular perturbation parameter is small. Numerical experiments are conducted to validate the theoretical results.     

The Interior Layer Problem for the Strongly Nonlinear Singularly Perturbed System

The nonlinear strongly singularly perturbed systemεyi" = fi(t, y,y'), a < t < b, i = 1, 2, …, n, yi(a,ε)-pily'i(a,ε) = Ai, yi(b,ε) pi2y'i(b,ε) = Biare considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of interior solution for the problems are studied.     

Robust a Posteriori Error Estimation for a Singularly Perturbed Reaction–Diffusion Equation on Anisotropic Tetrahedral Meshes

We consider a singularly perturbed reaction–diffusion problem and derive and rigorously analyse an a posteriori residual error estimator that can be applied to anisotropic finite element meshes. The quotient of the upper and lower error bounds is the so-called matching function which depends on the anisotropy (of the mesh and the solution) but not on the small perturbation parameter. This matching function measures how well the anisotropic finite element mesh corresponds to the anisotropic problem. Provided this correspondence is sufficiently good, the matching function is O(1). Hence one obtains tight error bounds, i.e. the error estimator is reliable and efficient as well as robust with respect to the small perturbation parameter. A numerical example supports the anisotropic error analysis.     

The “Duck Survival” Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables

We consider the system of ordinary differential equations x = f(x,y), y = g(x,y), where x ², y , 0 < 1, and f,g C. It is assumed that the equation g = 0 determines two different smooth surfaces y = (x) and y = (x) intersecting generically along a curve l. It is further assumed that the trajectories of the corresponding degenerate system lying on the surface y = (x) are ducks, i.e., as time increases, they intersect the curve l generically and pass from the stable part {y =(x),g'_y < 0} of this surface to the unstable part {y =(x),g'_y > 0}. We seek a solution of the so-called duck survival problem, i.e., give an answer to the following question: what trajectories from the one-parameter family of duck trajectories for = 0 are the limits as 0 of some trajectories of the original system.     

The Singularly Perturbed Nonlinear Nonlocal Initial Boundary Value Problems for Reaction Diffusion Equations

The singularly perturbed nonlinear nonlocal initial boundary value problem for reaction diffusion equations is discussed. Under suitable conditions, the outer solution of the original problem is obtained. By using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. By using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied, and by educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are considered.     

Nonmonotone Interior Layer Solutions for Singularly Perturbed Semilinear Boundary Value Problems with a Turning Point

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On a Generalized Dirichlet Problem for Analytic Family

In this paper, we give the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron in Cn, and get a necessary and sufficient condition for a generalized Dirichlet problem to be solvable for an analytic family on a bounded holomorphic domain. Especially, we derive that this condition is just that the continuous real boundary value is prescribed on and only on the Silov boundary for an analytic family on a bounded strictly pseudoconvex domain or an analytic polyhedron.     

A Finite Element Method for Singularly Perturbed Reaction—diffusion Problems

A finite element method is proposed for the sing ularly perturbed reaction-diffusion problem.An optimal error bound is derived,independent of the perturbation parameter.     

RESEARCH ANNOUNCEMENTS On a Spectrum Problem for Analytic Function

1 Introduction Let Ω be the unit disk {z: |z| ＜ 1} in the complex plane C. In this paper, we consider the following boundary value problem: Find out a λ and the analytic function u(z) in the unit disk Ω such that they satisfy the boundary condition     

RESEARCH ANNOUNCEMENTS On a Spectrum Problem for Analytic Function

1 IntroductionLet Ω be the unit disk {z: |z| < 1} in the complex plane C. In this paper, we consider thefollowing boundary value problem: Find cot a λ and the analytic function u(z) in the unit diskΩ such that they satisfy the boundary conditionwhere denotes the outer normal derivative of μ.The problem (l) has background in physics and mechanics [2,3]. To our knowledge it seemsthat few work have been done. In some cases one usually assumes that λ is sufficiently small,e.g. for second or…     

Lp Decay Rate for a Nonlinear Convection Diffusion Reaction Equation in R~n

This paper studies the asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in Rⁿ.Firstly,the global existence and uniqueness of classical solutions for small initial data are established.Then,we obtain the L^p,2≤p≤+∞decay rate of solutions.The approach is based on detailed analysis of the Green function of the linearized equation with the technique of long wave-short wave decomposition and the Fourier analysis.     

Pseudo Almost Periodic Solutions of a Singularly Perturbed Differential Equation with Piecewise Constant Argument

Under suitable assumptions, the existence and the uniqueness of the pseudo-almost periodic solution for a singularly perturbed differential equation with piecewise constant argument are obtained. In addition, the stability properties of these solutions are characterized by the construction of manifolds of initial data.     

The Center Problem for a Linear Center Perturbed by Homogeneous Polynomials

The centers of the polynomial differential systems with homogeneous polynomials have been studied for the degrees s = 2, 3, 4, 5. for s = 2, 3, and partially classified for s = 4, 5. In this paper we recall and we give new centers for s = 6, 7 a linear center perturbed by They are completely classified these results for s = 2, 3, 4, 5,     

L~P Estimate for Klein-Gordon Equation with a Perturbed Potential

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