Singularly perturbed solution to semilinear reaction diffusion equations with two parameters

A class of singularly perturbed initial boundary value problems for semilinear reaction diffusion equations with two parameters is considered, Under suitable conditions and using the theory of differential inequalities, the existence and the asymptotic behavior of the solution to the initial boundary value problem are studied.     

A second order uniform difference scheme for a singularly perturbed turning point problem

We construct a positive type difference scheme.for a singularly perturbed boundary value problem with a turning point. It's proved that this scheme is the second order convergence, uniformly in ε, to the solution of the singularly perturbed B.V.P. Numerical examples are provided.     

Singularly perturbed solution for weakly nonlinear equations with two parameters

A class of singularly perturbed boundary value problems of weakly non- linear equation for fourth order on the interval[a,b]with two parameters is considered. Under suitable conditions,firstly,the reduced solution and formal outer solution are con- structed using the expansion method of power series.Secondly,using the transformation of stretched variable,the first boundary layer corrective term near x=a is constructed which possesses exponential attenuation behavior.Then,using the stronger transfor- mation of stretched variable,the second boundary layer corrective term near x=a is constructed,which also possesses exponential attenuation behavior.The thickness of second boundary layer is smaller than the first one and forms a cover layer near x=a. Finally,using the theory of differential inequalities,the existence,uniform validity in the whole interval[a,b]and asymptotic behavior of solution for the original boundary value problem are proved.Satisfying results are obtained.     

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

A CLASS OF NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS WITH BOUNDARY PERTURBATION

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions, first, the outer solution of the original problem was obtained. Secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer was constructed. Finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems was studied, and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation were discussed.     

A UNIFORMLY CONVERGENT DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF A HIGH ORDER ELLIPTIC DIFFERENTIAL EQUATION

ＡＵＮＩＦＯＲＭＬＹＣＯＮＶＥＲＧＥＮＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＴＨＥＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＰＲＯＢＬＥＭＯＦＡＨＩＧＨＯＲＤＥＲＥＬＬＩＰＴＩＣＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮ（刘国庆）（苏煜城）ＡＵＮＩＦＯ...     

On singularly perturbed quasilinear systems

In this paper, the objective is to give sufficient conditions for the existence of solution of the nonlinear two-point boundary value problem(1.1). And we employ these results to consider the boundary layer phenomena of the quasilinear weakly coupled singularly perturbed system (DP)_q.     

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.     

Construction of an asymptotic solution of one optimal control problem

We construct an asymptotic solution of a singularly perturbed optimal control problem in the case of a regular pencil of boundary matrices.     

ON SINGULAR PERTURBATION FOR A NONLINEAR INITIAL-BOUNDARY VALUE PROBLEM (Ⅱ)

In this paper, we consider a singularly perturbed problem of a kind of quasilinear hyperbolic-parabolic equations, subject to initial-boundary value conditions with moving boundary. When certain assumptions are satisfied and e is sufficiently small, the solution of this problem has a generalized asymptotic expansion (in the Van der Corput sense), which takes the sufficiently smooth solution of the reduced problem as the first term, and is uniformly valid in domain Q where the sufficiently smooth solutioh exists. The layer exists in the neighborhood of t=0. This paper is the development of references.     

Third-order nonlinear singularly perturbed boundary value problem

ＩｎｔｒｏｄｕｃｔｉｏｎＳｉｎｃｅ 1 958,Ｌｅｖｓｉｏｎａｎｄｓｅｖｅｒａｌｏｔｈｅｒａｕｔｈｏｒｓ[1]- [4 ]ｈａｖｅｍａｄｅａｓｕｂｓｔａｎｔｉａｌｓｔｕｄｙｆｏｒｅｘｉｓｔｅｎｃｅａｎｄｅｓｔｉｍａｔｅｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｔｈｉｒｄ＿ｏｒｄｅｒｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｗｉｔｈｔｗｏｐｏｉｎｔｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓｘ( 0 ) =Ａ ,ｘ′( 0 ) =Ｂ ,ｘ′( 1 ) =Ｃ ,ｏｒｓｌｉｇｈｔｌｙｅｘｔｅｎｓｉｖｅｌｙｌｉｎｅａｒｂｏ…     

Eigenoscillations of an elastic body with a rough surface

Explicit presentations for the initial terms of the asymptotic solution of the spectral problem of the elasticity theory in a plane region with a rapidly oscillating boundary are obtained. Based on asymptotic formulas, two methods for problem modeling are proposed: with the use of Wenzel’s boundary conditions and with the use of the principle of a smooth image of a singularly perturbed boundary. Various approaches to justification of asymptotic presentations are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 103–114, November–December, 2007.     

The Asymptotic Behavior of Solution for the Singularly Perturbed Initial Boundary Value Problems of the Reaction Diffusion Equations in a Part of Domain

Ｔｈｅａｕｔｈｏｒｓｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1 ] -[7] .Ｎｏｗｗｅｒａｉｓｅａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｏｎａｐａｒｔｏｆｄｏｍａｉｎ .Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｉｎｉｔｉａｌｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｔｈｅｒｅａｃｔｉｏｎｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｓ　　ｕｔ-λε(ｘ) ( μ(ｕ)ｕｘ) ｘ Ｋｘ(ｕ) ｆ(ｘ ,ｔ,ｕ) =0 ,(ｔ,ｘ) ∈ ( 0 ,Ｔ)× ( ( 0 ,α) ∪ (…     

A Class of Singularly Perturbed Generalized Boundary Value Problems for Quasi-Linear Elliptic Equation of Higher Order

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅａｕｔｈｏｒｓｔｕｄｉｅｄａｃｌａｓｓｏｆｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓｉｎ [1]- [8].Ｔｈｉｓｐａｐｅｒｉｎｖｏｌｖｅｓｔｈｅｇｅｎｅｒａｌｉｚｅｄｓｏｌｕｔｉｏｎｏｆｔｈｅｓｉｎｇｕｌａｒｌｙｐｅｒｔｕｒｂｅｄｐｒｏｂｌｅｍｓ.ＮｏｗｗｅｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇＤｉｒｉｃｈｌｅｔｐｒｏｂｌｅｍｆｏｒｔｈｅｑｕａｓｉ＿ｌｉｎｅａｒｅｌｌｉｐｔｉｃｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｏｆｈｉｇｈｅｒｏｒｄｅｒεＬ2ｍ[ｕ] ∑ｎ…     

A class of singularly perturbed reaction diffusion systems

A class of singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems is studied. Project supported by the National Natural Science Foundation of China     

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.     

Existence of solutions for a singularly perturbed boundary value problem for the third order semi-linear differential equation with a turning point

In this paper,by using the techniques of differential inequalities,we prove the existence of the solutions of a singularly perturbed boundary value problem for the third order semilinear differential equation with a turning point.     

Asymptotic solution of nonlocal nonlinear reaction-diffusion Robin problems with two parameters

In this paper, the nonlocal nonlinear reaction-diffusion singularly perturbed problems with two parameters are studied. Using a singular perturbation method, the structure of the solutions to the problem is discussed in relation to two small parameters. The asymptotic solutions of the problem are given.     

Singular Perturbation of Boundary Value Problem for Quasilinear Third-Order Ordinary Differential Equations Involving Two Small Parameters

ＩｎｔｒｏｄｕｃｔｉｏｎＡｂｏｕｔｓｉｎｇｕｌａｒｐｅｒｔｕｒｂａｔｉｏｎｏｆｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍｆｏｒｓｅｃｏｎｄ＿ｏｒｄｅｒｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｗｏｒｋｓｏｆａｌａｒｇｅｎｕｍｂｅｒｗｅｒｅｃｏｎｓｉｄｅｒｅｄｏｎｌｙｆｏｒｔｈｅｃａｓｅｏｆｉｎｖｏｌｖｉｎｇｏｎｅｓｍａｌｌｐａｒａｍｅｔｅｒ[1- 4].Ｏｎｌｙａｆｅｗｗｏｒｋｓｗｅｒｅｃｏｎｓｉｄｅｒｅｄｆｏｒｔｈｅｃａｓｅｏｆｉｎｖｏｌｖｉｎｇｔｗｏｓｍａｌｌｐａｒａｍｅｔｅｒｓ[5 - 8]…     

The nonlinear nonlocal singularly perturbed problems for reaction diffusion equations

A class of nonlinear nonlocal for singularly perturbed Robin initial boundary value problems for reaction diffusion equations is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed. Foundation items: the National Natural Science Foundation of China (10071048); the “Hunfred Talents Project” by Chinese Academy of Sciences Biography: Mo Jia-qi (1937−)