A singularly perturbed reaction diffusion problem for the nonlinear boundary condition with two parameters

A class of singularly perturbed initial boundary value problems of reaction diffusion equations for the nonlinear boundary condition with two parameters is considered. Under suitable conditions, by using the theory of differential inequalities, the existence and the asymptotic behaviour of the solution for the initial boundary value problem are studied. The obtained solution indicates that there are initial and boundary layers and the thickness of the boundary layer is less than the thickness of the initial layer.

Singular analysis of bifurcation systems with two parameters

Bifurcation properties of dynamical systems with two parameters are investigated in this paper. The definition of transition set is proposed, and the approach developed is used to investigate the dynamic characteristic of the nonlin- ear forced Duffing system with nonlinear feedback controller. The whole parametric plane is divided into several persistent regions by the transition set, and then the bifurcation dia- grams in different persistent regions are obtained.

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 constructed 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 transformation of stretched variable, the second boundary layer corrective term near x = a is constructed, wtfich 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.

ON THE SINGULAR PERTURBATION OF A NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH TWO PARAMETERS

In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated.

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.

Singularity analysis of Duffing-van der Pol system with two bifurcation parameters under multi-frequency excitations

Bifurcation properties of a Duffing-van der Pol system with two parameters under multi-frequency excitations are studied. Three cases are discussed： （1） λ 1 is considered as bifurcation parameter, （2） λ 2 is considered as bifurcation parameter, and （3） λ 1 and λ 2 are both considered as bifurcation parameters. According to the definition of transition sets, the whole parametric space is divided into several different persistent regions by the transition sets for different cases. The bifurcation diagrams in different persistent regions are obtained, which provides a theoretical basis for optimal design of the system.

SINGULARLY PERTURBED SOLUTION TO SEMILINEAR HIGHER ORDER REACTION DIFFUSION EQUATIONS WITH TWO PARAMETERS

A boundary value problem for semilinear higher order reaction diffusion equa-tions with two parameters is considered. Under suitable conditions, by the theory of differential inequalities, the existence and asymptotic behavior of solutions to initial boundary value problem are studied.

A SINGULARLY PERTURBED PROBLEM OF THIRD ORDER EQUATION WITH TWO PARAMETERS

A singularly perturbed problem of third order equation with two parameters is studied. Using singular perturbation method, the structure of asymptotic solutions to the problem is discussed under three possible cases of two related small parameters. The results obtained reveal the different structures and limit behaviors of the solutions in three different cases. And in comparison with the exact solutions of the autonomous equation they are relatively perfect.

Singularly perturbed problem for non-local reaction-diffusion equations involving two small parameters

The problem of two small parameters in ordinary differential equations were extended to that in partial differential equations. The initial boundary problem for the singularly perturbed non-local reaction-diffusion equation was solved. Under suitable conditions, the formal asymptotic solutions were constructed using the method of two-step expansions and the uniform validity of the solutions was proved using the differential inequality.

QUASI-DIAGONALIZATION FOR A SINGULARLY PERTURBED DIFFERENTIAL SYSTEM WITH TWO PARAMETERS

By two successive linear transformations,a singularly perturbed differential system with two parameters is quasi-diagonalized. The method of variation of constants and the principle of contraction map are used to prove the existence of the transformations.

A class of singular perturbation solutions to semilinear equations of fourth order

A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.

SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM FOR NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS INVOLVING TWO SMALL PARAMETERS

In this paper, a class of singular perturbation of boundary value problem for nonlinear second order ordinary differential equation involving two parameters is considered. For the three cases ε/μ2→ 0(μ→0),μ2/ε→0(ε→0) and ε=μ2, uniformly valid asymptotic solutions are obtained using the method of two steps expansions and a fixed-point theorem.

Reaction Diffusion Equations for Nonlinear Boundary Conditions

A class of singularly perturbed initial boundary value problems of reaction diffusion equations for nonlinear boundary conditions is considered.Under suitable conditions,by using the theory of differential inequalities,the existence and asymptotic behavior of solution for initial boundary value problem are studied.Moreover,the obtained solution indicates that there are initial and boundary layers,and the thickness of the initial layer is less than that of the boundary layer.