Asymptotic solution for a class of weakly nonlinear singularly perturbed reaction diffusion problem

Under appropriate conditions, with the perturbation method and the theory of differential inequalities, a class of weakly nonlinear singularly perturbed reaction diffusion problem is considered. The existence of solution of the original problem is proved by constructing the auxiliary functions. The uniformly valid asymptotic expansions of the solution for arbitrary mth order approximation are obtained through constructing the formal solutions of the original problem, expanding the nonlinear terms to the power in small parameter ε and comparing the coefficient for the same powers of ε. Finally, an example is provided, resulting in the error of 0（ε^2）.

Asymptotic solution of a weak nonlinear model for the mid-latitude stationary wind field of a two-layer barotropic ocean

A weak nonlinear model of a two-layer barotropic ocean with Rayleigh dissipation is built.The analytic asymptotic solution is derived in the mid-latitude stationary wind field,and the physical meaning of the corresponding problem is discussed.

Asymptotic solution for a perturbed mechanism of western boundary undercurrents in the Pacific

This paper consider a class of perturbed mechanism for the western boundary undercurrents in the Pacific. The model of generalized governing equations is studied. Using the perturbation method, it constructs the asymptotic solution of the model. And the accuracy of asymptotic solution is proved by the theory of differential inequalities. Thus the uniformly valid asymptotic expansions of the solution are obtained.

ON THE ASYMPTOTIC SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR A CLASS OF SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS (Ⅰ)

A new method is applied to study the asymptotic behavior of solutions of boundary value problems for a class of systems of nonlinear differential equations u' = nu, epsilon nu' + f(x, u, u')nu' - g(x, u, u') nu = 0 (0 < epsilon much less than 1). The asymptotic expansions of solutions are constructed, the remainders are estimated. The former works are improved and generalized.

ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS

In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution.

Asymptotic solutions for the asymmetric flow in a channel with porous retractable walls under a transverse magnetic field

The self-similarity solutions of the Navier-Stokes equations are constructed for an incompressible laminar flow through a uniformly porous channel with retractable walls under a transverse magnetic field. The flow is driven by the expanding or contracting walls with different permeability. The velocities of the asymmetric flow at the upper and lower walls are different in not only the magnitude but also the direction. The asymptotic solutions are well constructed with the method of boundary layer correction in two cases with large Reynolds numbers, i.e., both walls of the channel are with suction, and one of the walls is with injection while the other one is with suction. For small Reynolds number cases, the double perturbation method is used to construct the asymptotic solution. All the asymptotic results are finally verified by numerical results.

ASYMPTOTIC SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH TURNING POINTS

Boundary value problem; for third-order ordinary differential equations with turning points are studied as follows : epsilon gamma ' ' + f(x ; epsilon) gamma ' + g(x ; epsilon) gamma ' +h(x ; epsilon) gamma = 0 (- a < x < b, 0 epsilon 1), where f(x ; 0) has several multiple zero points in ( - n, b). the necessary conditions for exhibiting resonance is given, and the uniformly valid asymptotic solutions and the estimations of remainder terms are obtained.

UNIFORMLY VALID ASYMPTOTIC SOLUTIONS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR ORTHOTROPIC RECTANGULAR THIN PLATE OF FOUR CLAMPED EDGES WITH VARIABLE THICKNESS

By using “the method of modified two-variable”,“the method of mixing perturbation” and introducing four small parameters,the problem of the nonlinear unsymmetrical bending for orthotropic rectangular thin plate with linear variable thickness is studied.And the uniformly valid asymptotic solution of Nth-order for ε_1 and Mth-order for ε_2 of the deflection functions and stress function are obtained.

HIGHER ORDER ASYMPTOTIC SOLUTION OF MODE Ⅰ CRACK GROWTH IN ELASTIC-PERFECTLY PLASTIC MEDIA

A higher order asymptotic solution of near-tip field is studied for plane-atrain Mode-Ⅰ quasi-static steady crack growth in the incompressible (v=1/2) elastic perfectly-plastic media.The results show that the near-tip stress and strain are fully continuous,and the strain possesses In (A/r) singularity at the crack tip.The expressions of the stress,strain and velocity in each region are also given.

A COMPLETE EXPRESSION OF THE ASYMPTOTIC SOLUTION OF DIFFERENTIAL EQUATION WITH THREE_TURNING POINTS

In this paper, a second order linear ordinary differential equation with three-turning points is studied. This equation is as follows (dx2)/(d2y) + [lambda(2)q(1)(x) + lambda q(2)(x, lambda)]y = 0, where q(1) (x) = (x - mu(1))(x - mu(2))(x - mu(3))f(x), f(x) not equal 0, mu(1) < mu(2) < mu(3) and lambda is a large parameter, but q(2)(x, lambda) = Sigma(i = 0)(epsilon a) g(i)(x)lambda(-i) (here g(0)(x) not equivalent to 0). By using JL function, the complete expression of the formal uniformly, valid asymptotic solutions of the equation near turning point is obtained.

AN ASYMPTOTIC SOLUTION OF THE NONLINEAR REDUCED WAVE EQUATION

This paper uses the boundary layer theory to obtain an asymptotic solution of the nonlinear educed wave equation. This solution is valid in the secular region where the geometrical optics result fails. However it agrees with the geometrical optics result when the field is away from the secular region. By using this solution the self-focusing length can also be obtained.

ASYMPTOTIC SOLUTION FOR SINGULAR PERTURBATION PROBLEMS OF DIFFERENCE EQUATION

In this paper, we constructed a new asymptotic method for singular perturbation problems of difference equation with a small parameter.

ON THE ASYMPTOTIC SOLUTIONS FOR A CLASS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SLOWLY VARYING COEFFICIENTS

In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted, and the results in [1 - 4] are improved and extended by means of the modified method of multiple scales.

AN ASYMPTOTIC SOLUTION OF VELOCITY FIELD IN SHIP WAVES

The stationary phase method in conventional Lighthill＇s two-stage scheme to get the expressions of the velocity field was given up in this paper. The method that Ursell had used in deducing the elevation expression of ship wave was adopted, and an asymptotic solution of velocity field of ship waves on an inviscid fluid that is perfectly fit for the region inside and outside the critical lines was obtained. It is very convenient to be used in SAR technique.

ASYMPTOTIC SOLUTION FOR A CLASS OF WEAKLY NONLINEAR SINGULARLY PERTURBED REACTION DIFFUSION EQUATION

In this paper a class of nonlinear reaction diffusion problems are considered. Using the perturbed method, the asymptotic solution for corresponding problem is obtained.

A PDE Approach to the Long-Time Asymptotic Solutions of Contact Hamilton-Jacobi Equations

We study the long-time asymptotic behaviour of viscosity solutions u(x,t)of the Hamilton-Jacobi equation u_(t)(x,t)+H(x,u(x,t),Du(x,t))=0 in T^(n)×(0,∞)with a PDE approach,where H=H(x,u,p)is coercive in p,non-decreasing in u and strictly convex in(u,p),and establish the uniform convergence of u(x,t)to an asymptotic solution u_(∞)(x)as t→∞.Moreover,u_(∞) is a viscosity solution of Hamilton-Jacobi equation H(x,u(x),Du(x))=0.

A COMPLETE EXPRESSION OF THE ASYMPTOTIC SOLUTION OF DIFFERENTIAL EQUATION WITH A N－TH ORDER TURNING POINT

A second order linear ordinary differential equation has been studied,and thecomplete expression.of the formal uniformly valid asymptotic solutions to the equationnear turning point is obtained by using extended Airy function.

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR NONLOCAL REACTION-DIFFUSION SYSTEM

This paper deals with reaction-diffusion system with nonlocal source. It is proved that there exists a unique classical solution and the solution either exists globally or blows up in finite time. Furthermore, its blow-up set and asymptotic behavior are obtained provided that the solution blows up in finite time.

FLUID-SOLID COUPLING MATHEMATICAL MODEL OF CONTAMINANT TRANSPORT IN UNSATURATED ZONE AND ITS ASYMPTOTICAL SOLUTION

The process of contaminant transport is a problem of multicomponent and multiphase flow in unsaturated zone. Under the presupposition that gas existence affects water transport, a coupled mathematical model of contaminant transport in unsaturated zone has been established based on fluid_solid interaction mechanics theory. The asymptotical solutions to the nonlinear coupling mathematical model were accomplished by the perturbation and integral transformation method. The distribution law of pore pressure, pore water velocity and contaminant concentration in unsaturated zone has been presented under the conditions of with coupling and without coupling gas phase. An example problem was used to provide a quantitative verification and validation of the model. The asymptotical solution was compared with Faust model solution. The comparison results show reasonable agreement between asymptotical solution and Faust solution, and the gas effect and media deformation has a large impact on the contaminant transport. The theoretical basis is provided for forecasting contaminant transport and the determination of the relationship among pressure_saturation_permeability in laboratory.

Asymptotical solutions of coupled nonlinear Schrodinger equations with perturbations

In this paper Lou＇s direct perturbation method is applied to the perturbed coupled nonlinear Schrodinger equations to obtain their asymptotical solutions, which include not only the zero-order solutions but also the first-order modifications. Based on the asymptotical solutions, the effects of perturbations on soliton parameters and the collision between two solitons are then discussed in brief. Furthermore, we directly simulate the perturbed coupled nonlinear SchrSdinger equations by split-step Fourier method to check the validity of the direct perturbation method. It turns out that our analytical results are well supported by the numerical calculations.