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The nonlocal nonlinear boundary value problem with turning point is considered. Leading the stretched variables the formal asymptotic solution of the original problem is constructed. And by using the theory of differential inequalities the existence and uniformly validity of solution for the original boundary value problems are proved. The contribution of this paper is that, the asymptotic behavior of solution is studied by using a simple and special method.

The impacts of initial perturbations on the computational stability of nonlinear evolution equations for non-conservative difference schemes and non-periodic boundary conditions are studied through theoretical analysis and numerical experiments for the case of onedimensional equations. The sensitivity of the difference scheme to initial values is further analyzed. The results show that the computational stability primarily depends on the form of the initial values if the difference scheme and boundary conditions are determined. Thus, the computational stability is sensitive to the initial perturbations.

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.

A class of nonlinear coupled system for E1 Niňo-southern Oscillation (ENSO) model is considered. Using the asymptotic theory and method of variational iteration, the asymptotic expansion of the solution for ENSO models is obtained.

A time delay equation for the sea–air oscillator model is studied. The aim is to create an asymptotic solving method of nonlinear equation for the El Nino–Southern Oscillation model. And based on a class of oscillators of the model, employing the method of ENSO singular perturbation, the asymptotic solution of corresponding problem is obtained. It is proven from the results that the method of singular perturbation can be used for analysing the sea surface temperature anomaly in the equatorial eastern Pacific of the atmosphere–ocean oscillation for ENSO model.

In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and its asymptotic behavior of the solution for the problem are studied.

The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered. Under suitable conditions, the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.

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.

A class of initial boundary value problems for the reaction diffusion equations are considered. The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.

The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.

For several difference schemes of linear and non-linear evolution equations, taking the one-dimensional linear and non-linear advection equations as examples, a comparative analysis for computational stability is carried out and the relationship between non-linear computational stability, the construction of difference schemes, and the form of initial values is discussed. It is proved through comparative analysis and numerical experiment that the computational stability of the difference schemes of the non-linear evolution equation are absolutely different from that of the linear evolution equation.

In this paper, we consider a more general form of weak vector variational inequalities and prove some results on the existence of solutions of our new class of weak vector variational inequalities in the setting of Hausdorff topological vector topological space. We also discuss the connectedness of a solution set to some vector variational inequalities, and obtain a useful sufficient condition.

Based on the forced dissipative nonlinear evolution equations for describing the motion of atmosphere and ocean, the computational stability of the explicit difference schemes of the forced dissipative nonlinear atmospheric and oceanic equations is analyzed and the computationally stable explicit complete square con-servative difference schemes are constructed. The theoretical analysis and numerical experiment prove that the explicit complete square conservative difference schemes are computationally stable and deserve to be disseminated.

Some problems of nonlinear computational instability are discussed in this article, which are shown as follows: 1) Three types of representative evolution equations are analyzed, and the close relationship between the nonlinear computational stability or instability in their corresponding difference equations and the properties of their solution is revealed. 2) The problem of nonlinear computational instability in conservative differencing equations with the periodic boundary condition is further discussed, and some effective ways to avoid nonlinear computational instability are proposed. 3) The problem of nonlinear computational instability in non-conservative difference equations with the aperiodic boundary condition is focused on by using nonlinear advection equations as examples, and “synthetic analysis method” is given to judge their computational stability.

The El Nino–Southern Oscillation (ENSO) is an interannual phenomenon involved in the tropical Pacific Ocean–atmosphere interactions. In this paper, an asymptotic method of solving the nonlinear equation for the ENSO model is used. And based on a class of oscillator of ENSO model, the approximate solution of a corresponding problem is studied by employing the perturbation method. Firstly, an ENSO model of nonlinear time delay equation of equatorial Pacific is introduced, Secondly, by using the perturbed method, the zeroth and first order asymptotic perturbed solutions are constructed. Finally, from the comparison of the values for a figure, it is seen that the first asymptotic perturbed solution using the perturbation method has a good accuracy. And it is proved from the results that the perturbation method can be used as an analytic operation for the sea surface temperature anomaly in the equatorial Pacific of the atmosphere-ocean oscillation for the ENSO model.

The solitary wave approximate solutions for a class of nonlinear-disturbed time delay long-wave system are considered. First, we introduce into exact solution of a non-disturbed typical long-wave system. Then, by using the homotopic mapping and an improved technique, the approximate expansions of the traveling wave solutions for the nonlinear-disturbed time delay long-wave systems are constructed.

A class of nonlinear generalized Duffing equation for disturbed oscillator is considered. Firstly, the typical Duffing equation is solved. Then approximate solutions to the nonlinear Duffing equation for disturbed oscillators in stochastic resonance is obtained using the generalized functional variation principle, and the uniform validity is proved.

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.

A class of sea-air coupled oscillator dynamic model is studied. Using the parameter modification and average method for the singular perturbation, the vibrating approximate solution corresponding to El Nino-Southern Oscillation (ENSO) coupled oscillation dynamic model equations is obtained.

The approximate solitary-like wave solution for a class of disturbed evolution equation is considered using a simple and valid technique. First, an approximate solution of a corresponding typical differential equation is introduced, and then the approximate solution for an original disturbed evolution equation is obtained using the functional mapping method. It is pointed out that the series of approximate solution is convergent. The accuracy of the approximate solution is also discussed using the analytic method.

A class of delayed oscillator of El Nio-Southern Oscillation (ENSO) coupling systems is considered. Using the delayed theory, perturbation and other methods, the asymptotic expansion of the solution for ENSO model is obtained and the asymptotic behavior of solution for the corresponding problem is studied.

A solving problem for the Lorenz system in atmospheric physics is considered. First, a set of variational iterations is constructed by using the generalized variation principle. Then, the initial approximate solution is determined. Finally, using the variational iteration, each approximate solution for corresponding model is found. The generalized variational iteration method is an analytic method, and the obtained solution can be analytically operated further.

A reduces equation of the Kelvin wave is considered. By using the homotopic mapping solving method, the approximate solution is obtained. The homptopic mapping method is an analytic method, the obtained solution can analyse operations sequentially.

A box model of the interhemispheric thermohaline circulation in the disturbed sea-air for globe climate is considered. By using the homotopic mapping method，a class of nonlinear model is studied, and the approximate solution is obtained. The homotopic mapping solving method is an analytic method，the solution obtained can be analytically operated sequentially. And then we can also study the diversified qualitative and quantitative behaviors of corresponding physical quantities.

An equation for the equatoral Kelvin wave is considered. By using the variational iteration method, the approximate solution is studied. The variational iteration method is an analytic method, the obtained solution can be analytic operation sequentially.

The model for gain fluence of a laser pulse amplifier is studied. Firstly, using the generalized variational iteration method, the Lagrange multipicator is determined. Then the iteration expansions are constructed. Thus approximate solutions for the corresponding model are obtained.

The problem of solving the Lorenz equation is considered. Firstly，a set of homotopic mapping is constructed. Then the initial approximate solution is determined. Finally，using the homotopic mapping，the approximation for corresponding model is found. The homotopic mapping method is an analytic method，the obtained solution can be used in analytic operations subsequently.

A three dimentional equation for the equatoral sea-air oscillator Kelvin wave is considered. By using the variational iteration method，the approximate solution of the corresponding model is studied. The variational iteration method is an analytic method，the obtained solution can be used in further analytical operations.

A class of coupled systems in oscillation of the El Nio-Southern Oscillation (ENSO) mechanism has been studied. Using the homotopic mapping and perturbed theory, approximate expansions of the solution for a class of ENSO model are obtained in a simple manner.

The El Nio atmospheric physics mechanism is studied. Using the variational iteration theory, the approximation of the solution for its model is obtained for shortcut calculation. And compared to the special case for the exact solution of the model, the first approximate solution from the variational iteration method is in excellent agreement with the exact solution in degree of accuracy.