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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.

In this paper, a 3D shape measurement system based on a digital fringe projection and phase-shifting technique was described. The fringe generated by a computer was projected onto the object by utilizing a digital video projector. Four steps phase shifting algorithm was used to calculate the 2π wrapping phase, and a heterodyne principle was used to unwrap the phase to obtain the real phase. Then the 3D coordinates object was calculated base on stereo vision algorithm by using the real phase as a constraint. Experiments demonstrated that the system was not sensitive to the phase error caused by the DVP’s nonlinearity.

In this paper, the influence of the El Nnio-Southern Oscillation (ENSO) cycle on the sensitivity of nonlinear factors in the numerical simulation is investigated by conducting numerical experiments in a simple air-sea coupled model for ENSO prediction. Two sets of experiments are conducted in which zonal nonlinear factors, meridional nonlinear factors, or both are incorporated into the governing equations for the atmosphere or ocean. The results suggest that the ENSO cycle is very sensitive to the nonlinear factor of the governing equation for the atmosphere or ocean. Thus, incorporating nonlinearity into air-sea coupled models is of exclusive importance for improving ENSO simulation.

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.

The EI Nino and Southern Oscillation (ENSO) is an interannual phenomenon involved in the tropical Pacific sea–air interactions. In this paper, an asymptotic method of solving nonlinear equations for the ENSO model is proposed. And based on a class of oscillator of the ENSO model and by employing the method of homotopic mapping, the approximate solution of equations for the corresponding ENSO model is studied. It is proved from the results that homotopic method can be used for analysing the sea surface temperature anomaly in the equatorial Pacific of the sea–air oscillator for the ENSO model.

A class of nonlinear global climate oscillation models is considered. Using perturbation theory and its methods, solutions to the asymptotic expansions of some related problems are constructed. These asymptotic ex- pansions of the solutions for the original problem possess a higher approximation. The perturbed asymptotic method is an analyt method. ic

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.

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.

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 the nonconservative schemes of the nonlinear evolution equations, taking the one-dimensional shallow water wave equation as an example, the necessary conditions of computational stability are given. Based on numerical tests, the relationship between the nonlinear computational stability and the construction of difference schemes, as well as the form of initial values, is further discussed. It is proved through both theoretical analysis and numerical tests that if the construction of difference schemes is definite, the computational stability of nonconservative schemes is decided by the form of initial values.

For the conservative and non-conservative schemes of nonlinear evolution equations, by taking the two-dimensional shallow water wave equations as an example, a comparative analysis on computational stability is carried out. The relationship between the nonlinear computational stability, the structure of the difference schemes, and the form of initial values is also discussed.

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.

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.

The computational stability of the explicit difference schemes of the forced dissipative nonlinear evolution equations is analyzed and the computational quasi-stability criterion of explicit difference schemes of the forced dissipative nonlinear atmospheric equations is obtained on account of the concept of computational quasi-stability, Therefore, it provides the new train of thought and theoretical basis for designing computational stable difference scheme of the forced dissipative nonlinear atmospheric equations.

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.

A time delay equation for sea-air oscillator model is studied. The aim is to create an asymptotic soling method of nonlinear equation for the ENSO model. And based on a class of oscillators of ENSO model, employing the variational iteration method, the approximate solution of corresponding problem is obtained. It is proven from the results that the method of variational iteration method can be used for analyzing the sea surface temperature anomaly in the equatorial eastern Pacific of the atmosphere-ocean oscillation for ENSO model. *Supported by National Natural Science Foundation of China (Grant Nos. 40576012, 90111011), the National Program on Key Basic Research Project (Grant No. 2004CB418304) and in part by E-Institute of Shangai Municipal Education Commission (Grant No. E03004)

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.

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.

The EI Nino-southern oscillation (ENSO) is an interannual phenomenon involved in tropical Pacific ocean-atmosphere interactions. In this paper, we develop an asymptotic method of solving the nonlinear equation using the ENSO model. Based on a class of the oscillator of the ENSO model, a approximate solution of the corresponding problem is studied employing the perturbation method.

A class of nonlinear system is studied. Firstly, the reaction-diffusion system with speed, temperature and humidity for a atmospheric nonlinear force dissipative system is discussed using the atmospheric nonlinear theory of thermodynamics and dynamics. Secondly, the small disturbed solution of atmospheric nonlinear reaction-diffusion system in the neighborhood of homogeneous steady state solution is obtained from the Lyapunov stability theory. Finally, from the variations of the control parameters for transgenic processes, the states of ordered-unordered-ordered processes of atmospheric nonlinear reaction-diffusion system are found. Thus the corresponding local atmospheric nonlinear force dissipative system can be predicated and calculated.

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.

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.

In this paper, a class of wading wave equations are considered. Using the homotopic mapping theory and method, the corresponding approximate solution of the system is obtained.

A time delay equation for sea-air oscillator model is studied. We aim at the cre ateion of an approximate method for solving sea-air oscillator model of nonlinea r equation, employing the theory of homotopic mapping, with due colisideration o f the approximate solution of the corresponding problem. This is an approximate analytical method, which can be further used for analyzing other behaviors of th e sea surface temperature anomaly of the atmosphere-ocean oscillator model.

Atmospheric physics deals with very complicated natural phenomena. Basic models have to be implemented for the sea-air and land-air oscillators and then solve them using the approximate method. In this paper, a class of nonlinear coupling system for land-air oscillator model is studied. Using the homotopic mapping method, it is translated into solving linear problems, and the approximate expressions for the original nonlinear problem are obtained.

A sea-air oscillator model of the nonlinear coupling system is studied.Using the variational iteration principle,a series of corresponding functional is constructed.Then the Lagrange operators are selected.Finally,using the generalized variational iteration method,the approximate sequences of a series of solutions for the sea-air oscillator model are obtained.

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.

A class of models of equator sea-air coupled oscillators is studied. Using the perturbative method the asymptotic solution of the corresponding model is obtained.

The oscillation for a class of coupled systems in the El Nio-Southern Oscillatio n (ENSO) mechanism is studied. Using the perturbation theory and perturbative me thod, the uniformly valid asymptotic expansions of the solution for a class of E NSO model are obtained easily.

A class of ENSO perturbed model is studied. Using the perturbation theory and method, the asymptotic expansions of solution of corresponding problem are constructed.