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Using the improved LP method, the authors seek to salve a class of problems of square strongly nonlinear free oscillations and of strongly nonlinear nonoscillations . Their first-order approximate solutions which has high accuracy are obtained. The method of this paper is different from the known L-P methods.

In this paper, using the integration method, it is sought to solve the problem for the laminar boundary-layer on a flat plate. At first, a trial function of the velocity profile which satisfies the basical boundary conditions is selected. The coefficients in the trial function awaiting decision are decided by using some numerical results of the boundary-layer differential equations. It is similar to the method proposed by Peng Yichuan, but the former is simpler. According to the method proposed by Peng, when the awaiting decision coefficients of the trial function are decided, it is sought to solve a third power algebraic equation. On the other hand, in this paper, there is only need for solving a linear algebraic equation. Moreover, the accuracy of the results of this paper is higher than that of Peng.

In this paper, using the interpolation perturbation method, the author seeks to solve several nonlinear problems. Numerical examples show that the method of this paper has good accuracy.

When there is small quantity term contained in differential equation, we may solve it by using the M. E. Shvez iterative method. If the small quantity term has singularity, or in the certain region where the small quantity term is not small, using this method to solve such differential equation, we may meet difficulties. In this paper, the author improves the M. E. Shvez method. Several examples are given to demonstrate the algorithm.

In this paper, on the basis of Ref. [1], the author studies the boundary value problems of the second-order differential equations, the highest order derivatives of which contain the small parameters. The numerical examples show that the calculating process of this method is quite simple and its accuracy is even higher than that of the multiple scales method.

In reference [1], the wave solutions of nonlinear heat conduction equation are studied. In it, the similar variable ξ is wave variable and it is assumed that the heat conduction coefficient is only the function of the similar variable ξ.

In this paper, the author obtains the more general displacement solutions for the isotropic plane elasticity problems. The general solution obtained in ref. [1] is merely the particular case of this paper. In comparison with ref. [1], the general solutions of this paper contain more arbitrary constants. Thus they may satisfy more boundary conditions.

The present article researches the problems of the lateral instability of cantilever rectangular plates under a concentrated force or a uniformly distributed load respectively. We select the polynomial (2.1) instead of the cosecant function in Ref. [1] as the flexural functions. The minimum critical load obtained here is more exact than the results obtained in Ref.[1]

Using the pressure gradient as the new variable instead of the ordinary longitudinal coordinate x. Liu transformed the ordinary laminar boundary equations into a new form. On this base Liu obtained the frictional stress factor by using the graphical method.

In this paper, we propose some formulas for seeking a part of the particular solutions of the heavy harmonics and Tricomi equations and obtain the precise polynomial solutions of the finite-item number for rectangular strip bending problem when the intensity of the distributed load varies with the fourth, power of longitudinal coordinote.

In ref. l. under the condition that the components of velocity are only the functions of time and polar angle 0. Dvornikov solved eqs. (1.1)–(1.3) of the ideal gas unsteady planar parallel potential flow. It was pointed out in ref.[1] that in general cases, the evident solutions could not be obtained. Only for two especial cases, the evident solutions were obtained. In this paper, the author studies the same problem as that in ref. [1]. In the first section we obtain the evident solution of equations (1.1)–(1.3) under the condition that the sonic velocity is restricted by some complemental conditions. In the second section, we obtain the first-order approximate solutions of the fundamental equation for the case that

Chernyi's series method[1] is not proper for the case that(γ−1)/(γ+1)〈2/(γ+1)×M2sin2Β (γ=cp/cv — adiabatic index number, M — Mach number, Β — shock incidence). In this paper, we only suppose that in the neighbour of the shock, there exists a shock layer in which the density of the gas is very big, but we do not remove the case that (γ−1)/(γ+1)〈2/(γ+1)M2sin2Β.

In [1], under the condition that all the perfectly plastic stress components at a crack tip are functions of ϕ only, making use of equilibrium equations, stress-strain rate relations, compatibility equations and yield condition. Lin derived the general analytical expressions of the perfectly plastic stress field at a mixed-mode crack tip under plane and anti-plane strain. But in [1] there were several restrictions on the proportionality factor γ in the stress-strain rate relations, such as supposing that γ is independent of ϕ and supposing that γ=c or cr−1. In this paper, we abolish these restrictions. The cases in [1], γ=crd (n=0 or-1) are the special cases of this paper.

For the practical metallic material, preceding the fracture, there always exist the domains of plasticity. For the plasticity limited in scope, the linear elastic mechanics is still applicable, but it is neccesary to correct the influence of the domain of plasticity. The traditional method of correction (Irwin’s method) is to introduce the conception of the effective fracture length, namely, owing to the existence of the domain of plasticity, the practical length of the fracture increases. If we take the length of the fracture to be equal to the effective fracture length 2(a + ry), where 2a is the original length of the fracture and 2ry is the added length, we may not regard the existence of the domain of plasticity and still use the linear elastic fracture mechanics to deal with.

In ref. [1], Lin obtained an approximate analytical solution of the ideal potential flow around a circular cylinder between two parallel flat flates.

In this paper, we obtain a third-order approximate solution for the laminar boundary layer between two planes perpendicular to each other. In boundary layer equations, the viscous and the inertial terms have the same quantity step. In this paper, at first, supposing that the inertial terms are bigger than the viscous terms, we solve the boundary layer equations, and then we suppose that the viscous terms are bigger than the inertial terms. At last, we take the mean value as the valid solution of the boundary layer equations. The first- and the second-order approximate solutions obtained in this paper coincide with the results in ref. [1], while the third-order solution obtained in this paper is better than that in ref. [1].

In ref. [1], V. E. Najenov studied the conditions that when the viscosity of the liquid is an exponential function of temperature, the pipe flow, having steady heat transfer, is one dimensional and with nonuniform temperature. For plane canal and circular pipe he still studied the velocity and the temperature fields.

In this paper, the same problem in ref. [1] is studied. The author’s solution approximately satisfies the whole fundamental equations (1.1) and (1.2) and the whole hound values conditions (1.3-1.5). But the Liu’s solution[1] does not satisfy the equation of continuity (1.2).