Homogenous Balance Method and Exact Analytical Solutions for Whitham-Broer-Kaup Equations in Shallow Water

Based on the homogenous balance method and with the help of mathematica, the Backlund transformation and the transfer heat equation are derived. Analyzing the heat-transfer equation, the multiple soliton solutions and other exact analytical solution for Whitham-Broer-Kaup equations(WBK) are derived. These solutions contain Fan's, Xie's and Yan's results and other new types of analytical solutions, such as rational function solutions and periodic solutions. The method can also be applied to solve more nonlinear differential equations.

THE EXACT ANALYTICAL SOLUTIONS TO THE NEW MODEL OF RESERVOIR FILTRATION PROBLEM

The present study has obtained the new model of the reservoir filtration problem by taking into account the effect of wellbore storage and skin and by making use of the coupled equations of doubled porous media filtration and consequently has got, through various forms of limits, the exact analytical solutions of the three common reservoirs (fissure, homogeneous and the two-layered) pressure distribution under the conditions of three boundaries, i.e., infinite boundary, sealed finite boundary and the finite boundary at constant pressures.

STUDY ON EXACT ANALYTICAL SOLUTIONS FOR TWO SYSTEMS OF NONLINEAR EVOLUTION EQUATIONS

The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.

Exact analytical solutions to the mean-field model depicting microcavity containing semiconductor quantum wells

By using a two-mode mean-field approximation, we study the dynamics of the microcavities containing semiconductor quantum wells. The exact analytical solutions are obtained in this study. Based on these solutions, we show that the emission from the microcavity manifests periodic oscillation behaviour and the oscillation can be suppressed under a certain condition.

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.

Analytical Study of Nonlinear Dust Acoustic Waves in Two-Dimensional Dust Plasma with Dust Charge Variation

The nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation is analytically investigated by using the formally variable separation approach. New analytical solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma for the first time. We derive exact analytical expressions for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation.

Exact Analytic Solutions for the Caudrey Dodd-Gibbon-Kotera-Sawada Equation

The Caudrey-Dodd-Gibbon-Kotera Sawada (CDGKS) equation has attracted many physicists and mathematicians. In this paper, based on the idea of variable-coefficient balancing-act method and the computerized .symbolic compu tation, some exact analytic solutions for the CDGKS equation have been obtained.

AN EXACT METHOD OF BENDING OF ELASTIC THIN PLATES WITH ARBITRARY SHAPE

This paper presents a new method exactly to solve the bending of elastic thinplates with arbitrary shape. First the analytic solution of differential equation ofelastic thin plate is derived in polar coordinate, then the analytic solution is substituted into the boundary conditions of elastic thin plate with arbitrary shape. The boundaryequations are expanded along the boundary by the use of Fourier series, all unknown coefficients can be decided. The results are exact.

Six types of spin solitons in three-component Bose-Einstein condensates

Exact analytical solutions are good candidates for studying and explaining the dynamics of solitons in nonlinear systems.We further extend the region of existence of spin solitons in the nonlinearity coefficient space for the spin-1 Bose-Einstein condensate.Six types of spin soliton solutions can be obtained,and they exist in different regions.Stability analysis and numerical simulation results indicate that three types of spin solitons are stable against weak noise.The non-integrable properties of the model can induce shape oscillation and increase in speed after the collision between two spin solitons.These results further enrich the soliton family for non-integrable models and can provide theoretical references for experimental studies.

An exact analytical solution for convective heat transfer in rectangular ducts

An exact analytical solution is obtained for convective heat transfer in straight ducts with rectangular cross-sections for the first time.This solution is valid for both H1 and H2 boundary conditions,which are related to fully developed convective heat transfer under constant heat flux at the duct walls.The separation of variables method and various other mathematical techniques are used to find the closed form of the temperature distribution.The local and mean Nusselt numbers are also obtained as functions of the aspect ratio.A new physical constraint is presented to solve the Neumann problem in non-dimensional analysis for the H2 boundary conditions.This is one of the major innovations of the current study.The analytical results indicate a singularity occurs at a critical aspect ratio of 2.4912 when calculating the local and mean Nusselt numbers.