Exact Solutions Of Nonlinear Equations Research Articles

Exact solution of a KdV equation with variable coefficients

A simple direct method is developed for finding exact solutions of nonlinear equations with variable coefficients. The (1 + 1)-dimensional KdV equation is used as an example to elucidate the solution procedure, and its exact solution is obtained.

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.

New solitons and periodic wave solutions for some nonlinear physical models by using the sine–cosine method

In this paper, we establish exact solutions for nonlinear equations. The sine–cosine method is used to construct periodic and soliton solutions of nonlinear physical models. Many new families of exact travelling wave solutions of the symmetric regularized long-wave (SRLW) and the Klein–Gordon–Zakharov (KGZ) equations are successfully obtained. These solutions may be important for the explanation of some practical physical problems. It is shown that the sine–cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

Study of exact solutions of nonlinear heat equations

One of the ways that energy transports in fluid is electron thermal conduction. The aim of Inertial Confinement Fusion (ICF) is to show that the thermal conductivity is strongly dependent on temperature and the equation of heat condition is a nonlinear equation. In this article, we analyze the exact solutions of the nonlinear equation of heat conduction problem with variable transfer coefficients which is the problem of ICF.

Self-diffraction of light waves by a local photorefractive grating in a crystal of the symmetry group bar 4̄3m in the transverse configuration

The exact solution of nonlinear equations describing the symmetric stationary interaction of two light waves of an arbitrary polarisation on a local photorefractive grating in a crystal of the symmetry group 4̄3m in the transverse configuration of crystal faces is obtained in the paraxial limit. The polarisation of light waves and efficiency of energy transfer between them are studied as functions of the reduced length of a sample and the intensity ratio of incident waves.

New classes of exact solutions to general nonlinear diffusion-kinetic equations

New classes of exact solutions of nonlinear equations and systems of equations that are encountered in the mass and heat transfer theory of reacting media and mathematical biology are described. General equations and systems of equations where the rates of chemical reactions depend on one or several arbitrary functions are the focus of attention. Solutions with functional arbitrariness are found (they are expressed through the solutions of a linear heat transfer equation and the solutions of nonlinear systems of ordinary differential equations); solutions with generalized separation of variables and other solutions are described. The transfer coefficients and the kinetics of chemical reactions are considered as functions of temperature (concentration).

Exact solutions of nonlinear equations

Based on the homogeneous balance method, a general method is suggested to obtain several kinds of exact solutions for some kinds of nonlinear equations. The validity and reliability of the method is tested by applying it to the Bousseneq equation.

The Nonclassical Group Analysis of the Heat Equation

The nonclassical method of reduction was devised originally by Bluman and Cole in 1969, to find new exact solutions of the heat equation. Much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance. However, the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved, although particular solutions have been given. Recently, Arrigo, Goard, and Broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical Lie symmetry reduction solutions. Their arguments leave open the problem of what is the general nonclassical group action, and its effect on the relevant solution of the heat equation. In this article, both these problems are solved. In the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of Burgers' equation.

Lie symmetries and exact solutions of nonlinear equations of heat conductivity with convection term

The Lie symmetries of nonlinear diffusion equations with convection term are completely described. The Lie ansatzes and exact solutions of a certain nonlinear generalization of the Murray equation are constructed. An example of the family of non-Lie solutions of the Murray equation is given.

A constructive method for obtaining new exact solutions of nonlinear evolution equations

New constructive method is suggested for obtaining exact solutions of nonlinear equations. The method is based on the consideration of a fixed nonlinear PDE (system of PDEs) together with additional condition in the form of a linear ODE (system of ODEs). With the help of this method new exact solutions are obtained for a nonlinear generalizations of the Bürgers equations and some nonlinear evolution systems, which describe real processes in physics and chemistry.

Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics

The solitary wave solutions of the approximate equations for long water waves, the coupled KdV equations and the dispersive long wave equations in 2 + 1 dimensions are constructed by using a homogeneous balance method.

Exact solutions of nonlinear equations and the method of operator identities

Exact solutions and wave functions of integrable nonlinear equations are constructed by the method of operator identities.

Exact solutions of a family of Fisher equations

A family of Kolmogorov-Petrovskii-Piskunov-Fisher equations is considered. Transformations of solutions that generalize devices used earlier to find special classes of exact solutions of nonlinear equations are proposed. Exact solutions of two types of Fisher equation are obtained.

On exact solutions of nonlinear integrable equations via integral linearising transforms and generalised Backlund-Darboux transformations

A new approach for constructing the exact solutions of nonlinear equations via integral transforms which convert them into their linear limits is discussed. Generalised Backlund-Darboux transformations are introduced.

Application of Finite Expansion in Elliptic Functions to Solve Differential Equations

In order to solve certain class of ordinary differential equation, its solution is expanded in finite power series of p ( x ) and d p /d x , where p ( x ) is any function satisfying (d p /d x ) 2 =Σ k b k p ( x ) k . This power series systematically yields exact solutions of non-linear equations, such as the 5th KdV, Kuramoto, Fisher and other dispersive and diffusive equation.

Turbulence measurements at low velocities with symmetrically bent V-shaped hot-wires. (2nd report Measuring velocity component fluctuations)

A method of turbulence mesurements at low velocities with V-shaped hot-wires is described along with associated probe arrangements. The present method is based on the simultaneous solution of exact nonlinear equations for the effective cooling velocities, thus providing instantaneous values of velocity components. The validity of the technique was tested for the measurement of turbulent velocity components which were artificially produced by shaking the hot-wire probe with an accurately known motion in a steady flow. Instantaneous velocities obtained with the proposed method were in almost perfect agreement with the corresponding imposed values.

Dissipation mechanics and exact solutions for nonlinear equations of dissipative type—Principle and application of dissipation mechanics (I)

This work is the continuation and the distillation of the discussion of Refs. [1-4]. (A) From complementarity principle we build up dissipation mechanics in this paper. It is a dissipative theory of correspondence with the quantum mechanics. From this theory we can unitedly handle problems of macroscopic non-equilibrium thermodynamics and viscous hydrodynamics, and handle each dissipative and irreversible problems in quantum mechanics. We prove the basic equations of dissipation mechanics to eigenvalues equations of correspondence with the Schrodinger equation or Dirac equation in this paper. (B) We unitedly merge the basic nonlinear equations of dissipative type, especially the Navier-Stokes equation as a basic equation for macroscopic non-equilibrium thermodynamics and viscous hydrodynamics into integrability condition of basic equation of dissipation mechanics. And we can obtain their exact solutions by the inverse scattering method in this paper.

Additional nonperturbative contributions to pion-pion scattering

A cross-section for pion-pion scattering is calculated which includes contributions made by exact solutions of nonlinear equations. Results of this calculation are compared with cross-sections extrapolated from experimental data.

On the Exact Solution of Certain Nonlinear Dispersive Wave Equations

The exact solutions of some nonlinear equations with higher order dispersive terms have been found using the properties of the Weierstrass elliptic function. The equations have been generalized to the (2 r +1) th order which will have exact solutions.

Locally ?simple? unsteady deformations

Various procedures by which exact solutions of nonlinear equations may be used as local approximations to more general deformations are described. The theory of modulated simple waves and the Whitham/Luke description of modulations in periodic wavetrains are compared. The role of material idealisations is also discussed. Incompressibility and fibre inextensibility introduce kinematic constraints which lead to simply described deformations. These are local approximations to nearly incompressible and almost inextensible deformations.