Exact Solutions Of Equations Research Articles

A Class of New Exact Solutions of Navier-Stokes Equations with Body Force For Viscous Incompressible Fluid

This paper is to indicate a class of new exact solutions of the equations governing the two-dimensional steady motion of incompressible fluid of variable viscosity in the presence of body force. The class consists of the stream function $\psi$ characterized by equation $\theta=f(r)+ a \psi + b $ in polar coordinates $r$, $\theta$ , where a continuously differentiable function is $f(r)$ and $a\neq 0 , b $ are constants. The exact solutions are determined for given one component of the body force, for both the cases when $f(r)$ is arbitrary and when it is not. When $f(r)$ is arbitrary, we find $a=1$ and we can construct an infinite set of streamlines and the velocity components, viscosity function, generalized energy function and temperature distribution for the cases when $R_{e}P_{r}=1$ and when $R_{e}P_{r}\neq 1$ where $R_{e}$ represents Reynolds number and $P_{r}$Prandtl number. For the case when $f(r)$ is not arbitrary we can find solutions for the cases $R_{e}P_{r}\neq a$ and $R_{e}P_{r}=a$ where $"a"$ remains arbitrary.Â

Off-diagonal deformations of Kerr metrics and black ellipsoids in heterotic supergravity

Geometric methods for constructing exact solutions of equations of motion with first order alpha ^{prime } corrections to the heterotic supergravity action implying a nontrivial Yang–Mills sector and six-dimensional, 6-d, almost-Kähler internal spaces are studied. In 10-d spacetimes, general parametrizations for generic off-diagonal metrics, nonlinear and linear connections, and matter sources, when the equations of motion decouple in very general forms are considered. This allows us to construct a variety of exact solutions when the coefficients of fundamental geometric/physical objects depend on all higher-dimensional spacetime coordinates via corresponding classes of generating and integration functions, generalized effective sources and integration constants. Such generalized solutions are determined by generic off-diagonal metrics and nonlinear and/or linear connections; in particular, as configurations which are warped/compactified to lower dimensions and for Levi-Civita connections. The corresponding metrics can have (non-) Killing and/or Lie algebra symmetries and/or describe (1+2)-d and/or (1+3)-d domain wall configurations, with possible warping nearly almost-Kähler manifolds, with gravitational and gauge instantons for nonlinear vacuum configurations and effective polarizations of cosmological and interaction constants encoding string gravity effects. A series of examples of exact solutions describing generic off-diagonal supergravity modifications to black hole/ellipsoid and solitonic configurations are provided and analyzed. We prove that it is possible to reproduce the Kerr and other type black solutions in general relativity (with certain types of string corrections) in the 4-d case and to generalize the solutions to non-vacuum configurations in (super-) gravity/string theories.

Exact solutions of three-dimensional equations of static transversely isotropic elastic model

We found the conditions under which the system of the equations of the three-dimensional static transversely isotropic elastic model has a gradient of a harmonic function as a partial solution. In this case, parameters of the elasticity modulus tensor satisfy the Gassmann conditions. The Gassmann conditions are widely used in geophysics in the research of transversely isotropic elastic media. We fulfilled a group foliation of the system of the equations of the static transversely isotropic elastic model with the Gassmann conditions with respect to the infinite subgroup generated by the gradient of a harmonic function and contained in a normal subgroup of the main group of this system. We obtained a general solution of the automorphic system. This solution is a three-dimensional analogue of the Kolosov–Muskhelishvili formula. We found the main Lie group of transformations of the resolving system of this group foliation. With the help of this group foliation, we obtained non-degenerate exact solutions of the equations of the static transversely isotropic elastic model with the Gassmann conditions. For the found exact solutions, we have depicted the corresponding deformations arising in an elastic body for particular values of the elastic moduli.

Local fractional differential equations by the Exp-function method

Purpose – The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation. The exact solutions for the equation are successfully established. The paper aims to discuss these issues. Design/methodology/approach – Use the chain rule of the local fractional derivative and the exp-function method. Findings – Some new exact solutions for the fractional differential equation are successfully established, and the process of the solution is extremely simple and remarkably accessible. Originality/value – The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation.

Steady vortex flows of a self-gravitating gas

An exact solution of equations of steady motion of a self-gravitating gas is found. This solution describes a vortex flow of the gas from the surface of a spherical source and is partly invariant with respect to the group of rotations (Ovsyannikov vortex, singular vortex). The factor-system of the solution is reduced to finite formulas and one ordinary differential equation of the third order. Various regimes of gas motion described by this solution are determined: unlimited spreading of the gas with swirling from the surface of a spherical source and gas exhaustion with formation of a sphere with elevated density at a finite distance from this source.

Exact Solutions for a New Nonlinear KdV-Like Wave Equation Using Simplest Equation Method and Its Variants

The simplest equation method presents wide applicability to the handling of nonlinear wave equations. In this paper, we focus on the exact solution of a new nonlinear KdV-like wave equation by means of the simplest equation method, the modified simplest equation method and, the extended simplest equation method. The KdV-like wave equation was derived for solitary waves propagating on an interface (liquid-air) with wave motion induced by a harmonic forcing which is more appropriate for the study of thin film mass transfer. Thus finding the exact solutions of this equation is of great importance and interest. By these three methods, many new exact solutions of this equation are obtained.

Decomposition and exact solutions of equations of a weakly compressible barotropic fluid

Decomposition and exact solutions of equations of a weakly compressible barotropic fluid

Further Results about Traveling Wave Exact Solutions of the Drinfeld-Sokolov Equations

We employ the complex method to obtain all meromorphic exact solutions of complex Drinfeld-Sokolov equations (DS system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all constant and simply periodic traveling wave exact solutions of the equations (DS) are solitary wave solutions, the complex method is simpler than other methods and there exist simply periodic solutionsvs,3(z)which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

Third harmonic generation in the field of a backward pump wave

Third harmonic generation in a nonlinear medium the refractive index of which is negative at the pump frequency and positive at the harmonic frequency with allowance for self-action effects is considered theoretically. An exact solution of equations that describe the parametric interaction of continuous waves has been obtained analytically. The behavior of the solution is analyzed depending on the degree of violation of the phase matching condition. It is shown that, in a definite domain of values of nonlinear susceptibilities of the third order, there exists a critical value of the phase detuning at which the monotonic transform of the pump into the harmonic is changed by a periodical one. This phenomenon does not occur when the third harmonic is generated in ordinary media with a positive refractive index.

Some New Exact Solutions for Prescribed Vorticity Distribution of Couple Stress Fluids in the Presence of Unknown Body Force

In the present paper, we indicate some new exact solutions of equations of motion for plane steady incompressible couple stress fluids flows in the presence of unknown body force for which vorticity distribution is defined by Eq. (12). All solutions involve arbitrary real constants and arbitrary real constants depending on the parameter èindicating that a large number of streamfunctions and expressions for body force can be constructed. Some streamfunctions ø are constructed for some values of parameters. The streamlines patterns of these streamfunctions are presented and discussed.

EXACT SOLUTIONS OF COUPLED KdV EQUATION DERIVED FROM THE COUPLED NLS EQUATION USING MULTIPLE SCALES METHOD

In this paper, we have considered a generalized coupled higher-order nonlinear Schrödinger equations. We applied multiple scales method to coupled higher-order nonlinear Schrödinger equations to derive the coupled KdV equation. Then, the extended tanh method is applied to derived coupled KdV equation for exact solutions. The obtained results include solitary wave solutions.

Group properties and exact solutions of equations for vibrational convection of a binary mixture

A model of vibrational convection of a binary mixture with the thermodiffusion effect is considered. The symmetries of model equations are found, depending on the values of physical parameters. A partially invariant solution, which describes separation of the binary mixture in a thermodiffusion column, is constructed and studied. The influence of streamwise vibrations on the flow regime and separation of the mixture is investigated.

New Exact Solutions of (1 + 1)-Dimensional Coupled Integrable Dispersionless System

This paper applies the exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear evolution equations, to the Riccati equation, and some exact solutions of this equation are obtained. Based on the Riccati equation and its exact solutions, we find new and more general variable separation solutions with two arbitrary functions of (1+1)-dimensional coupled integrable dispersionless system. As some special examples, some new solutions can degenerate into variable separation solutions reported in open literatures. By choosing suitably two independent variables p(x) and q(t) in our solutions, the annihilation phenomena of the flat-basin soliton, arch-basin soliton, and flat-top soliton are discussed.

New infinite sequence exact solutions of nonlinear evolution equations with variable coefficients by the second kind of elliptic equation

In the paper, to construct new infinite sequence exact solutions of nonlinear evolution equations, several kinds of new solutions of the second kind of elliptic equation Bäcklund transformation are proposed. The KdV equation containing variable coefficients and forcible term, combined with (2+1)-dimensional and (3+1)-dimensional Zakharov-Kuznetsov equation with variable coefficients is taken as example to construct new infinite sequence exact solutions of these equations with the help of symbolic computation system Mathematica, which include infinite sequence compact soliton solutions of Jacobi elliptic function and triangular function, and infinite sequence peak soliton solutions.

The contribution of a lateral wave in simulating low-frequency sound fields in an irregular waveguide with a liquid bottom

For a two-dimensional irregular waveguide with a bottom in the form of a liquid half-space simulating the coastal zone of a sea shelf, we present calculations on sound propagation at certain low frequencies taking into account the contribution of the integral over the Pekeris branch cut approximated according to the Zavadsky-Krupin technique. Calculations were conducted on the basis of causal matrix equations for the modes obtained in previous studies and that are equivalent to the equations of the cross section method. It is shown how, with a lowering of sound frequency, there is an increase in the contribution to the full field of the branch-line integral corresponding to a lateral wave. Features of transformation of the first propagating mode are established as the section of the cutoff is passed; in such a situation, we have an idea of the wave pattern of the exposure region of the waveguide beyond this section, where propagating modes are absent. As earlier, we perform a comparative analysis of the curves of losses during propagation, corresponding to the solution of exact equations and the description of approximating one-way propagation when allowing for and ignoring coupling of modes.

On the velocity of the Rayleigh wave propagation along curvilinear surfaces

To investigate the propagation of Rayleigh waves on curvilinear boundaries, wave propagation along cylindrical and spherical surfaces is considered. For elastic media with indicated boundaries, exact solutions of equations of elasticity theory are constructed and the asymptotics of Hankel and Legendre functions are used. On the basis of the results obtained, a conjecture is made concerning the dependence of the velocity of the Rayleigh wave on a small curvature of the route and on a small curvature in the perpendicular direction. Bibliography: 7 titles.

Exact Solutions of Equations for the Strongly-Conductive and Weakly-Conductive Magnetic Fluid Flow in a Horizontal Rectangular Channel

This paper presents the results of exact solutions and numerical simulations of strongly-conductive and weakly-conductive magnetic fluid flows. The equations of magnetohydrodynamic (MHD) flows with different conductivity coefficients, which are independent of viscosity of fluids, are investigated in a horizontal rectangular channel under a magnetic field. The exact solutions are derived and the contours of exact solutions of the flow for magnetic induction modes are compared with numerical solutions. Also, two classes of variational functions on the flow and magnetic induction are discussed for different conductivity coefficients through the derived numerical solutions. The known results of the phenomenology of magnetohydrodynamics in a square channel with two perfectly conducting Hartmann-walls are just special cases of our results of magnetic fluid.

Relatively undistorted waves and waveforms in nonlinear acoustics: Exact solutions of equations

We demonstrate the physical conditionality and efficiency of using relatively undistorted waves as an anzats for determining exact solutions to equations of nonlinear acoustics.

Exact Solutions for a Class of Variable Coefficients Nonlinear Evolution Equations

A class of Variable Coefficients nonlinear evolution equations With . By using the general process of working out the exact solution of the nonlinear evolution equations in the homogeneous balance method, and the exact solutions of the equations are obtained.

Extended simplest equation method for nonlinear differential equations

The modified simplest equation method to look for the exact solutions of the nonlinear differential equations is presented. Our approach is applied to search for the exact solutions of the Sharma–Tasso–Olver and the Burgers–Huxley equations. The new exact solutions of these equations are obtained.