Some exact solutions of an elastico-viscous fluid

The exact solutions for the unsteady flow of an elastico-viscous fluid caused by general periodic oscillations are obtained. Further, the plate is assumed to be rigid as well as porous executing periodic rotary oscillations. The velocity field and shear stress are established.     

Some exact solutions of the variable coefficient Schrodinger equation

By constructing appropriate transformations, we find some exact solutions of variable coefficient Schrodinger equation, which include elliptic function and trigonometric solutions and so on.     

Some exact solutions for nonlinear desorptive diffusion

The existence is pointed out of a class of exact solutions for desorption which can be obtained by means of a simple inversion method. This method is related to that proposed by Philip (1960) for sorption and redistribution and it consists of the inversion of the diffusivity equation of Matano (1933) and Bruce and Klute (1952). Some of the exact solutions can be useful in the study of problems encountered in flow in porous media.

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Résumé On démontre l'existence d'une classe de solutions exactes pour le problème de désorption qui peuvent être obtenues par une méthode d'inversion. Cette méthode correspond à la méthode proposée par Philip (1960) pour sorption et rédistribution, et elle consiste de l'inversion de l'équation pour la diffusivité de Matano (1933) et de Bruce et Klute (1952). Quelques unes de ces solutions peuvent être utiles dans l'étude du mouvement de fluides en milieux poreux.

     

Black holes in the brane world: Some exact solutions

We briefly review methods for and results from constructing multidimensional solutions with an event horizon generated by matter localized on a brane. We investigate a particular exact solution of the black star type with the Schwarzschild-Tangherlini geometry in the bulk. We study the extrinsic curvature tensor, which predicts the matter distribution on the brane in accordance with the Israel conditions. We calculate all components of the four-dimensional Riemann tensor describing the intrinsic geometry on the brane surface and also investigate their asymptotic forms at a large distance from the center, limits on the horizon, and the asymptotic forms in the singularity neighborhood of a five-dimensional black hole in the case where the brane intersects it.     

Some exact solutions of KdV equation with variable coefficients

In this paper, the extended mapping transformation method is used to obtain some new exact solutions of a variable-coefficient KdV equation arising in arterial mechanics. The obtained solutions include soliton solutions, periodic solutions and rational solutions.     

Some exact solutions of the equations of the mechanics of nonhomogeneous media

We consider problems of the torsion of nonhomogeneous prismatic rods and the oscillation of plats of variable thickness operating in shear. We obtain an inequality characterizing the dependence of the maximum tangential stress on the shear modulus. An estimate from below is obtained for the torsional moment. An optimization problem for plates of variable thickness for a specified class of thicknesses is reduced to a problem on characteristic values for the Helmholtz equation.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 87–90, 1987.     

Some exact Wick type stochastic generalized Boussinesq equation solutions

The resolution of the stochastic generalized Boussinesq equation driven by a white noise is undertaken. Explicit solutions are found thanks to a white noise functional approach and the F-expansion method. Among these solutions, periodic and solitonic ones are pointed out.     

Some exact pseudo-plane solutions of the first kind for the Navier-Stokes equations

We study pseudo-plane flows of the first kind generated by a stream function =f(x, z, t) +g(y, z, t) which are generalized Beltrami flows in every plane parallel to thexy-plane. This problem is solved completely resulting in several new families of exact solutions for the Navier-Stokes equations.     

Some exact solutions of the ideal MHD equations through symmetry reduction method

We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3+1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1?r?4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.     

Magnetohydrodynamics with regard to electron inertia: Some exact solutions

Magnetohydrodynamic (MHD) equations are derived with the finite electron mass taken into account. The low-frequency spectrum of a cold two-component plasma can be described using these equations. In the linear approximation, the MHD equations lead to results coinciding with those of multifluid hydrodynamics. An exact solution to the generalized MHD equations is obtained as a stationary nonlinear wave. A soliton-like solution is found for a wide range of Mach numbers and external magnetic field intensities. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 116, No. 3, pp. 349–361, September, 1998.     

Some exact solutions to Stefan problems with fractional differential equations

Some exact solutions to the first, second and extended Stefan problems with fractional time derivative described in the Caputo sense are given by means of fractional Green's function and Wright function in this paper. By the aid of simple calculations, many results of differential equations of integer order can be obtained as special cases of the results given by this paper.     

Galilei invariant non-linear equations of schrödinger type and their exact solutions. I

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1349–1357, October, 1989.     

CTE method and exact solutions for modified Boussinesq system

In this paper, the truncated Painlevé analysis and the consistent tanh expansion method are developed for the modified Boussinesq system, and new exact solutions such as the single‐soliton, the two‐soliton, the rational solutions, and the explicit interaction solutions among a soliton and the cnoidal periodic waves are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms

We study a nonlinear reaction-diffusion system that is modeled by a system of parabolic equations with power-law nonlinear terms. The proposed construction of exact solutions enables us to split the process of finding the components depending on time and the spatial coordinates. We construct multiparametric families of exact solutions in elementary functions. The cases are elaborated of blow-up solutions as well as exact solutions time-periodic but spatially anisotropic.     

Some new exact analytical solutions for helical flows of second grade fluids

The helical flow of a second grade fluid, between two infinite coaxial circular cylinders, is studied using Laplace and finite Hankel transforms. The motion of the fluid is due to the inner cylinder that, at time t = 0⁺ begins to rotate around its axis, and to slide along the same axis due to hyperbolic sine or cosine shear stresses. The components of the velocity field and the resulting shear stresses are presented in series form in terms of Bessel functions J₀(•), Y₀(•), J₁(•), Y₁(•), J₂(•) and Y₂(•). The solutions that have been obtained satisfy all imposed initial and boundary conditions and are presented as a sum of large-time and transient solutions. Furthermore, the solutions for Newtonian fluids performing the same motion are also obtained as special cases of general solutions. Finally, the solutions that have been obtained are compared and the influence of pertinent parameters on the fluid motion is discussed. A comparison between second grade and Newtonian fluids is analyzed by graphical illustrations.     

Some open problems in non-linear extended thermodynamics and their possible solutions

In this article we aim to furnish arguments for further considerations on some procedures commonly used in Extended Thermodynamics, such as the Taylor’s expansions around equilibrium or the transition to subsystems. The initial impulse for these considerations lies in the fact that we have found, for a 14 moments model, the exact closure to the conditions arising from the entropy principle and the material objectivity principle, without using Taylor’s expansions. These generated some problems; one of these concerns the relationship between system and subsystems, another one concerns the convexity of entropy and hyperbolicity; a third problem deals with the correct moments to be used as independent variables, those suggested by the classical limit of the relativistic moment theory. We will give suggestions for the solution of these problems.