In this paper a 1-D model for the evolution of inelastic particles in a thermal bath is studied. We derive an equation for the evolution of the probability density of the particles in phase space. The analysis of the homogeneous equation is the main objective of this paper. The existence and uniqueness of this equation is addressed. Finally, we prove that the asymptotic behavior is governed by ...

We study the nonhogeneous heat equation under the form: u t ? u xx = '(t)f(x), where the unknown is the pair of functions (u; f). Under various assumptions about the the function ' and the nal value in t = 1 i.e. g(x), we propose diierent regularizations on this ill-posed problem based on the Fourier transform associated with a Lebesgue measure. For ' 6 6 0 the solution is unique. I. Introducti...

Shannon's entropy has been characterized in several ways. Kaminski and Mikusihski [5] simplified Fadeev's [3] approach by considering, what they called the entropy equation: (1.1) H(x, y, z) = H(x + y, 0, z) + H(x, y, 0) , (x ^ 0 , y ^ 0 , z > 0 , xy + yz + zx > 0) . A general continuous symmetric solution of (1.1) given by (1.2) H(x, y, z) = (x + y + z) log (x + y + z) — x log x y log y — z lo...

Using the projection operator technique, the exact homogeneous generalized master equation (HGME) for the relevant part of a distribution function (statistical operator) is derived. The exact (mass) operator governing the evolution of the relevant part of a distribution function and comprising arbitrary initial correlations is found. Neither the Bogolyubov principle of weakening of initial corr...

We consider Poisson’s equation in an n-dimensional exterior domain G (n≥ 2) with a sufficiently smooth boundary. We prove that for external forces and boundary values given in certain Lq(G)-spaces there exists a solution in the homogeneous Sobolev space S2,q(G), containing functions being local in Lq(G) and having second-order derivatives in Lq(G). Concerning the uniqueness of this solution we ...

We study a method for solving the homogeneous Bethe-Salpeter equation. By introducing a ‘fictitious’ eigenvalue λ the homogeneous BetheSalpeter equation is interpreted as a linear eigenvalue equation, where the bound state mass is treated as an input parameter. Using the improved ladder approximation with the constant fermion mass, we extensively study the spectrum of the fictitious eigenvalue ...

in this paper, the modified simplest equation method is successfully implemented to find travelling wave solutions of the generalized forms $b(n,1)$ and $b(-n,1)$ of burgers equation.this method is direct, effective and easy to calculate, and it is a powerful mathematical tool for obtaining exact travelling wave solutions of the generalized forms $b(n,1)$ and $b(-n,1)$ of burgers equation and c...

In this work, we successfully construct the new exact traveling wave solutions of the generalized Sinh–Gordon equation by new application of the homogeneous balance method. The idea introduced in this paper can be applied to other nonlinear evolution equations.

The homogeneous balance method can be used to construct exact traveling wave solutions of nonlinear partial differential equations. In this paper, this method is used to construct new soliton solutions of the (3+1) Jimbo--Miwa equation.