A method for solving the functional equation with variable delay

Mathematical simulation of physical conditions at a moving boundary in problems of wave reflection leads to a finite-difference equation with variable delay. If the reflector velocity is smaller than the wave-propagation velocity, the existence and uniqueness of the equation solution is proved using the principle of contraction maps and the method of successive approximations. It is shown that the solution can be expressed as a function of the composite argument that depends on one variable. The required accuracy of numerical calculations is ensured by partial sums when the argument is represented by an infinite converging series. N. I. Lobachevsky State University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 9, pp. 900–905, September, 1999     

Equations for generating functionals in classical and quantum statistical mechanics

This paper develops a formalism for the generating functionals for partial distribution functions in classical statistical mechanics and partial density matrices in quantum statistical mechanics. For the case of a large canonical ensemble, functional equations are written with respect to the functionals introduced. Each functional system creates a system of integral equations for distribution functions or density matrices.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 9, pp. 98–102, September, 1971.     

Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics

In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the K(m, n) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.     

Sinc-collocation method for solving the Blasius equation

Sinc-collocation method is applied for solving Blasius equation which comes from boundary layer equations. It is well known that sinc procedure converges to the solution at an exponential rate. Comparison with Howarth and Asaithambi's numerical solutions reveals that the proposed method is of high accuracy and reduces the solution of Blasius' equation to the solution of a system of algebraic equations.     

On Ruch's Principle of Decreasing Mixing Distance in classical statistical physics

Ruch's Principle of Decreasing Mixing Distance is reviewed as a statistical physical principle and its basic suport and geometric interpretation, the Ruch-Schranner-Seligman theorem, is generalized to be applicable to a large representative class of classical statistical systems.This paper is dedicated to Ernst Ruch on the occasion of his 70 th birthday.     

Substantiating the Gibbs method in classical statistical mechanics

An approach for the substantiation of the Gibbs method in equilibrium statistical thermodynamics is described; this approach is based not on the quasiergodicity hypothesis, but on the weaker assumption of macroscopic determinacy of thermodynamic systems. A generalized microcanonical Gibbs distribution is obtained. An electron gas in a homogeneous magnetic field is taken as an example. It is shown that the classical diamagnetism of the given system is not zero in the sense of quasimean nor of generalized Gibbs ensemble distributions. The equation of state of an electron as in a magnetic field is obtained, and hence it is shown that classical diamagnetism only vanishes if isotropy of the pressure at the vessel wall is assumed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 65–68, June, 1978.     

复摆方程的一种求解方法

利用复摆测量物体的转动惯量是一种重要方法,本文对复摆的转动惯量公式进行拉普拉斯变化,用能量和时间来表达转动惯量,丰富了不规则物体转动惯量的测量方法.     

A method of solving the Dirac equation

Electromagnetic fields are found for which the Dirac equation reduces to a system of independent second-order differential equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp.41–43, December, 1980.     

Variational iteration method for solving the multi-pantograph delay equation

In this Letter, the variational iteration method is applied to solve the multi-pantograph delay equation. Sufficient conditions are given to assure the convergence of the method. Examples show that the method is effective.     

Spectral density method in classical statistical mechanics

The spectral density method in classical statistical mechanics is formulated and application to a linear classical spin system is made in the simplest approximation.     

The operator method for solving the fractional Fokker-Planck equation

The operator method has been used to solve the fractional Fokker-Planck equation (FPE) which recently formulated as a model for the anomalous transport process. Two classes of special interest of fractional F-P equations coming from plasma physics and charged particle transport problem has been considered. It is shown that the mean square-displacement 〈x²(t)〉 satisfy the universal power law characterized the anomalous time evolution i.e. .     

Chebyshev collocation method for solving the radiative transfer equation

A Chebyshev collocation method is presented for solving the spherical harmonics approximation to the equation of radiative transfer in a plane-parallel, homogeneous medium. As a result of test computations performed for Rayleigh and Henyey-Greenstein phase functions, it was found that the proposed method can be used to solve transfer problems stably and accurately, even for an optically thick case, and that the accuracy of computations is greatly improved by increasing the order of Chebyshev polynomials in expansion.     

Application of the projection operator method in equilibrium statistical physics

In the framework of the Liouville equation formalism the projecton operator method is used in equilibrium statistical physics.     

Derivation of classical Markovian kinetic equations via generating functional technique

An exact Markovian kinetic equation for a classical system of particles interacting via pair forces is derived by means of Bogoljubov's Generating Functional method. A nondiverging collision integral is obtained for Coulomb plasmas, which takes into account the screening effects and the energy and momentum transfer between two colliding particles and their surroundings.     

Modified spherical harmonics method for solving the radiative transport equation

A new effective approach to solving the three-dimensional radiative transport equation with an arbitrary phase function is proposed. The solution depends on eigenvectors and eigenvalues of several symmetrical tridiagonal matrices of infinite size. The matrices must be truncated and diagonalized numerically. Then, given eigenvectors and eigenvalues of these matrices, the dependence of the solution on position and direction is found analytically. The approach is based on expanding the angular part of the specific intensity in q-dependent spherical functions for each spatial Fourier component characterized by the vector q. Apart from the truncation of the matrices, no other approximations are made.     

A polynomial expansion method for solving the laser Fokker-Planck equation

Distribution functions of the laser amplitude and intensity can be determined by solving the laser Fokker-Planck equation. By a suitable expansion of the distribution functions in Laguerre polynomials, a system of ordinary differential equations for the coefficients of the expansion is derived and is shown to have the form of a recurrence relation with length four. Applying it to the transient solution, the averaged amplitude and the first four cumulants of the intensity distribution are obtained even for those pump parameters where the hitherto known numerical solution is not applicable.     

Some moments method of solving the Boltzmann equation

An approximate method to obtain time dependent solutions of the Boltzmann equation (BE) is proposed. Here we restrict our consideration to the homogeneous BE only. Approximate evolution equations for mean values are formulated.The work was supported by the Polish Ministry of Science, Higher Education and Technology, Project M.R.I.7     

Modified spherical harmonics method for solving the radiative transport equation

A new effective approach to solving the three-dimensional radiative transport equation with an arbitrary phase function is proposed. The solution depends on eigenvectors and eigenvalues of several symmetrical tridiagonal matrices of infinite size. The matrices must be truncated and diagonalized numerically. Then, given eigenvectors and eigenvalues of these matrices, the dependence of the solution on position and direction is found analytically. The approach is based on expanding the angular part of the specific intensity in q-dependent spherical functions for each spatial Fourier component characterized by the vector q. Apart from the truncation of the matrices, no other approximations are made.     

Lax-type equation unifying gradient systems in quantum and classical statistical models

In a previous paper, Y. Uwanoet al.: nlin/SI:0512004, the author made a short remark on a similarity between gradient systems on a quantum information space and on a classical one: One is on the space of regular relative-configurations of multi-qubit states of ordered tuples and the other is on the space of multinomial distributions studied by Y. Nakamura: Japan J. Indust. Appl. Math.10 (1993) 179. This paper aims to report very briefly a Lax-type equation unifying the gradient systems on the information spaces with quantum and with classical features.