Models of a linearized Boltzmann collision integral

For rarefied gas flows at moderate and low Knudsen numbers, model equations are derived that approximate the Boltzmann equation with a linearized collision integral. The new kinetic models generalize and refine the S-model kinetic equation.     

Wavelet approximations of a collision operator in kinetic theory

This Note is devoted to the derivation of conservative and entropic fast wavelet approximations for the isotropic Fokker–Planck–Landau collision operator arising in the modeling of charged particles in plasma physics. The present approach combines the advantages of both the finite difference schemes (conservation and entropy) and the spectral methods (accuracy) which are developed in the literature. Furthermore, the wavelet approach provides a fast algorithm for the evaluation of such a collision operator. The present work is a first step to the development of wavelet approximations to more complex collision operators in kinetic theory. To cite this article: X. Antoine, M. Lemou, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

About one system of integral equations in kinetic theory

One system of integral convolution equations is considered on a half-line with an noninvertible matrix integral operator whose symbol has a fourth-order zero. The application of a special factorization method makes it possible to distinguish noninvertible factors in the original noninvertible operator and reduce the system to a new system with a nonsingular integral operator. The structural theorem of the existence of a solution to the original system is proved.     

An initial-value method for integral operators. II. Complex eigenfunctions

Combining classical results from the theories of integral equations and complex variables with a recent technique for transforming Fredholm integral equations into Cauchy systems yields a new method for characterizing complex eigenfunctions. Numerical examples show the computational efficacy of the approach.This work was supported by the US Atomic Energy Commission.     

An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory

Summary. We propose a finite difference scheme to approximate the Fokker-Planck collision operator in 3 velocity dimensions. The principal feature of this scheme is to provide a decay of the numerical entropy. As a consequence, it preserves the collisional invariants and its stationary solutions are the discrete Maxwellians. We consider both the whole velocity-space problem and the bounded velocity problem. In the latter case, we provide artificial boundary conditions which preserve the decay of the entropy. Received October 18, 1993     

On the eigenvalues and eigenfunctions of some integral operators

Some properties of the eigenvalues of the integral operator K_gt defined as K_τf(x) = ∫₀^τK(x ? y) f (y) dy were studied by Vittal Rao (J. Math. Anal. Appl.53 (1976), 554–566), with some assumptions on the kernel K(x). In this paper the eigenfunctions of the operator K_τ are shown to be continuous functions of τ under certain circumstances. Also, the results of Vittal Rao and the continuity of eigenfunctions are shown to hold for a larger class of kernels.     

A quasilinear method in the theory of small eigenfunctions for nonlinear periodic boundary-value problems

Small eigenfunctions of a nonlinear periodic boundary-value problem are studied for the case of double degeneration of the eigenvalue of the linearized problem; the quasilinear representation is used.     

On the divergence free eigenfunctions of the magnetostatic integral operator

For the eigenvalues from (−1,1) it is shown that the divergence free eigenfunctions of the adjoint magnetostatic integral operator can be explicitly obtained from the eigenfunctions of the adjoint electrostatic integral operator to the same eigenvalue.     

Fast algorithms for computing the Boltzmann collision operator

The development of accurate and fast numerical schemes for the five-fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed.

     

Discrete model collision operators of Boltzmann type

In the present paper we develop a new kind of discrete velocity models to discretize the Boltzmann collision operator. The chosen approach is situated between the macroscopic ansatz of the BGK-Model and the microscopic ansatz of usual discrete velocity models. Beside questions of the solvability and the form of the solutions of the arising optimization problems, the weak convergence of the discrete collision operators to the original operator is proved.     

The Barnett approximation in the theory of hydrodynamic fluctuations

The Langevin dynamics and fluctuational-dissipative relationships for the hydrodynamic fluctuations for systems which are described in the third Barnett order with respect to the gradients of the hydrodynamic variables are generalized on the basis of a kinetic approach.     

The equiconvergence of expansions in eigenfunctions and associated functions of an integral operator with involution

In this paper we study an integral operator with involution. We solve the problem on the exact inversion of this operator, we obtain and study the integro-differential system for the Fredholm resolvent and, finally, we prove the theorem on the equiconvergence of expansions in eigenfunctions and associated functions, in the usual trigonometric system.     

An integral representation for eigenfunctions of linear ordinary differential operators

This paper generalizes an integral representation formula for eigenfunctions of Sturm-Liouville operators, known as the Volterra transformation operator in the theory of the inverse scattering problem, to higher-order differential operators. A specific fourth-order initial value problem is considered: $\text{Lφ = k}{}^4\text{φ, L =}\text{d}{}^4\text{dx}{}^4\text{+}\text{d}\text{dx}\text{(q}\text{d}\text{dx}\text{) + r}$φ(0) = 1, φ′(0) = 0, φ″(0) = ?k², φ? = 0 The solution for complex k is expressed as an inverse-Laplace-Borel transform. Jump formulae are obtained relating the representing kernel directly to the coefficients of L. The result admits obvious generalization to operators of arbitrary order.     

Kinetic equations in the method of two-time finite-temperature Green's functions. I. Renormalization of the collision integral

A system of equations that includes a generalized kinetic equation and equations for the static correlation functions is constructed for a normal quantum system of interacting Bose and Fermi particles with two-body interaction on the basis of the method of two-time finite-temperature Green's functions. The equations are in general valid for systems with arbitrary density of the particles. A method of successive approximation that makes it possible to go beyond the usual low-density expansion is discussed. The proposed method leads to a renormalization of the collision integral and makes it possible to obtain correlation functions for the total energy density, including its potential part.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 3, pp. 351–372, September, 1993.     

Experimental method of solving an integral equation of the theory of viscoelasticity

A solution of an integral equation of the theory of viscoelasticity is proposed. This solution, based on a creep test on a polymer specimen with an included elastic element, improves the convergence of the method of approximations.Moscow M. V. Lomonosov State University. Translated from Mekhanika Polimerov, No. 4, pp. 584–587, July–August, 1969.     

The use of integral operators in number theory

It is known that all of the non-real zeros of the Riemann zeta-function lie in the critical strip $\text{L}$={s=σ+it: O<σ<1} and important arithmetic consequences would follow from the assertion that all such zeros, in fact, lie on the line $\text{σ =}\text{1}\text{2}$. This assertion, known as the Riemann Hypothesis, is not yet established. In the present article, it is shown that the knowledge of the spectrum of any member of a certain class of self-adjoint, integral operators yields an explicit region of $\text{l}$ devoid of zeros of the Riemann zeta-function. Moreover, if a certain non-linear functional, Φ, is unbounded, then the Riemann hypothesis would follow.