Spectral representation of the isotropic Coulomb collisional operator

1997 ◽  
Vol 58 (3) ◽  
pp. 475-484 ◽  
Author(s):  
I. K. KHABIBRAKHMANOV ◽  
D. SUMMERS
Keyword(s):  
Thermal Equilibrium ◽  
Recurrence Relations ◽  
Spectral Representation ◽  
Particle Distribution ◽  
Laguerre Polynomials ◽  
Planck Equation ◽  
Isotropic Part ◽  
Particle Distributions ◽  
Generalized Laguerre Polynomials ◽  
Spectral Coefficients

A spectral representation for the isotropic part of the Coulomb collisional operator is given. The particle distribution function is expanded in a series of generalized Laguerre polynomials, and the Coulomb collisional operator is expressed in terms of the spectral amplitudes. When the spectral representation is applied to the Fokker–Planck equation, a system of coupled ordinary differential equations for the spectral amplitudes is obtained. The spectral coefficients related to the Coulomb operator are defined through recurrence relations, which we reduce to simplified form. This makes possible accurate and efficient analytical and numerical evaluations of the interaction matrices. The results presented can be used in analytical investigations of the properties of the Coulomb collisional operator as well as in numerical calculations for plasmas far from thermal equilibrium. The method can also be generalized to include angular dependencies for non-isotropic particle distributions.

scholarly journals On the connection coefficients and recurrence relations arising from expansions in series of modified generalized Laguerre polynomials: Applications on a semi-infinite domain

2019 ◽  
Vol 8 (1) ◽  
pp. 318-327 ◽  
Author(s):  
E.H. Doha ◽  
Y.H. Youssri
Keyword(s):  
Recurrence Relations ◽  
Laguerre Polynomials ◽  
Infinite Domain ◽  
Laguerre Expansion ◽  
Varying Coefficients ◽  
Generalized Laguerre Polynomials ◽  
General Order ◽  
In Series ◽  
Expansion Coefficients ◽  
Derivatives Of

Abstract Herein, three important theorems were stated and proved. The first relates the modified generalized Laguerre expansion coefficients of the derivatives of a function in terms of its original expansion coefficients; and an explicit expression for the derivatives of modified generalized Laguerre polynomials of any degree and for any order as a linear combination of modified generalized Laguerre polynomials themselves is also deduced. The second theorem gives new modified generalized Laguerre coefficients of the moments of one single modified generalized Laguerre polynomials of any degree. Finally, the third theorem expresses explicitly the modified generalized Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its modified generalized Laguerre coefficients. Some spectral applications of these theorems for solving ordinary differential equations with varying coefficients and some specific applied differential problems, by reducing them to recurrence relations in their expansion coefficients of the solution are considered.

scholarly journals Two variables generalized Laguerre polynomials

2021 ◽  
Vol 13 (1) ◽  
pp. 134-141
Author(s):  
A. Asad
Keyword(s):  
Differential Equations ◽  
Hypergeometric Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Laguerre Polynomial ◽  
Laguerre Polynomials ◽  
Explicit Representation ◽  
Generalized Hypergeometric Function ◽  
Generalized Laguerre Polynomials

The objective of this paper is to introduce and study the generalized Laguerre polynomial for two variables. We prove that these polynomials are characterized by the generalized hypergeometric function. An explicit representation, generating functions and some recurrence relations are shown. Moreover, these polynomials appear as solutions of some differential equations.

scholarly journals Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 984
Author(s):  
Pedro J. Miana ◽  
Natalia Romero
Keyword(s):  
Integral Representation ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Function Spaces ◽  
Laguerre Functions ◽  
Addition Formula ◽  
Lebesgue Spaces ◽  
Rodrigues Formula ◽  
Generalized Laguerre Polynomials ◽  
New Family

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.

SPATIOTEMPORAL STRONGLY NONLOCAL SPHERICAL LIGHT BULLETS

2013 ◽  
Vol 22 (01) ◽  
pp. 1350006
Author(s):  
WEI-PING ZHONG
Keyword(s):  
Numerical Simulations ◽  
Legendre Polynomials ◽  
Schrodinger Equation ◽  
Laguerre Polynomials ◽  
Three Dimensional ◽  
Integration Constant ◽  
Generalized Laguerre Polynomials ◽  
Existence And Stability ◽  
Light Bullets ◽  
Nonlocal Nonlinear

The general spherical beam solution of the three-dimensional (3D) spatiotemporal strongly nonlocal nonlinear (NN) Schrödinger equation in the form of light bullets is presented. The 3D spatiotemporal spherical beams are built by the products of generalized Laguerre polynomials and associated Legendre polynomials. By the choice of a specific integration constant, the spherical beam becomes an accessible soliton, which can exist in various forms. We confirm the existence and stability of these solutions by numerical simulations.

Sign in / Sign up

Export Citation Format

Share Document