In the paper we analyze singularly perturbed telegraph systems applying 
the newly developed compressed asymptotic method and show that the 
diffusion equation is an asymptotic limit of singularly perturbed telegraph 
system of equations. The results are applied to the random walk theory 
for which the relationship between correlated and uncorrelated random 
walks is explained in asymptotic terms.

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Singularly perturbed telegraph equations with applications in the random walk theory

In this paper we present the asymptotic analysis of the linear Boltzmann equation for neutrons with a small positive parameter ϵ related to the mean free path, based upon the Chapman–Enskog procedure of the kinetic theory. We prove that if proper initial conditions derived by considering initial layer solutions are used, the diffusion equation gives the uniform approximation to the neutron density function with the O(ϵ2) accuracy.

Diffusion limit for the linear boltzmann equation of the neutron transport theory

We apply the new algorithm developed by Mika and Palczewski for singularly perturbed systems of ordinary differential equations to systems of second order equations closely related to a discretised version of the telegraph equation.

First-order asymptotic expansion method for singularly perturbed systems of second-order ordinary differential equations

Viewing the prompt jump approximation (PJA) of nuclear reactor dynamics as the zeroth‐order approximation of an asymptotic expansion to singularly perturbed systems of ordinary differential equations, we derive here the equations describing its first‐order approximation, the improved PJA, and show it may be used to supply an estimate of the error of the classical PJA.

The improved prompt jump approximation

We present three different asymptotic expansion procedures for a linear kinetic equation, of which only two lead to a diffusion equation. We show that the diffusion equation gives a first-order approximation to the kinetic equation if it is supplemented by an initial condition derived from the initial layer considerations. As an example, we take a one-speed neutron transport equation in plane geometry.

Diffusion limit for a linear kinetic equation

Atmtract--Pre~mt ed in this paper is a new algorithm for the asymptotic expansion of a solution to an initial value problem for ~ngularly perturbed (stiff) systems of ordinary dlfferemtial eqtmtlons. This algorithm is related to the Chapman-Enskog asymptotic expamdon method mind in the kinetic theory to derive the equatiorm of hydrodynamics, whcxeas the standard algorithm pertains to the Hilbert approach known to give inferior results. In cases of systems of ordinary differential equatioms the new algorithm leads to the reduction of numerical effort needed to achieve a given accuracy M compm-ed with the st~mdard uymptotlc expansion method. The proof of the asymptotic c~vc=lpmce is given. The numerical example demonstrates the feasibility of the new approach.

Janus R. Mika

Papers

Asymptotic analysis of singularly perturbed systems of ordinary differential equations