The asymptotic interrelation of the Cauchy problem solutions corresponding to parabolic and telegraph type equations

Abstract

Mass and heat transport processes modelled by parabolic and telegraph type equations are discussed. In order to do this the fundamental solution of the Cauchy ProblemE(x, t) for the telegraph equation (ɛ2∂2/∂t2 + 2m ∂/∂t−c2Δ)E(x, t)=0 (xeRn,m andc are positive constants, ɛ is assumed to be a small one, the boundaries are absent) is considered. It is shown that its support may be subdivided into 4 subrogions according to the type of the asymptotic expansion. Within two of them the asymptotics ofE(x, t) is equivalent to the “Poisson kernel”. It is shown that the telegraph equation may be used to solve the above mentioned problems if and only ifn=1 together with the conditionsu(x, 0) ≥ 0 and ∂u(x, 0)/∂t=0 imposed on the initial values. Various types of solutions corresponding to the initial data of this kind are considered and sufficient conditions for the asymptotic transition to the traditional formalism based on parabolic equations are presented. Analogous results for the asymptotic expansion of the mass flow density are also given. It is shown that the presented methods are suitable to obtain an asymptotic expansion of the solution of the Cauchy problem if the initial data functions belong toL1(−∞, ∞) and their supports are compact. The connection of the considered methods with those of the probability theory is outlined as well.