Telegraph-type Equation Research Articles

Heat conduction: a telegraph-type model with self-similar behavior of solutions

For the heat flux q and temperature T, we introduce a modified Fourier–Cattaneo law qt + lq/(t + τ) = −kTx. The consequence of it is a non-autonomous telegraph-type equation. This model already has a typical self-similar solution which may be written as a product of two traveling waves modulo a time-dependent factor and might play a role of intermediate asymptotics.

A TRANSIENT LOCAL ENERGY APPROACH AS AN ALTERNATIVE TO TRANSIENT SEA: WAVE AND TELEGRAPH EQUATIONS

This paper is concerned with the high-frequency dynamics of structures in the time domain. In this frequency range, a differential equation governing the vibrational energy density is proposed. This simplified equation leads to the prediction of large scale space and time evolution of the energy quantities averaged over time, frequency and space. For one-dimensional undamped structures, the energy equation is the classical wave equation whereas its generalization for damped structures leads to a telegraph-type equation. The discussion is focussed on the comparison with the diffusion equation sometimes encountered in the literature.

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

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.