Some results on a heat conduction problem by Myshkis

Abstract

An infinite homogeneous d-dimensional medium initially is at zero temperature. A heat impulse is applied at the origin, raising the temperature there to a value greater than a constant value u 0 > 0 . The temperature at the origin then decays, and when it reaches u 0 , another equal-sized heat impulse is applied at a normalized time τ 1 = 1 . Subsequent equal-sized heat impulses are applied at the origin at the normalized times τ n , n = 2 , 3 , … , when the temperature there has decayed to u 0 . This sequence of normalized waiting times τ n can be defined recursively by a difference equation and its asymptotic behavior was known recently. This heat conduction problem was first studied in [J. Difference Equations Appl. 3 (1997) 89–91]. A natural subsequent question is what happens if the problem is set in a finite region, like in a laboratory, with the temperature at the boundary being kept zero forever. In this paper we obtain the asymptotic behavior of the heating times for the one-dimensional case.