Mathematical and physical aspects of the initial value problem for a nonlocal model of heat propagation with finite speed

Jerzy A. Gawinecki¹ ; Agnieszka Gawinecka² ; Jarosław Łazuka¹ ; J. Rafa¹

¹ Institute of Mathematics and Cryptology Faculty of Cybernetics Military University of Technology Kaliskiego 2 00-908 Warszawa, Poland
² Central Military Bureau of Design and Technology Kaliskiego 2 00-908 Warszawa, Poland

Applicationes Mathematicae, Tome 40 (2013) no. 1, pp. 31-61

Résumé

Theories of heat predicting a finite speed of propagation of thermal signals have come into existence during the last 50 years. It is worth emphasizing that in contrast to the classical heat theory, these nonclassical theories involve a hyperbolic type heat equation and are based on experiments exhibiting the actual occurrence of wave-type heat transport (so called second sound). This paper presents a new system of equations describing a nonlocal model of heat propagation with finite speed in the three-dimensional space based on Gurtin and Pipkin's approach. We are interested in the physical and mathematical aspects of this new system of equations. First, using the modified Cagniard–de Hoop method we construct a fundamental solution to this system of equations. Next basing on this fundamental solution, we obtain explicit formulae for the solution of the Cauchy problem to this system. Applying the methods of Sobolev space theory, we get an $L^p$-$L^q$ time decay estimate for the solution of the Cauchy problem. For a special form of the source we perform analytical and numerical calculations of the distribution of the temperature for the nonlocal model of heat with finite speed. Some features of the propagation of heat for the nonlocal model are illustrated in a figure together with the comparison of the solution of this model with the solution of the classical heat equation.

Détail

DOI : 10.4064/am40-1-3

Keywords: theories heat predicting finite speed propagation thermal signals have come existence during years worth emphasizing contrast classical heat theory these nonclassical theories involve hyperbolic type heat equation based experiments exhibiting actual occurrence wave type heat transport called second sound paper presents system equations describing nonlocal model heat propagation finite speed three dimensional space based gurtin pipkins approach interested physical mathematical aspects system equations first using modified cagniard hoop method construct fundamental solution system equations basing fundamental solution obtain explicit formulae solution cauchy problem system applying methods sobolev space theory get p l time decay estimate solution cauchy problem special form source perform analytical numerical calculations distribution temperature nonlocal model heat finite speed features propagation heat nonlocal model illustrated figure together comparison solution model solution classical heat equation