A new mathematical model of heat conduction processes
In the paper a fourth-order PDE invariant under the Galileo's group G(1,3) is studied. The equation has so called biparabolic form \(L(u)=\alpha_ 1L_ 1(u)+\alpha_ 2L_ 2(u)=0\), where \(L_ 2=L_ 1L_ 1\) and \(L_ 1\) is the classic heat conduction operator [see the author, Dokl. Akad. Nauk Ukr. SSR, Ser. A 1988, No.8, 22-27 (1988; Zbl 0682.35052)]. It is noted that this equation describes heat and diffusion processes in a more adequate way than hyperbolic heat equations. This comes from the fact that for most of the PDEs of second order with respect to time neither the Galileo's relativistic principle, nor the Poincaré-Einstein principle are fulfilled. An integral representation of the solution of a corresponding boundary value problem, as well as solutions of the Cauchy problem and of the travelling wave type are obtained. Also, a power boundary regime and some peaking mode [see, for example, \textit{A. A. Samarskij}, \textit{V. A. Galaktionov}, \textit{S. P. Kurdyumov} and \textit{A. P. Mikhajlov}, Processes with strain for quasilinear parabolic equations (Russian) (1987; Zbl 0631.35002)] are analyzed.
scientific article; zbMATH DE number 4169230