Solution of the fourth and fifth boundary value problems for a one-dimensional second-order parabolic equation in a curvilinear region

Abstract

CLASSICAL theory of heat transmission in a body is usually confined to three types of linear boundary condition (see [1]): specification of the temperature u( x, t) on the body surface (boundary condition of the 1st kind), specification of the derivative ∂u ∂x , which corresponds to specifying the heat flow through the surface (boundary condition of the 2nd kind), and specification of a linear combination of u and ∂u ∂x , which corresponds to heat exchange obeying Newton's law at the body surface with the ambient medium of given temperature (boundary condition of the 3rd kind). The heat capacity of the medium is assumed to be so large that the heat flow through the surface has no effect on its temperature. If the heat capacity of the medium is moderate, its temperature change due to the body in turn becomes an unknown function of time; then the heat balance at the body surface can be assumed given. To find the temperatures of the body and medium, we then have to solve new boundary value problems, with boundary conditions of the 4th and 5th kinds (we use the terminology of the authors of [2–5], who discussed similar problems). Notice that boundary conditions of the same kind are obtained when we consider the heating of a body by point sources (problem with concentrated heat capacities), which was first investigated for the one-dimensional heat conduction equation in [6–8]. In [2–4], boundary value problems were discussed for the equation of heat conduction only; the results of [2, 3], referring to the one-dimensional case, are to be found in [6–8]. Existence and uniqueness theorems are proved in [4] for the multidimensional case in a cylindrical region. The uniqueness of the solution of the new problems is examined in [5] for a non-linear parabolic equation, though again, only for cylindrical regions. The method of heat potentials was used in [9, 10] for boundary value problems in a curvilinear region for a general linear one-dimensional second-order parabolic equation, both with discontinuous [9] and continuous coefficients (but with non-classical boundary conditions), and the existence of a solution of these problems was proved under minimal assumptions regarding smoothness of the relevant functions. We show in the present paper that, in the case of boundary value of the 4th and 5th kinds, the method of heat potentials enables existence theorems to be proved under minimal assumptions regarding the smoothness of the functions in a curvilinear region with lateral boundaries that merely satisfy the condition ( D) of [11]. There are three sections. In the first, we state the problems with boundary conditions of the 4th and 5th kinds, and formulate existence and uniqueness theorems. In Section 2 we prove the uniqueness theorem. In Section 3 we prove the existence theorems.