Boundary value problems for parabolic equations in noncylindrical domains

A mathematical theory is developed for constructing integral relations of a new type in analytical solutions of boundary value problems for parabolic equations in domains with boundaries moving in time (noncylindrical domains). For the uniform law of boundary displacement, a modification of the method of generalized thermal potentials of a simple and double layer is proposed, which leads to functional relations of a new (simple) form in comparison with the previously known results based on the transition and further solution of the Volterra integral equations when finding an unknown potential density. The developed method is based on preliminary determination of the operational (according to Laplace) form of the potential density, which significantly reduces the cumbersomeness and computational difficulties that occur in the traditional application of thermal potentials for solving parabolic type equations in noncylindrical domains. Numerous cases are considered for bounded and partially bounded domains, which are of practical interest for many applications. The theory of the Green's function method for noncylindrical domains is developed. Integral relations are proposed for writing analytical solutions of boundary value problems for parabolic type equations in terms of boundary functions in the original formulation of the problem and the corresponding Green's functions. The case of the root dependence of the moving boundary is studied. A number of specific features of model representations of nonstationary heat transfer in domains with moving boundaries are revealed.