Structure of solutions to linear unsteady boundary value problems in mechanics and mathematical physics

Abstract

We consider unidimensional and multidimensional linear unsteady nonhomogeneous boundary value problems for equations of the parabolic and hyperbolic types with coefficients being arbitrary functions of spatial coordinates and time. We derive general formulas allowing solutions to these problems to be expressed in terms of the Green’s functions in the case of boundary conditions of all basic types. These results can be used in the theory of heat transfer and mass transfer, in wave theory, and in other branches of mechanics and theoretical physics. The formulas obtained generalize results of a large number of studies (e.g., [1‐7]) in which special unsteady boundary value problems were considered. 1. Parabolic equations with one spatial variable. We consider a linear nonhomogeneous differential parabolic equation of the general form with variable coefficients