A temperature problem for a square: An exact solution

Abstract

We construct examples of exact solutions of the temperature problem for a square: the sides of the square are (i) free and (ii) firmly clamped. Initially, we solve the inhomogeneous problem for an infinite plane. The known exact solutions for a square, with which the boundary conditions on the sides of the square are satisfied, are added to this solution. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas.