### New analytic solutions of the one-dimensional heat equation for temperature and heat flow rate both prescribed at the same fixed boundary (with applications to the phase change problem)

David Langford
1967 Quarterly of Applied Mathematics
New solutions of the heat equation are exhibited for the case in which both the temperature and heat flow rate are prescribed at a single fixed boundary. The prescribed temperature and heat flow rate may be any arbitrary infinitely differentiable functions of time. The new solutions are applicable for one-dimensional (radial) heat flow in spheres, cylinders, and slabs. Special solutions may be obtained by choosing special forms for the prescribed boundary temperature and boundary heat flow
more » ... dary heat flow rate. These special solutions include the classical solutions of the heat equation, new sequences of polynomial and quasipolynomial solutions of the heat equation, and new closed-form solutions to constantvelocity phase change problems with spherical and cylindrical symmetry. A. Introduction. We treat the one-dimensional diffusion of heat perpendicular to the surfaces of parallel planes, coaxial cylinders, and concentric spheres. For constant density and constant thermal diffusivity, the indicated heat diffusion process is governed by the relation x~k-(xh-ux)x = u,(x, t). Here u(x, t) is the dimensionless temperature at point x at time t, x is the dimensionless radial position variable, t is the dimensionless time variable, and k = 0, 1, 2 for plane, cylinder, and sphere, respectively. We solve the Cauchy problem, that is, we determine u(x, t) when both the dimensionless temperature f(t) and the dimensionless heat flow rate q(t) are prescribed at some fixed boundary, say at x = xn . (The dimensionless temperature at x0 is just u(x0 , t); the dimensionless heat flow rate at x0 is just limitI_a.0 [ -xk-ux(x, £)])• We consider both x0 = 0 and x0 ^ 0, with k = 0,1, 2. *