English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 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. *Received December 10, 1965; revised manuscript received May 16, 1966. This is based in part on a dissertation [4] accepted by Rensselaer Polytechnic Institute in partial fulfillment of the requirements for the degree of Doctor of Engineering Science. The author wishes to acknowledge valuable discussions with Professor Bronis R. Onuf (who suggested the thesis topic), Professor Howard W. Butler (who acted as thesis adviser) and Professor Murray R. Spiegel (who served on the Doctoral Committee). The research was supported by an educational grant from Pratt & Whitney Aircraft Division of United Aircraft Corporation. "Now with Drexel Institute of Technology, Philadelphia, Pa. 316 DAVID LANGFORD [Vol. XXIV, No. 4 We assume that /(f) and q(t) are analytic. Existence and uniqueness of an analytic solution are then guaranteed (in the small) (for fixed k and fixed x0) by a Theorem of Cauchy and Kowalewsky [2]. Further, the nonexistence of any nonanalytic solution is guaranteed by a Theorem of Holmgren [2], The analytic solutions given here correspond to physical reality because [2] they depend continuously on the prescribed data. The use of these solutions is demonstrated for several cases of physical interest. We now outline the procedure used to obtain the solutions. For x0 = 0, we may assume that u(x, f) —£ (amn-\-bmn-\ne x)xmt", substitute into m, n the heat diffusion equation, and obtain recursion relations for the constants am„ and bmn . The recursion relations may be used to decompose u{x, f) into series of polynomial and quasi-polynomial solutions of the heat equation. These series may be summed to obtain the final solutions to the Cauchy problem. The final solutions are presented with the derivation being omitted; the elements of the series are exhibited as special cases of the final solutions. For x0 9^ 0: the solution u"(x, t) for the planar case is a linear transformation of the solution to the Cauchy problem for x0 = 0; the solution u'(x, f) for the spherical case is readily obtained from uv{x, f) when it is noted that (x-u') must satisfy the planar heat equation; the solution uc(x, t) for the cylindrical case is obtained in the Appendix. We now exhibit the solutions and illustrate their use. B. The fixed boundary at the origin of coordinates. Consider first the case x0 = 0. Let /(f) and q(t) be any infinitely differentiate functions. Define m = (dn/dt")Kl) and q\t) = (aT/dtn)q{t). (1) Then it may be verified by differentiation that the following are formal solutions of the heat equation: ff(0-x2" (2 n)\ (2n+l)!_ plane, u(x, t) = £ ~2n—1 sphere, u(x, t) = g + (2a) [ f{t)-x2n ^ q"(t) -x2 . (a) (2n)\ J ' cylinder, u(x, t) = £ (x/2 r fr!) n\-n\ f(0 l(2c) where the factor in braces is just {In, xj for n = 0. These solutions have the property that limitx_n [-xk-ux(x, f)] = q(t); this makes it possible to prescribe the heat flow rate at x0 as an arbitrary infinitely differentiate function q{t). For the planar case, the solution 2a has the property that u{0, f) = /(f); this makes it possible to prescribe the temperature at x„ as an arbitrary infinitely differentiate function /(f). For the spherical and cylindrical cases, if q(t) ^ 0, then it is physically necessary that u{0, t) — oo, as required by Eqs. 2b, c; if q{t) = 0, then u{0, f) = /(f).* The above solutions are general solutions of the heat equation in that they may be specialized to yield other solutions of the heat equation. For example, the classical *The solution 2a was given by Stefan [8] in 1890; he used it to compute the thermal conductivity of ice. The solutions 2b, c have not been previously published, though several of the leading terms of the solution 2c were given by Pekeris and Slichter [6] in 1939. 1967] ANALYTIC SOLUTIONS OF THE ONE-DIMENSIONAL HEAT EQUATION 317 Fourier series and Bessel series solutions of the heat equation may be obtained from Eqs. 2a, b, c by choosing ](t) = Am-exp and q(t) = X Bm-exp {—bit). (3) m = 0 m = 0 New solutions of the heat equation may also be obtained from Eq. 2. For example, if m = E (/,<>!) and q{t) = £ (g,f/»!), (4) n=0 «=0 then direct substitution into Eqs. 2a, b, c yields

quarterly of applied mathematics

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)