Spatial decay estimates in transient heat conduction

Abstract

Abstract. The spatial decay of solutions to initial-boundary value problems for the heat equation in a three-dimensional cylinder, subject to non-zero boundary conditions only on the ends, is investigated. It is shown that the spatial decay of end effects in the transient problem is faster than that for the steady-state case. Qualitative methods involving second-order partial differential inequalities for quadratic functionals are first employed. The explicit spatial decay estimates are then obtained by using comparison principle arguments involving solutions of the one-dimensional heat equation. The results give rise to versions of Saint-Venant's principle in transient heat conduction.1. Introduction. The spatial decay of solutions of parabolic differential equations has been the subject of much recent attention [1][2][3][4][5]. These studies were motivated by a desire to establish, for parabolic equations, decay estimates analogous to those obtained for elliptic equations in the investigation of Saint-Venant's principle in elasticity theory. (See [6] for a review of recent work on Saint-Venant's principle.)Boley [7][8][9][10] was apparently the first to raise the issue of the validity of a Saint-Venant principle for transient heat conduction. In [8,10] versions of Saint-Venant's principle for non-cylindrical domains, similar to those of von Mises and Sternberg for elasticity, were considered. Using explicit upper bounds for solutions of half-space problems obtained from appropriate fundamental solutions, Boley [10] observed that the spatial influence of transient effects was even more localized than that of the steady-state. The validity of a Saint-Venant principle of the more traditional type, for cylindrical domains subject to non-zero boundary conditions on the ends only, was considered in [7,9], An illustrative example involving the explicit solution of a specific initial-boundary value problem for a semi-infinite rectangular strip is discussed in [9], pp. 186-187. In this example, it is seen that the spatial decay of end effects at any time t in the transient problem is faster than that for the steady-state case [9],

Spatial decay of transient end effects in functionally graded heat conducting materials

“…In this section, we show that the function P(z,t) satisfying (3.3) or equivalently (4.25) can be bounded above by the solution to a related initial-boundary value problem for the one-dimensional heat equation. The argument here follows that of Horgan et al [16]. By virtue of its definition, P(z,t) satisfies the initial condition P(z, 0) = 0, 2>0, (5.1) and the boundary condition The maximum principle for the heat equation now yields v < w, z> 0, t > 0, (5.14) and so, from (5.5), we find that…”

“…A sharper result was obtained by Horgan et al [16] who showed that solutions decay at a rate that is faster than the exponential decay predicted by Knowles [21]. Our purpose here is to establish an analog of the result of Horgan et al [16] for the inhomogeneous material and to assess the effects of material inhomogeneity on the rate of decay of end effects.…”

“…A similar result was obtained by Horgan and Wheeler [18] using the maximum principle. A stronger result, analogous to that of Boley [3], Boley and Weiner [4], was established by Horgan et al [16], who showed that the spatial decay of end effects at each fixed time t in the transient problem is faster than that for the steady-state case. This work was extended to classes of nonlinear heat equations by Quintanilla [24], where references to related research may be found.…”

“…then reduces to the two-dimensional version of the result of Horgan et al [16] established for three-dimensional cylindrical bodies (cf. Ignaczak [19] for half-space problems).…”

“…Rather we wish to assess the spatial decay character of the estimate (6.5) and contrast it with the corresponding known result for homogeneous materials. The arguments in Horgan et al [16] can be used to show that the estimate (6.5) implies that the spatial decay of end effects in the transient problem is faster than that for the steady-state.…”