Non-steady-state heat conduction in composite walls

Abstract

The problem of heat conduction in one-dimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the one-dimensional heat equation in each domain. The location of the interfaces is known, but neither temperature nor heat flux is prescribed there. Instead, the physical assumptions of their continuity at the interfaces are the only conditions imposed. The problem of two semi-infinite domains and that of two finite-sized domains are examined in detail. We indicate also how to extend the solution method to the setting of one finite-sized domain surrounded on both sides by semi-infinite domains, and on that of three finite-sized domains.

Solving Wiener–Hopf problems without kernel factorization

“…Therefore, in combination with the general strategy put forward in this article, problems of Wiener-Hopf type arising for those equations should also be amenable to an analysis akin to that given here. It is interesting to note that a recent study of non-steady state heat conduction with composite walls [24] has several features in common with our approach.…”

Initial-to-Interface Maps for the Heat Equation on Composite Domains

“…Using the Fokas method [1,2] such solutions may be constructed. This has been done in the case of the heat equation with n interfaces in infinite, finite, and periodic domains as well as on graphs [3][4][5][6][7]. The method has also been extended to dispersive problems [8,9], and higher order problems [10].…”

Vector Riemann–Hilbert problem with almost periodic and meromorphic coefficients and applications

“…Here, H 15) and the Cauchy integral is explicitly evaluated by the theory of residues. Alternatively, this splitting can be obtained by representing the function H(s) as a sum of n integrals similar to (2.8) and then removing the poles.…”

“…In §2, we consider the heat equation for an infinite rod u t = a 2 (x)u xx + g(x, t) with a piecewise constant diffusivity a(x). The Fokas method [13] was recently applied [14,15] to the heat equation in a ring and a rod. Deconinck et al [15] found an exact solution in the homogeneous case, g ≡ 0, when the diffusivity is a piecewise constant function, and (i) the rod is finite and the diffusivity has one or two points of discontinuity and (ii) the rod is infinite and the function a(x) is discontinuous either at 0 and ∞ or at two finite points and infinity.…”

“…The Fokas method [13] was recently applied [14,15] to the heat equation in a ring and a rod. Deconinck et al [15] found an exact solution in the homogeneous case, g ≡ 0, when the diffusivity is a piecewise constant function, and (i) the rod is finite and the diffusivity has one or two points of discontinuity and (ii) the rod is infinite and the function a(x) is discontinuous either at 0 and ∞ or at two finite points and infinity. In particular, for an infinite rod with a(x) = a ± , ±x > 0, and g ≡ 0, they managed to represent the solution by double integrals with one-sided Fourier internal integrals and contour external integrals over the boundary of infinite wedge-like domains.…”