An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab

Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 RESEARCH Open Access An initial-boundary value problem for the one-dimensional non-classical heat equation in a slab Natalia Nieves Salva1,2, Domingo Alberto Tarzia1,3* and Luis Tadeo Villa1,4 * Correspondence: DTarzia@austral. edu.ar 1 CONICET, Rosario, Argentina Full list of author information is available at the end of the article Abstract Nonlinear problems for the one-dimensional heat equation in a bounded and homogeneous medium with temperature data on the boundaries x = 0 and x = 1, and a uniform spatial heat source depending on the heat flux (or the temperature) on the boundary x = 0 are studied. Existence and uniqueness for the solution to non-classical heat conduction problems, under suitable assumptions on the data, are obtained. Comparisons results and asymptotic behavior for the solution for particular choices of the heat source, initial, and boundary data are also obtained. A generalization for non-classical moving boundary problems for the heat equation is also given. 2000 AMS Subject Classification: 35C15, 35K55, 45D05, 80A20, 35R35. Keywords: Non-classical heat equation, Nonlinear heat conduction problems, Volterra integral equations, Moving boundary problems, Uniform heat source 1. Introduction In this article, we will consider initial and boundary value problems (IBVP), for the one-dimensional non-classical heat equation motivated by some phenomena regarding the design of thermal regulation devices that provides a heater or cooler effect [1-6]. In Section 2, we study the following IBVP (Problem (P1)): ut − uxx = −F(ux (0, t), t), 0 < x < 1, u(0, t) = f (t), (P1) t>0 u(1, t) = g(t), u(x, 0) = h(x), t>0 (1:1) (1:2) t>0 0 ≤ x ≤ 1, (1:3) (1:4) where the unknown function u = u(x,t) denotes the temperature profile for an homogeneous medium occupying the spatial region 0 < x <1, the boundary data f and g are real functions defined on ℝ+, the initial temperature h(x) is a real function defined on [0,1], and F is a given function of two real variables, which can be related to the evolution of the heat flux ux(0,t) (or of the temperature u(0,t)) on the fixed face x = 0. In Sections 6 and 7 the source term F is related to the evolution of the temperature u(0,t) when a heat flux ux(0,t) is given on the fixed face x = 0. © 2011 Salva et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 2 of 17 Non-classical problems like (1.1) to (1.4) are motivated by the modelling of a system of temperature regulation in isotropic media and the source term in (1.1) describes a cooling or heating effect depending on the properties of F which are related to the evolution of the heat ux(0,t). It is called the thermostat problem. A heat conduction problem of the type (1.1) to (1.4) for a semi-infinite material was analyzed in [5,6], where results on existence, uniqueness and asymptotic behavior for the solution were obtained. In other frameworks, a class of heat conduction problems characterized by a uniform heat source given as a multivalued function from ℝ into itself was studied in [3] with results regarding existence, uniqueness and asymptotic behavior for the solution. Other references on the subject are [2,4,7,8]. Recently, free boundary problems (Stefan problems) for the non-classical heat equation have been studied in [9-11], where some explicit solutions are also given. Section 2 is devoted to prove the existence and the uniqueness of the solution to an equivalent Volterra integral formulation for problems (1.1) to (1.4). In Section 3, 4 and 5, boundedness, comparisons results and asymptotic behavior regarding particular initial and boundary data are obtained. In Section 6, a similar problem to (P1) is presented: the heat source F depends on the temperature on the fixed face x = 0 when a heat flux boundary condition is imposed on x = 0, and we obtain the existence of a solution through a system of three second kind Volterra integral equations. In Section 7, we solve a more general problem for a non-classical heat equation with a moving boundary x = s(t) on the right side which generalizes the boundary constant case and it can be useful for the study of free boundary problems for the classical heat-diffusion equation [12]. 2. Existence and uniquenes of problem (P1) For data h = h(x), g = g(t), f = f(t) and F in problems (1.1) to (1.4) we shall consider the following assumptions: (HA) g and f are continuously differentiable functions on ℝ+; (HB) h is a continuously differentiable function in [0,1], which verifies the following compatibility conditions: h(0) = f (0), h(1) = g(0); (2:1) (HC) The function F = F(V,t) verifies the following conditions: (HC1) The function F is defined and continuous in the domain ℝ × ℝ+; (HC2) For each M >0 and for |V| ≤ M, the function F is uniformly Hölder continuous in variable t for each compact subset of R+0; (HC3) For each bounded set B of ℝ × ℝ +, there exists a bounded positive function L0 = L0(t), which is independent on B, defined for t > 0, such that   F(V2 , t) − F(V1 , t) |≤ LO (t) |V2 − V1 , ∀(V2 , t), (V1 , t) ∈ B; (HC4) The function F is bounded for bounded V for all t ≥ 0; (HD) F(0,t) = 0, t >0. Under these assumptions, from Th. 20.3.3 of [13] an integral representation for the function u = u(x,t), which satisfies the conditions (1.1) to (1.4), can be written as below:  t  t  θ (x − ξ , t) − θ (x + ξ , t) h(ξ )dξ −2 θx (x, t − τ )f (τ )dτ + 2 θx (x − 1, t − τ )g(τ )dτ 0 0 0  (2:2)  t  1   − θ (x − ξ , t − τ ) − θ (x + ξ , t − τ ) dξ F(V(τ ), τ )dτ  1 u(x, t) = 0  0 Salva et al. Boundary Value Problems 2011, 2011:4 http://www.boundaryvalueproblems.com/content/2011/1/4 Page 3 of 17 where θ = θ (x,t) is the known theta function defined by θ (x, t) = K(x, t) + ∞  [K(x + 2j, t) + K(x − 2j, t)] (2:3) j=1 and K = K(x,t) is the fundamental solution to the heat equation defined by: x2 − 1 K(x, t) = √ e 4t , 2 πt (2:4) t > 0. Moreover the function V = V(t), defined by t > 0, V(t) = ux (0, t), (2:5) as the heat flux on the face x = 0, must satisfy the following second kind Volterra integral equation  t (2:6) V(t) = V0 (t) − K(t − τ )F(V(τ ), τ )dτ 0 where 1 t (θξ (−ξ , t) − θξ (ξ , t))h(ξ )dξ − 2 Vo (t) = 0 1 =2 t θ (0, t − τ )ḟ (τ )dτ + 2 0 θ (ξ , t)h (ξ )dξ − 2 0 θ (−1, t − τ )ġ(τ )dτ 0 t (2:7) t θ (0, t − τ )ḟ (τ )dτ + 2 0 θ (−1, t − τ )ġ(τ )dτ , t > 0, 0 with K = K(t) and K1 (x, t; ξ, τ) defined by  1 K(t) = K1 (0, t; ξ , 0)dξ , t > 0, (2:8) 0 K1 (x, t; ξ , τ ) = θx (x − ξ , t − τ ) − θx (x + ξ , t (...truncated)