A Stefan Problem for a Non-Classical Heat Equation^{∗}

Domingo A. TARZIA

Depto de Matemática and CONICET, FCE, Universidad Austral, Paraguay 1950, S2000FZF Rosario, Argentina.

E-mail: Domingo.Tarzia@fce.austral.edu.ar; tarzia@uaufce.edu.ar

Abstract

We review some recent results concerning to the heat equation with a heat source depending on the heat flux occurring at the fixed x=0 of a semi-infinite material. We also present a new free boundary problem (one-phase Stefan-like problem) for a non-classical heat equation, and we obtain the temperature and the free boundary (the phase-change interface) through the solution of a system of two Volterra integral equations.

Resumen: Se da una revisión de algunos recientes resultados concernientes a la ecuación del calor con una fuente que depende del flujo de calor que ocurre en la frontera fija x=0 de un cuerpo semi-infinito. También se presenta un nuevo problema de frontera libre (problema de tipo Stefan a una fase) para una ecuación no clásica para la cual se obtiene la temperatura y la frontera libre (la frontera de cambio de fase) a través de la solución de un sistema de dos ecuaciones integrales de Volterra.

Key words: Non-classical heat equation, asymptotic behavior, Stefan problem, phase-change problem, free boundary problem, Volterra integral equation.

Palabras claves: Ecuación del calor no-clásico, comportamiento asintótico, problema de Stefan, problema de cambio de fase, problemas de frontera libre, ecuación integral de Volterra.

AMS Subject classification: 35R35, 80A22, 35K05, 45D05, 35B40.

1 Introduction

The following non-classical heat conduction problem for a semi-infinite material was studied in [17]

\{ \begin{matrix}u_{t}(x, t) − u_{xx}(x, t) = Φ(x)F (u_{x}(0, t)), & x > 0, & t > 0, \\ u(0, t) = g(t), & t > 0, \\ u(x, 0) = h(x), & x > 0, \end{matrix}  (1)

∗MAT - Serie A, 3 (2001), 21-26.

where Φ , g , h are real functions defined on ℝ^{+} and F is defined on ℝ^{+} × ℝ which depends on the heat flux at the extremum x = 0 . Non-classical problems like (1) are motivated by the modelling of a system of temperature regulation in isotropic media and the source term Φ(x) F (u_{x}(0, t)) describes a cooling or heating effect depending on the properties of F which are related to the evolution of the heat flux u_{x}(0, t) . It is called the thermostat problem. Related problems are considered in [4],[6],[9]. Under suitable assumptions on data, existence, uniqueness and monotone-continuous dependence on the data are established in [17] for problem (1).

It was consider in [2] the simple instance of problem (1) given by

\{ \begin{matrix}u_{t} − u_{xx} = −F (u_{x}(0, t)), & x > 0, & t > 0, \\ u(0, t) = 0, & t > 0, \\ u(x, 0) = h(x), & x > 0, \end{matrix}  (2)

where h(x) , x > 0, and F (v) , v ∈ ℝ, are continuous functions. The function F, referred as control function, was assumed to fulfill the following condition:

A) v F (v) ≥ 0, F (0) = 0,

which intuitively means that the control attempts to stabilize the process at every time.

As it is shown in [18] (see also [17]), the solution to problem (2) can be represented by

u(x, t) = u_{0}(x, t) − ∫_{0}^{t} erf (\frac{x}{2\sqrt{t − τ}} ) F (V (τ)) dτ,  (3)

where u_{0} = u_{0}(x, t), defined by

u_{0}(x, t) = ∫_{0}^{+∞} G(x, t; ξ, 0) h(ξ) dξ,  (4)

is the solution to problem (2) with null source term F = 0. Function V = V (t) in (3) represents the heat flux at the extremum of the slab, i.e.

V (t) = u_{x}(0, t), t > 0,  (5)

and it satisfies the following Volterra integral equation

V (t) = V_{0}(t) − ∫_{0}^{t} \frac{F (V (τ))}{\sqrt{π(t − τ)}} dτ,  (6)

where the forcing function V_{0}(t) is given by

V_{0}(t) = \frac{1}{2\sqrt{π}t^{\frac{3}{2}}} ∫_{0}^{+∞} ξ exp (−\frac{ξ^{2}}{4t}) h(ξ) dξ, t > 0.  (7)

Function G in (4) denotes the Green function of the heat equation in the quarter plane and, as it is well-known, it can be written as G(x, t; ξ, τ) = K(x, t; ξ, τ)−K(−x, t; ξ, τ), x, ξ > 0, t > τ > 0, where

K(x, t; ξ, τ) = \frac{1}{\sqrt{4π(t − τ)}} exp (−\frac{(x − ξ)^{2}}{4(t − τ)}) ,

is the one-dimensional heat kernel. Moreover, we also define the Neumann function of the heat equation in the quarter plane as N(x, t; ξ, τ) = K(x, t; ξ, τ) + K(−x, t; ξ, τ), x, ξ > 0, t > τ > 0.

From now on, we suppose that h is a non-negative and non-identically null function which, in view of (7), implies V_{0}(t) > 0, t > 0. When the control function F satisfies condition (A) and, moreover, the initial temperature h is non-negative, then the solution u(x, t) to problem (2) tends to zero when t → +∞ (see [17], [18]). In [2] was studied the problem of ”controlling” problem (2) through F so that, by the stabilizing effect of the control, its solution should converge to zero (when the time goes to infinity) faster than that corresponding to problem (2) in absence of control; i.e.lim_{t→+∞} u(x, t)/u_{0}(x, t) = 0. The heat flux w(x, t) = u_{x}(x, t) satisfies a classical heat conduction problem with a nonlinear convective condition at x = 0. The first papers in this direction are [10] and [13]. Other related problems are considered in [1], [8] and [12]. In [2], a general study of the above stated control problem for (2) was done finding spatially uniform bounds for the quotient u(x, t)/u_{0}(x, t) which depend on the solution V (t) to integral equation (6), from which becomes apparent that condition (A) is not sufficient to attain the objective of the control; i.e., to obtain lim_{t→+∞} u(x, t)/u_{0}(x, t) = 0. In Section 2, for linear control functions F (v) = λ v , we give an example to illustrate that there exists an exact solution to problem (6) providing u(x, t)/u_{0}(x, t) ≅ 1/(2 λ^{2}t), t → +∞. In Section 3, we present a one-phase Stefan problem for a semi-infinite material for a non-classical heat equation with a source term F which depends of the evolution of the heat flux at the extremum x = 0. Its solution is given by the solution of a system of two Volterra integral equations [3],[7],[11].

2 Constant initial temperature and their control function

We shall consider the instance of problem (2) corresponding to a constant initial temperature h(x) = h_{0} > 0, x ≥ 0. The solution to problem (2) is represented by (3) with

u_{0}(x, t) = h_{0} erf (\frac{x}{2\sqrt{t}}) , x > 0, t > 0,  (8)

while V = V (t) becomes the solution to the Volterra integral equation

V (t) = \frac{h_{0}}{\sqrt{πt}} − ∫_{0}^{t} \frac{F (V (τ))}{\sqrt{π(t − τ)}} dτ, t > 0.  (9)

Therefore, the following inequalities

\frac{\sqrt{πt}}{h_{0}} V (t) ≤ \frac{u(x, t)}{u_{0}(x, t)} ≤ \frac{1}{h_{0}} ∫_{0}^{t} \frac{V (τ)}{\sqrt{π(t − τ)}} dτ = 1 − \frac{1}{h_{0}} ∫_{0}^{t} F (V (τ)) dτ,  (10)

hold for x > 0, t > 0 [2].

From now on we suppose the case of linear controls: i.e.,

F (v) = λv, (λ > 0),  (11)

and in order to obtain the explicit solutions u and V of the problems (2) and (9) respectively, we define the real function Q(x) = \sqrt{π}x exp(x^{2}) erfc(x), defined for x > 0 [15] which satisfies the following properties: Q(0) = 0, Q(+∞) = 1, Q′(x) > 0, x > 0. The most important facts on the behavior of the solution V (t) to equation (9) corresponding to a linear control (11) are collected in the following result (See [2]).

Lemma 1 If F is given by (11), then we have

0 < V (t) = \frac{h_{0}}{\sqrt{πt}} [1 − Q(λ\sqrt{t)}] < \frac{h_{0}}{\sqrt{πt}},  (12)

1 − \frac{1}{h_{0}} ∫_{0}^{t} F (V (τ)) dτ = exp(λ^{2}t) erfc(λ\sqrt{t)},  (13)

for all t > 0 and lim_{t→+∞} u(x, t)/u_{0}(x, t) = 0, uniformly in x > 0. Furthermore, we have the estimates

\frac{1}{πλ^{2}t} ≤ \frac{u(x, t)}{u_{0}(x, t)} ≤ \frac{1}{λ\sqrt{πt}} ,  (14)

as t → +∞ . Moreover, the temperature u is given by

u(x, t) = h_{0} exp(λ^{2}t) [erfc (λ\sqrt{t}) − exp(λx) erfc (λ\sqrt{t} + \frac{x}{2\sqrt{t}})] ,  (15)

and a more accurate estimation \frac{u(x, t)}{u_{0}(x, t)} ∼ 1/(2λ t^{2}), when t → +∞, uniformly in x > 0

is also obtained.

3 A Stefan problem for a non-classical heat equation.

We consider the following free boundary problem (one-phase Stefan problem) for the temperature u = u(x, t) and the free boundary x = s(t) (see [16]) with a control function F which depends on the evolution of the heat flux at the extremum x = 0 given by the following conditions:

\{ \begin{matrix}u_{t} − u_{xx} = −F (u_{x}(0, t)), & 0 < x < s(t), 0 < t < T, \\ u(0, t) = f(t) ⩾ 0, & 0 < t < T, \\ u(s(t), t) = 0, u_{x}(s(t), t) = − s^{.} (t) , & 0 < t < T, \\ u(x, 0) = h(x), & 0 ⩽ x ⩽ b = s(0) . \end{matrix}  (16)

Theorem 2 The solution of the free boundary problem (16) is given by

u(x, t) = ∫_{0}^{b} G(x, t; ξ, 0)h(ξ)dξ + ∫_{0}^{t} G_{ξ}(x, t; 0, τ)f(τ)dτ + ∫_{0}^{t} G_{ξ}(x, t; s(τ), τ)v(τ)dτ
− ∫∫_{D(t)} G(x, t; ξ, τ)F (V (τ))dξdτ ,
s(t) = b − ∫_{0}^{t} v(τ)dτ

where D(t) = {(x, τ)/ 0 < x < s(τ), 0 < τ < t} , and v(t) = u_{x}(s(t), t) = − s^{.} (t) and

V (t) = u_{x}(0, t) must satisfy the following system of two Volterra integral equations

v(t) = 2 [h(0) − f(0)] N(s(t), t; 0, 0) + ∫_{0}^{b} N(s(t), t; ξ, 0)h′(ξ)dξ

−2 ∫_{0}^{t} N(s(t), t; 0, τ) f^{.} (τ)dτ + 2 ∫_{0}^{t} G_{x}(s(t), t; s(τ), τ)v(τ)dτ

+2 ∫_{0}^{t} [N(s(t), t; s(τ), τ) − N(s(t), t; 0, τ)] F (V (τ))dτ ,

V (t) = [h(0) − f(0)] N(0, t; 0, 0) + ∫_{0}^{b} N(0, t; ξ, 0)h′(ξ)dξ − ∫_{0}^{t} N(0, t; 0, τ) f^{.} (τ)dτ

+ ∫_{0}^{t} G_{x}(0, t; s(τ), τ)v(τ)dτ + ∫_{0}^{t} [N(0, t; s(τ), τ) − N(0, t; 0, τ)] F (V (τ))dτ ,

where G and N are the Green and Neumann functions of the heat equation in the quarter plane, defined previously in Section 1.

Proof. We compute u_{x}(x, t), and their corresponding limits as x ⟶ 0^{+} and x ⟶ s(t)^{−}. By using the jump relations [5], [14] the system of two Volterra integral equations holds.

The corresponding study of the existence and uniqueness of the solution will be given in a forthcoming paper.

Acknowledgments: This paper has been partially sponsored by CONICET - UA (Rosario, Argentina). This financial support was granted to the Project PIP No. 4798/96 “Free Boundary Problems for the Heat Equation”.

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