A Heat Conduction Problem with Sources Depending on the Average of the Heat Flux on the Boundary

TL;DR

This paper studies a non-classical heat conduction problem where the internal energy source depends on the average heat flux at the boundary, deriving integral equations and explicit solutions in specific cases.

Contribution

It introduces a novel heat conduction model with boundary-dependent sources and provides existence, uniqueness, and explicit solutions for the problem.

Findings

The heat flux satisfies a Volterra integral equation of the second kind.

A unique local solution exists and can be extended globally.

Explicit solutions are obtained in the one-dimensional case using Laplace transform and Adomian decomposition.

Abstract

Motivated by the modeling of temperature regulation in some mediums, we consider the non-classical heat conduction equation in the domain $D=\mathbb{R}^{n-1}\times\br^{+}$ for which the internal energy supply depends on an average in the time variable of the heat flux $(y, s)\mapsto V(y,s)= u_{x}(0 , y , s)$ on the boundary $S=\partial D$. The solution to the problem is found for an integral representation depending on the heat flux on $S$ which is an additional unknown of the considered problem. We obtain that the heat flux $V$ must satisfy a Volterra integral equation of second kind in the time variable $t$ with a parameter in $\mathbb{R}^{n-1}$. Under some conditions on data, we show that a unique local solution exists, which can be extended globally in time. Finally in the one-dimensional case, we obtain the explicit solution by using the Laplace transform and the Adomian decomposition method.

Equations

$$D={\mathbb{R}}^{n-1}\times {\mathbb{R}}^{+}=\{(x,y)\in {\mathbb{R}}^n:x=x_1>0,y=(x_2,\cdots ,x_n)\in {\mathbb{R}}^{n-1}\},$$

$$S=\partial D={\mathbb{R}}^{n-1}\times \{0\}=\{(x,y)\in {\mathbb{R}}^n:x=0,y\in {\mathbb{R}}^{n-1}\}.$$

$$u_t-\Delta u$$

$$u(0,y,t)$$

$$u(x,y,0)$$

$$G_1(x,y,t;\xi ,\eta ,\tau )=\frac{\exp \left[-\frac{\Vert y-\eta {\Vert }^2}{4(t-\tau )}\right]}{{\left(2\sqrt{\pi (t-\tau )}\right)}^{n-1}}G(x,t,\xi ,\tau ),$$

$$G(x,t,\xi ,\tau )=\frac{e^{-\frac{(x-\xi {)}^2}{4(t-\tau )}}-e^{-\frac{(x+\xi {)}^2}{4(t-\tau )}}}{2\sqrt{\pi (t-\tau )}}t>\tau .$$

$$u(x,y,t)=u_0(x,y,t)$$

$$-{\int }_0^t\frac{\mathrm{erf}\left(\frac{x}{2\sqrt{t-\tau }}\right)}{(2\sqrt{\pi (t-\tau )}{)}^{n-1}}\left[{\int }_{{\mathbb{R}}^{n-1}}{e}^{\left[-\frac{\Vert y-\eta {\Vert }^2}{4(t-\tau )}\right]}F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)d\eta \right]d\tau$$

$$u_0(x,y,t)={\int }_D{G}_1(x,y,t;\xi ,\eta ,0)h(\xi ,\eta )d\xi d\eta$$

$$V(y,t)=V_0(y,t)$$

$$-2{\int }_0^t\frac{1}{(2\sqrt{\pi (t-\tau )}{)}^n}\left[{\int }_{{\mathbb{R}}^{n-1}}{e}^{\left[-\frac{\Vert y-\eta {\Vert }^2}{4(t-\tau )}\right]}F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)d\eta \right]d\tau$$

$$V_0(y,t)={\int }_D{G}_{1,x}(0,y,t;\xi ,\eta ,0)h(\xi ,\eta )d\xi d\eta .$$

$$u(x,y,t)$$

$$u_x(x,y,t)$$

$${\int }_D{G}_{1,x}(0,y,t;\xi ,\eta ,\tau )F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)d\xi d\eta =$$

$${\int }_{{\mathbb{R}}^{n-1}}\frac{F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)e^{-\frac{\Vert y-\eta {\Vert }^2}{4(t-\tau )}}}{(t-\tau {)}^{\frac{n+2}{2}}(2\sqrt{\pi }{)}^n}\left({\int }_0^{+\infty }\xi e^{-\frac{{\xi }^2}{4(t-\tau )}}d\xi \right)d\eta$$

$$=\frac{2}{(2\sqrt{\pi (t-\tau )}{)}^n}{\int }_{{\mathbb{R}}^{n-1}}F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)e^{-\frac{\Vert y-\eta {\Vert }^2}{4(t-\tau )}}d\eta .$$

$${\int }_D{G}_1(x,y,t;\xi ,\eta ,\tau )F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)d\xi d\eta =$$

$$\frac{1}{(2(\sqrt{\pi (t-\tau )}{)}^n}\times {\int }_D{e}^{\frac{-\Vert y-\eta {\Vert }^2}{4(t-\tau )}}\left[e^{-\frac{(x-\xi {)}^2}{4(t-\tau )}}-e^{-\frac{(x+\xi {)}^2}{W4(t-\tau )}}\right]F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)d\xi d\eta =$$

$$\frac{1}{(2(\sqrt{\pi (t-\tau )}{)}^n}{\int }_{{\mathbb{R}}^{+}}\left[e^{-\frac{(x-\xi {)}^2}{4(t-\tau )}}-e^{-\frac{(x+\xi {)}^2}{4(t-\tau )}}\right]d\xi {\int }_{{\mathbb{R}}^{n-1}}{e}^{\frac{-\Vert y-\eta {\Vert }^2}{4(t-\tau )}}F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)d\eta$$

$${\int }_0^{+\infty }{e}^{\frac{-(x-\xi {)}^2}{4(t-\tau )}}d\xi$$

$${\int }_0^{+\infty }{e}^{\frac{-(x+\xi {)}^2}{4(t-\tau )}}d\xi$$

$${\int }_D{G}_1(x,y,t;\xi ,\eta ,\tau )F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)d\xi d\eta =$$

$$\frac{\mathrm{erf}\left(\frac{x}{2\sqrt{t-\tau }}\right)}{(2\sqrt{\pi (t-\tau )}{)}^{n-1}}{\int }_{{\mathbb{R}}^{n-1}}{e}^{-\frac{\Vert y-\eta {\Vert }^2}{4(t-\tau )}}F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)d\eta .$$

$$V(y,t)=\frac{1}{t(2\sqrt{\pi t}{)}^n}{\int }_{{\mathbb{R}}^{+}}\xi e^{-\frac{{\xi }^2}{4t}}\left({\int }_{{\mathbb{R}}^{n-1}}{e}^{-\frac{\Vert y-\eta {\Vert }^2}{4t}}h(\xi ,\eta )d\eta \right)d\xi$$

$$-\frac{2}{(2\sqrt{\pi }{)}^n}{\int }_0^t\frac{1}{(t-\tau {)}^{n/2}}{\int }_{{\mathbb{R}}^{n-1}}F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)e^{-\frac{\Vert y-\eta {\Vert }^2}{4(t-\tau )}}d\eta d\tau .$$

$$V(y,t)=f(y,t)+{\int }_0^{t}g(y,\tau ,V(y,\tau ))d\tau$$

$$f(y,t)=\frac{1}{t(2\sqrt{\pi t}{)}^n}{\int }_{{\mathbb{R}}^{+}}\xi e^{-\frac{{\xi }^2}{4t}}\left({\int }_{{\mathbb{R}}^{n-1}}{e}^{-\frac{\Vert y-\eta {\Vert }^2}{4t}}h(\xi ,\eta )d\eta \right)d\xi$$

$$g(t,\tau ,y,V(y,\tau ))=-\frac{2(t-\tau {)}^{-n/2}}{(2\sqrt{\pi }{)}^n}{\int }_{{\mathbb{R}}^{n-1}}F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right)e^{-\frac{\Vert y-\eta {\Vert }^2}{4(t-\tau )}}d\eta .$$

$$V(\eta ,\tau )\mapsto F\left(\frac{1}{\tau }{\int }_0^{\tau }V(\eta ,s)ds\right),$$

$$|g(y,t,\tau ,X)|$$

$$m(t,\tau )=\frac{1}{\sqrt{\pi }}\underset{X\in B}{\sup }|F(X)|\frac{1}{\sqrt{(t-\tau )}}$$

$$|g(y,t,\tau ,X)|\le m(t,\tau )\forall 0\le \tau \le t\le k,X\in B$$

$$\underset{t\in [0,k]}{\sup }{\int }_0^{t}m(t,\tau )d\tau$$

$$\underset{t\to 0^{+}}{\lim }{\int }_0^{t}m(t,\tau )d\tau$$

Full text

Keywords: Non-classical n-dimensional heat equation, Non local sources, Volterra integral equation, Existence and uniqueness of solution, Integral representation of the solution, Explicit solution, Adomian decomposition method.

2010 Mathematics Subject Classification : 35C15, 35K05, 35K20, 35K60, 45D05, 45E10, 80A20.

1 Introduction

Let consider the domain $D$ and its boundary $S$ defined by

[TABLE]

The aim of this paper is to study the following problem 1.1 on the non-classical heat equation, in the semi-n-dimensional space domain $D$ with non local sources, for which the internal energy supply depends on the average $\frac{1}{t}\int_{0}^{t}u_{x}(0,y,s)ds$ of the heat flux on the boundary $S$.

Problem 1.1*.*

Find the temperature $u$ at $(x,y,t)$ satisfying the following conditions

[TABLE]

where $\Delta$ denotes the Laplacian in $\mathbb{R}^{n}$. This problem is motivated by modeling the temperature in an isotropic medium with the average of non-uniform and non local sources that provide cooling or heating system, according to the properties of the function $F$ with respect to the heat flow $(y,s)\mapsto V(y,s)=u_{x}(0,y,s)$ at the boundary $S$, see [11, 13]. Some references on the subject are [6] where $F\left(\frac{1}{t}\int_{0}^{t}u_{x}(0,y,s)ds\right)$ is replaced by $F(u_{x}(0,y,t))$, or [7] where is replaced by $F\left(\int_{0}^{t}u_{x}(0,y,s)ds\right)$; see also [4], [14], [23], [24] where the semi-infinite case of this nonlinear problem with $F(u_{x}(0,y,t))$ have been considered. The non-classical one-dimensional heat equation in a slab with fixed or moving boundaries was studied in [14], [22]. See also other references on the subject [8]-[10], [12], [16]-[19]. To our knowledge, it is the first time that the solution to the average of a non-classical heat conduction of the type of Problem 1.1 is given. Other non-classical problems can be found in [5].

In [6] basic solution to the n-dimensional heat equation, and a technical Lemma was established. We prove in Section 2 the local existence of a solution for the considered Problem 1.1 under some conditions on data $F$ and $h$ which can be extended globally in times. Moreover, in Section 3 we consider the corresponding one dimensional problem and we obtain its explicit solution for the heat flux and the average of the total heat flux at the face $x=0$, by using the Laplace transform and also the Adomian decomposition method [1, 2, 3, 7, 25, 26].

2 Existence results

In this Section, we give first in Theorem 2.1, the integral representation (2.1) of the solution of the considered Problem 1.1, but it depends on the heat flow $V$on the boundary $S$, which satisfies the Volterra integral equation (2.1) with initial condition (2.5). Then we prove, in Theorem 2.3, under some assumptions on the data, that there exists a unique solution of the problem locally in times which can be extended globally in times.

We first recall here the Green’s function for the n-dimensional heat equation with homogenuous Dirichlet’s boundary conditions, given the following expression

[TABLE]

where $G$ is the Green’s function for the one-dimensional case given by

[TABLE]

Theorem 2.1**.**

The integral representation of a solution of the Problem 1.1 is given by the following expression

[TABLE]

where $\zeta\mapsto{\rm erf}\left(\zeta\right)=\left({2\over\sqrt{\pi}}\int_{0}^{\zeta}e^{-X^{2}}dX\right)$ is the error function,

[TABLE]

and the heat flux $(y,t)\mapsto V(y,t)=u_{x}(0,y,t)$ on the surface $x=0$, satisfies the following Volterra integral equation

[TABLE]

in the variable $t>0$, with $y\in\mathbb{R}^{n-1}$ is a parameter where

[TABLE]

Proof.

As the boundary condition in Problem (1.1) is homogeneous, we have from [15, 20]

[TABLE]

and therefore

[TABLE]

From (2.1) (the definition of $G_{1}$) by derivation with respect to $x$, taking $x=0$ we obtain

[TABLE]

Thus taking $x=0$ in (2.7) with (2) we get (2.1).

Also by (2.1) we obtain

[TABLE]

using

[TABLE]

and

[TABLE]

so we get

[TABLE]

Taking this formula in (2.6) we obtain (2.1). ∎

Lemma 2.2**.**

The simplified form of Volterra integral equation (2.1) is given by

[TABLE]

Proof.

Using the derivative, with respect to $x$, of (2.1), then taking $x=0$ and $\tau=0$, then taking the new expression of $V_{0}(y,t)$ in the Volterra integral equation (2.1) we obtain (2.2). ∎

Theorem 2.3**.**

Assume that $h\in\mathcal{C}(D)$, $F\in\mathcal{C}(\mathbb{R})$ and locally Lipschitz in $\mathbb{R}$, then there exists a unique solution of the problem 1.1 locally in time which can be extended globally in time.

Proof.

We know from Theorem (2.1) that, to prove the existence and uniqueness of the solution (2.1) of Problem (1.1), it is enough to solve the Volterra integral equation (2.2). So we rewrite it as follows

[TABLE]

with

[TABLE]

and

[TABLE]

So we have to check the conditions $H1$ to $H4$ in Theorem 1.1 page 87, and $H5$ and $H6$ in Theorem 1.2 page 91 in [21].

$\bullet$ The function $f$ is defined and continuous for all $(y,t)\in\mathbb{R}^{n-1}\times\mathbb{R}^{+}$, so $H1$ holds.

$\bullet$ The function $g$ is measurable in $(t,\tau,y,x)$ for $0\leq\tau\leq t<+\infty$, $x\in\mathbb{R}^{+}$, $y\in\mathbb{R}^{n-1}$, and continuous in $x$ for all $(y,t,\tau)\in\mathbb{R}^{n-1}\times\mathbb{R}^{+}\times\mathbb{R}^{+}$, $g(y,t,\tau,x)=0$ if $\tau>t$, so here we need the continuity of

[TABLE]

which follows from the hypothesis that $F\in C(\mathbb{R})$. So $H2$ holds.

$\bullet$ For all $k>0$ and all bounded set $B$ in $\mathbb{R}$, we have

[TABLE]

thus there exists a measurable function $m$ given by

[TABLE]

such that

[TABLE]

and satisfies

[TABLE]

so $H3$ holds.

$\bullet$ Moreover we have also

[TABLE]

and

[TABLE]

$\bullet$ For each compact subinterval $J$ of $\mathbb{R}^{+}$, each bounded set $B$ in $\mathbb{R}^{n-1}$, and each $t_{0}\in\mathbb{R}^{+}$, we set

[TABLE]

[TABLE]

as the function $\tau\mapsto V(z,\tau)$ is continuous then

[TABLE]

is $\mathcal{C}^{1}(\mathbb{R}^{+})$ and is in the compact $B\subset\mathbb{R}$ for all $z\in\mathbb{R}^{n-1}$, so by the continuity of $F$ we get $F\left({1\over\tau}\int_{0}^{\tau}V(z\ ,s)ds\right)\subset F(B)$, that is there exists $M>0$ such that $\left|F\left({1\over\tau}\int_{0}^{\tau}V(z\ ,s)ds\right)\right|\leq M$ for all $(z,\tau)\in\mathbb{R}^{n-1}\times\mathbb{R}^{+}$. So

[TABLE]

using

[TABLE]

we obtain

[TABLE]

Thus we deduce that

[TABLE]

So $H4$ holds.

$\bullet$ For all compact $I\subset\mathbb{R}^{+}$, for all function $\psi\in\mathcal{C}(I,\mathbb{R}^{n})$, and all $t_{0}>0$,

[TABLE]

as $F\in\mathcal{C}(\mathbb{R})$ and $\psi\in\mathcal{C}(I,\mathbb{R}^{n})$ then there exists a constant $M>0$ such that

[TABLE]

Then we obtain as for H4, that

[TABLE]

So H5 holds.

$\bullet$ Now for each constant $k>0$ and each bounded set $B\subset\mathbb{R}^{n-1}$ there exists a measurable function $\varphi$ such that

[TABLE]

whenever $0\leq\tau\leq t\leq k$ and both $x$ and $X$ are in $B$. Indeed as $F$ is assumed locally Lipschitz function in $\mathbb{R}$ there exists constant $L>0$ such that

[TABLE]

then we have

[TABLE]

then $\varphi(t,\tau)={L\over\sqrt{\pi(t-\tau)}}$. We have also for each $t\in[0,k]$ the function $\varphi\in L^{1}(0,t)$ as a function of $\tau$ and we have also

[TABLE]

So H6 holds. All the conditions H1 to H6 are satisfied with (2.15) and (2.16).

Thus from [21] (Theorem 1.1 page 87, Theorem 1.2 page 91 and Theorem 2.3 page 97) there exists a unique solution, local in time, to the Volterra integral equation (2.1) which can be extended globally in time. Then the proof of this theorem is complete. ∎

3 The one-dimensional case of Problem 1.1

Let us consider now the one-dimensional case of Problem 1.1 for the temperature defined by

Problem 3.1*.*

Find the temperature $u$ at $(x,t)$ such that it satisfies the following conditions

[TABLE]

Taking into account that

[TABLE]

thus the solution of the Problem 3.1 is given by

[TABLE]

with

[TABLE]

and $V(t)=u_{x}(0,t)$ is the the solution of the following Volterra integral equation of the second kind

[TABLE]

where

[TABLE]

For the particular case

[TABLE]

then we have

[TABLE]

and the integral equation (3.4) becomes

[TABLE]

Then, we have

[TABLE]

where $W(t)$ is defined by

[TABLE]

By using the integral equation (3.8) for $V(t)$ we obtain for $W(t)$ the following Volterra integral equation of the second kind:

[TABLE]

by using that

[TABLE]

Therefore, we deduce the following results

Theorem 3.1**.**

Taking $h$ and $F$ as in (3.6), the solution of the non-classical heat conduction Problem 3.1 is given by (3.9) where $W(t)$ is the solution of the Volterra integral equation (3.11). Moreover, its Laplace transform $\mathcal{L}$ is given by the following expression:

[TABLE]

and $W(t)$is given by the following difference of two series with infinite radii of convergence:

[TABLE]

Proof.

By using the integral equation (3.11) for the real function $W(t)$, the Laplace transform $Q(s)$ of $W(t)$ satisfies the following first order ordinary differential problem

[TABLE]

whose solution is given by (3.12). From a series development of the exponential function we obtain

[TABLE]

and therefore we get

[TABLE]

that is the expression (3.13) for $W(t)$ holds by using that

[TABLE]

[TABLE]

[TABLE]

and the definition

[TABLE]

∎

Corollary 3.2**.**

The heat flux at the boundary $x=0$ of the solution of the Problem 3.1 is given by

[TABLE]

where $W(t)$ is given by (3.13).

Corollary 3.3**.**

The first terms of the development of the serie (3.13) of the average of the total heat flux at $x=0$ are given by

[TABLE]

which give us a singularity of $W$ of the type $t^{-1/2}$ at $t=0$.

Moreover, the first terms of the development of the heat flux at $x=0$ of the expression (3.15) are given by

[TABLE]

which give also us a singularity of $u_{x}(0,t)$ of the type $t^{-1/2}$ at $t=0$.

Proof.

It follows from the following results:

[TABLE]

which can be generalized to

[TABLE]

∎

Now, we will give a new proof of the serie (3.13) for the average of the total flux $W(t)$. We use the Adomian decomposition method [1, 2, 3, 7, 25, 26] through a serie expansion of the type

[TABLE]

in the Volterra integral equation (3.11). Taking

[TABLE]

we obtain the following recurrence formulas

[TABLE]

Then by (3.17) and (3.18) we get

[TABLE]

Theorem 3.4**.**

Moreover by a double induction principle we have

[TABLE]

and

[TABLE]

with $W_{0}$ and $W_{1}$ are given respectively by (3.17) and (3.19).

Proof.

Using (3.17) and (3.18) we get

[TABLE]

[TABLE]

[TABLE]

taking into account that

[TABLE]

and their generalizations by

[TABLE]

[TABLE]

The first step of the double induction principle is verified taking into account the above computations. For the second step, we suppose by induction hypothesis that we have (3.20) and (3.21). Therefore, we obtain

[TABLE]

and

[TABLE]

by using (3.22) and (3.23). Then, the proof by the induction principle holds. ∎

Conclusion: We have obtained the global solution of a non-classical heat conduction problem in a semi-n-dimensional space, in which the source depends of the average of the total heat flux on the face $x=0$. Moreover, for the one-dimensional case we have obtained the explicit solution by using the Laplace transform and also the Adomian decomposition method.

Acknowledgements: This paper was partially sponsored by the Institut Camille Jordan St-Etienne University for first author, and the projects PIP $\#$ 0275 from CONICET - Univ. Austral and ANPCyT PICTO Austral 2016 $\#$ 090 (Rosario, Argentina) for the second author.