Explicit solutions for a nonclassical heat conduction problem for a semiinfinite strip with a nonuniform heat source
 Andrea N Ceretani^{1},
 Domingo A Tarzia^{1}Email author and
 Luis T Villa^{2}
Received: 19 March 2015
Accepted: 13 August 2015
Published: 4 September 2015
Abstract
A nonclassical initial and boundary value problem for a nonhomogeneous onedimensional heat equation for a semiinfinite material with a zero temperature boundary condition is studied. It is not a standard heat conduction problem because a nonuniform heat source dependent on the heat flux at the boundary is considered. The purpose of this article is to find explicit solutions and analyze how to control their asymptotic temporal behavior through the source term.
Explicit solutions independent of the space or temporal variables, solutions with separated variables and solutions by an integral representation depending on the heat flux at the boundary are given. The controlling effects of the source term are analyzed by comparing the asymptotic temporal behavior of solutions corresponding to the same problem with and without source term. Finally, a relationship between the problem considered here with another nonclassical problem for the heat equation is established, and explicit solutions for this second problem are also obtained.
In this article, we give explicit solutions and analyze how to control them through the source term for several nonclassical heat equation problems. In addition, our results enable us to compute the asymptotic temporal behavior of the heat flux at the boundary for each explicit solution obtained. As a consequence of our study, several solved nonclassical problems for the heat equation that can be used for testing new numerical methods are given.
Keywords
MSC
1 Introduction
Our purpose is to find explicit solutions to Problem P and study how to control their asymptotic temporal behavior through the source term \(\Phi(x)F(u_{x}(0,t),t)\). Exact solutions to initial and boundary value problems for the heat equation allow us to better understand qualitative features of the thermal and diffusive process under them. In particular, this knowledge might give us some insights to develop numerical methods dealing with more complex phenomena related with more complicated equations. Even for cases where a physical interpretation is not obvious, exact solutions are important because of their use for testing accuracy, stability and convergence of numerical methods for solving partial differential equations without any known analytical solution. In addition, how to control the asymptotic behavior of solutions to Problem P through the source term in equation (1) gives us some insights about when it is possible to have stationary solutions. It also gives us a better understanding about how solutions to Problem P are related with the solutions to the same problem but in absence of source term, that is, with solutions to the problem where no cooler or heater term is considered.
Problem P for the slab \(0< x<1\) was studied in [15]. Recently, free boundary problems (Stefan problems) for the nonclassical heat equation have been studied in [16–21], where some explicit solutions are also given, and a first study of nonclassical heat conduction problem for an ndimensional material was given in [22]. There exists a large recent scientific production where exact solutions for heat transfer problems arising from a wide field of applications are given; see, for example, [23–36]. Numerical schemes for Problem P were studied in [37] when a nonhomogeneous boundary condition is considered and numerical solutions are given for two particular choices of data function corresponding to problems with known explicit solutions.
2 Explicit solutions for Problem P
2.1 Explicit solutions independent of space or temporal variables
Theorem 2.1
 (1)
Problem P does not admit any nontrivial solution independent of the space variable x.
 (2)If:
 (a)F is the zero function and h is defined bywith \(\eta>0\),$$ h(x)=\eta x,\quad x\geq0, $$(8)
 (b)F is a constant function defined bywith \(\nu\in\mathbb{R}\{0\}\), and h is a twice differentiable function such that \(h(0)\) exists and$$ F(V,t)=\nu,\quad V\in\mathbb{R}, t>0, $$(9)$$ h''(x)=\nu\Phi(x),\quad x>0, $$(10)
is a solution to Problem P independent of the temporal variable t.$$ u(x,t)=h(x),\quad x\geq0, t\geq0, $$(11)  (a)
Proof
 (1)If Problem P has a solution u independent of the space variable x, thenTherefore u is the zero function.$$ u(x,t)=u(0,t)=0,\quad x>0, t>0\quad \mbox{and} \quad u(0,0)= \lim _{x\to0^{+}}h(x)=0. $$(12)
 (2)
It is easy to check that the function u given in (11) is a solution to Problem P given in this item.
2.2 Explicit solutions with separated variables
Theorem 2.2
Proof
An easy computation shows that the function u given in (17) is a solution to Problem P. □
Remark 1
Under the hypothesis of the previous theorem, the problem of finding explicit solutions with separated variables to Problem P reduces to solving the initial value problem (15)(16).
With the spirit of exhibiting explicit solutions to Problem P, our next result summarizes explicit solutions to the initial value problem (15)(16) corresponding to three different definitions of the function F.
Proposition 2.1
If in Theorem 2.2 we consider:
2.3 Explicit solutions obtained from an integral representation
Our next theorem is a restatement of Theorem 1 in [13] for a particular choice of the function F in Problem P.
Theorem 2.3
 (1)h be a continuously differentiable function in \(\mathbb{R}^{+}\) such that \(h(0)\) exists and there exist positive numbers ϵ, \(c_{0}\) and \(c_{1}\) such that$$ \biglh(x)\bigr\leq c_{0}\exp{\bigl(c_{1}x^{2\epsilon} \bigr)},\quad \forall x>0, $$(30)
 (2)
Φ be a locally Hölder continuous function
 (3)F be the function defined bywith \(\nu>0\).$$ F=F(V,t)=\nu V, \quad V\in\mathbb{R}, t>0, $$(31)
Remark 2
The interest of the previous theorem is that it enables us to find an explicit solution \(u=u(x,t)\) to Problem P by finding the corresponding heat flux \(u_{x}(0,t)\) at the boundary \(x=0\) as a solution of the integral equation (38).
 (1)
F is given as in (31),
 (2)h is defined bywith \(\eta\in\mathbb{R}\{0\}\) and \(m\geq1\),$$ h(x)=\eta x^{m},\quad x>0, $$(41)
 (3)Φ is given by one of the following expressions:with \(\lambda>0\) and \(\mu>0\).$$ \varphi_{1}(x)=\lambda x, \qquad\varphi_{2}(x)= \mu\sinh{(\lambda x)} \quad\mbox{or} \quad\varphi_{3}(x)=\mu\sin{( \lambda x)},\quad x>0, $$(42)
It is easy to check that for this choice of functions F, h and Φ, Problem P is under the hypothesis of the previous theorem (see Appendix 1). Therefore, it has the solution \(u=u(x,t)\) given in (34).
Proposition 2.2
Proof
Corollary 2.1
Proof
The last corollary enables us to obtain the asymptotic behavior of the heat flux \(u_{x}(0,t)\) at the face \(x=0\) when t tends to +∞ for an odd number m. The next result is related to this topic. We do not reproduce here the computations involved in its proof, which follows by taking the limit when t tends to +∞ in the expression of \(u_{x}(0,t)\) given in Corollary 2.1.
Corollary 2.2
 (1)if \(m=1\), we have$$ \lim_{t\to+\infty}u_{x}(0,t)=0, $$(54)
 (2)if \(m=3\), we have$$ \lim_{t\to+\infty}u_{x}(0,t)=\frac{6\eta}{\nu\lambda}, $$(55)
 (3)if \(m\geq5\), we have$$ \lim_{t\to+\infty}u_{x}(0,t)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty& \textit{if }\eta< 0,\\ +\infty& \textit{if }\eta>0. \end{array}\displaystyle \right . $$(56)
Proposition 2.3
 (1)If \(\sigma\neq0\), then$$ u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{\eta}{\sigma} (\lambda+\nu\mu\exp{(\lambda\sigma t)} ) &\textit{if }m=1,\\ \frac{c\lambda}{\sigma}t^{(m1)/2}+\frac{c(m1)\nu\mu}{2\sigma}\exp {(\lambda\sigma t)} \\ \quad{}\times\int_{0}^{t}\tau^{(m3)/2}\exp{(\lambda \sigma\tau)}\,d\tau& \textit{if }m>1, \end{array}\displaystyle \right .\quad t>0. $$(59)
 (2)If \(\sigma=0\), then$$ u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \eta(1\lambda^{2}t)& \textit{if }m=1,\\ c t^{(m1)/2}\frac{2c\lambda^{2}}{m+1}t^{(m+1)/2}& \textit{if }m>1, \end{array}\displaystyle \right .\quad t>0. $$(60)
Proof
An easy computation shows that the expressions given in (59) and (60) satisfy the integral equation (38). Therefore, they correspond to the heat flux \(u_{x}(0,t)\) at the boundary \(x=0\) for the solution u of Problem P given in (34). □
Corollary 2.3
Proof
It follows by solving the integral in the expression of u given in (59) and the use of identity (53). □
Corollary 2.4
 (1)If \(\sigma\neq0\), then:
 (a)if \(m=1\), we have$$ \lim_{t\to+\infty}u_{x}(0,t)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty&\textit{if }\sigma>0,\eta< 0,\\ +\infty&\textit{if }\sigma>0,\eta>0,\\ \frac{\eta\lambda}{\sigma}&\textit{if }\sigma< 0, \end{array}\displaystyle \right . $$(64)
 (b)if \(m\geq3\), we have$$ \lim_{t\to+\infty}u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty& \textit{if }\sigma\eta< 0,\\ +\infty& \textit{if }\sigma\eta>0. \end{array}\displaystyle \right . $$(65)
 (a)
 (2)If \(\sigma=0\), then$$ \lim_{t\to+\infty}u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty& \textit{if }\eta>0,\\ +\infty& \textit{if }\eta< 0. \end{array}\displaystyle \right . $$(66)
Proposition 2.4
 (1)If \(\delta\neq0\), then$$ u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \frac{\eta}{\delta} (\lambda\nu\mu\exp{(\lambda\delta t)} )& \textit{if }m=1,\\ \frac{c\lambda}{\delta}t^{(m1)/2}\frac{c(m1)\nu\mu}{2\delta}\exp {(\lambda\delta t)}\\ \quad{}\times \int_{0}^{t}\tau^{(m3)/2}\exp{(\lambda \delta\tau)}\,d\tau& \textit{if }m>1, \end{array}\displaystyle \right .\quad t>0. $$(67)
 (2)If \(\delta=0\), then$$ u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \eta(1+\lambda^{2}t)& \textit{if }m=1,\\ c t^{(m1)/2}+\frac{2c\lambda^{2}}{m+1}t^{(m+1)/2}& \textit{if }m>1, \end{array}\displaystyle \right .\quad t>0. $$(68)
Proof
The proof of (67) and (68) follows by replacing \(\lambda^{2}\) by \(\lambda^{2}\) and σ by δ in the proof of Proposition 2.3. □
Corollary 2.5
Proof
It follows by solving the corresponding integral in expression (67) and the use of identity (53). □
Corollary 2.6
 (1)If \(\delta\neq0\), then:
 (a)if \(m=1\), we have$$ \lim_{t\to+\infty}u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty&\textit{if }\delta< 0,\eta< 0,\\ +\infty&\textit{if }\delta< 0,\eta>0,\\ \frac{\eta\lambda}{\delta} &\textit{if }\delta>0, \end{array}\displaystyle \right . $$(72)
 (b)if \(m=3\) or \(m=5\), we have$$ \lim_{t\to+\infty}u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty&\textit{if }\eta< 0,\\ +\infty&\textit{if }\eta>0 \end{array}\displaystyle \right . $$(73)
 (c)if \(m\geq7\), we have$$ \lim_{t\to+\infty}u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty&\textit{if }\delta\eta< 0,\\ +\infty&\textit{if }\delta\eta>0. \end{array}\displaystyle \right . $$(74)
 (a)
 (2)If \(\delta=0\), then$$ \lim_{t\to+\infty}u_{x}(0,t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \infty& \textit{if }\eta< 0,\\ +\infty& \textit{if }\eta>0. \end{array}\displaystyle \right . $$(75)
The next result is related to the behavior of the heat flux \(u_{x}(0,t)\) at the face \(x=0\) when t tends to 0^{+}, and shows that it is independent of the choice of Φ as any of the functions given in (42).
Corollary 2.7
Proof
It follows straightforwardly by computing the limit for the expression of \(u_{x}(0,t)\) given in Propositions 2.2, 2.3 or 2.4, according the definition of Φ. □
We end this section by giving explicit solutions to each Problem P. The proofs of the three following propositions follow from Theorem 2.3 and Corollary 2.1, 2.3 or 2.5, according to the definition of Φ (see Appendix 2).
Proposition 2.5
Remark 3
If \(m=1\), polynomial \(p_{1,m}(x)\) is defined by \(p_{1,m}(x)=0\), \(x>0\).
Proposition 2.6
Remark 4
If \(m=1\), polynomial \(p_{2,m}(x)\) is defined by \(p_{2,m}(x)=\frac{\nu \lambda}{\sigma}\), \(x>0\).
Proposition 2.7
Remark 5
If \(m=1\), polynomial \(p_{3,m}(x)\) is defined by \(p_{3,m}(x)=\frac{\nu \lambda}{\delta}\), \(x>0\).
3 The controlling problem
 (1)
\(VF(V)\geq0\), \(\forall V\in\mathbb{R}\),
 (2)
\(F(0)=0\),
 (3)
F is convex in \((0,+\infty)\),
 (4)
\(\lim_{V\to+\infty}F'(V)=\kappa>0\),
With the aim of supplementing the results given in [3], we will carry out our analysis under conditions which lead us to functions F depending on only one real variable, that is, \(F=F(V)\).
Next Theorems 3.1, 3.2 and 3.3 are respectively related with the results obtained in Sections 2.1, 2.2 and 2.3.
Remark 6
Theorem 3.1
Proof
Theorem 3.2
Proof
(2) It follows in the same manner as the proof of the previous item. □
We see from the previous theorem that it is possible to control a solution to Problem P through the parameters involved in the definition of the source term \(\Phi F\). When \(F(V)=\nu V\), we can increase (\(\gamma<0<\sigma\)) or decrease (\(0<\gamma<\sigma\)) the velocity of convergence to ∞ for u with respect to the velocity of convergence for \(u_{0}\). We also can stabilize the problem by doing u tending to a constant value (\(0<\sigma\leq\gamma\)) when \(u_{0}\) is going to ∞. When \(F(V)=\nu V^{n}\), we can decrease (\(\sigma>0\) and \(1=\frac{\lambda\mu\delta^{n}}{\sigma\eta^{1n}}\)) or maintain (\(\sigma >0\) and \(1\neq\frac{\lambda\mu\delta^{n}}{\sigma\eta^{1n}}\)) the velocity of convergence to ∞ for u with respect to the velocity of convergence for \(u_{0}\). We also can decrease the velocity of convergence to 0 for u with respect to the velocity of convergence for \(u_{0}\) (\(\sigma<0\)).
Theorem 3.3
From the previous theorem, we see again that there exist several cases where we can control a solution to Problem P through the source term \(\Phi F\).
4 Explicit solutions for Problem P̃
In this section we consider Problem P̃ given in (5)(7) with the aim of finding exact solutions. This problem corresponds to another temperature regulation problem where the temperature controller device depends on the temperature at the fixed boundary of the material instead of the heat flux on it, and a heat flux initial condition is known in place of a temperature condition.
The following theorem states a relationship between Problems P and P̃ given in (5)(7), and it was proved in [13].
Theorem 4.1
We see from Theorem 4.1 that we can find exact solutions to Problem P̃ from exact solutions to another temperature regulation problem which has the form of Problem P.
We end this section by giving explicit solutions for some particular cases of Problem P̃.
Proposition 4.1
Let g̃ be the zero function and:
 (a)
F̃ be the zero function and h̃ be a constant function, or
 (b)F̃ be a constant function defined bywith \(k\in\mathbb{R}\{0\}\), Φ̃ be a locally integrable function in \(\mathbb{R}^{+}\) and h̃ be a differentiable function such that$$ \widetilde{F}(V,t)=k,\quad V\in\mathbb{R}, t>0, $$(111)Then the function v defined by$$ \tilde{h}(x)=k \int_{0}^{x}\widetilde{\Phi}(\xi)\,d\xi,\quad x>0. $$(112)is a solution to Problem P̃ independent of the temporal variable t.$$ v(x,t)=\tilde{h}(x),\quad x\geq0, t\geq0 $$(113)

\(F(V,t)=\nu V\), \(V\in\mathbb{R}\), \(t>0\), with \(\nu\in\mathbb{R}\{0\}\),

\(F(V,t)=f_{1}(t)+f_{2}(t)V\), \(V\in\mathbb{R}\), \(t>0\), with \(f_{1}, f_{2}\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{+})\), or

\(F(V,t)=V^{n}f(t)\), \(V\in\mathbb{R}\), \(t>0\), with \(n<1\), \(f\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{+})\), \(f>0\) and \(\lambda, \delta, \eta>0\),
Proof
It follows from the previous theorem and the explicit solutions to Problem P obtained in Section 2. □
5 Conclusions
In this paper we consider a nonclassical initial and boundary value problem for a nonhomogeneous onedimensional heat equation which represents a temperature regulation problem for a semiinfinite homogeneous isotropic medium where the temperature controller device depends on the heat flux at the fixed boundary, an initial temperature distribution is known and the temperature at the fixed boundary is constant in time. We find explicit solutions for several cases of this problem, which, in particular, enables us to give explicit formulae for the heat flux at the boundary and to compute its asymptotic temporal behavior.
We also analyze how the source term affects the asymptotic temporal behavior of each explicit solution u obtained in this paper by comparing the limits of u and the solution \(u_{0}\) to the same problem but in absence of source term. As a result, we obtain conditions on the parameters involved in the definition of the source term that enables us to control the solutions u with respect to \(u_{0}\). In particular, we give conditions on data functions under which stationary solutions exist.
By giving a relationship between the problem considered here with another related nonclassical heat equation problem, we obtain explicit solutions for several particular cases of another temperature regulation problem where the thermostat depends on the temperature at the fixed boundary of the material instead on the heat flux on it, and a heat flux initial condition is known in place of a temperature condition.
As a consequence of our study, several solved nonclassical problems for the heat equation that can be used for testing new numerical methods are given. In addition, exact solutions given in this article also provide reference values for comparisons in laboratory experiments.
Declarations
Acknowledgements
This paper has been partially sponsored by the Project PIP No. 0534 from CONICETUA (Rosario, Argentina) and AFOSRSOARD Grant FA 95501410122. The authors would like to thank an anonymous referee and the editor for the helpful comments.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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