R E S E A R C H
Open Access
Non-classical heat conduction problem
with nonlocal source
Mahdi Boukrouche
1and Domingo A Tarzia
2**Correspondence:
2Departamento de
Matemática-CONICET, FCE, Universidad Austral, Paraguay 1950, Rosario, S2000FZF, Argentina Full list of author information is available at the end of the article
Abstract
We consider the non-classical heat conduction equation, in the domain
D=Rn–1×R+, for which the internal energy supply depends on an integral function
in the time variable of the heat flux on the boundaryS=
∂
D, with homogeneous Dirichlet boundary condition and an initial condition. The problem is motivated by the modeling of temperature regulation in the medium. The solution to the problem is found using a Volterra integral equation of second kind in the time variabletwith a parameter inRn–1. The solution to this Volterra equation is the heat flux (y,s)→ V(y,t) =ux(0,y,t) onS, which is an additional unknown of the considered problem.
We show that a unique local solution, which can be extended globally in time, exists. Finally a one-dimensional case is studied with some simplifications. We obtain the solution explicitly by using the Adomian method, and we derive its properties.
MSC: 35C15; 35K05; 35K20; 35K60; 45D05; 45E10; 80A20
Keywords: non-classicaln-dimensional heat equation; nonlocal sources; Volterra integral equation; existence and uniqueness of solution; integral representation of solution; explicit solution in one-dimensional case and its properties
1 Introduction
Let us consider the domainDand its boundarySdefined by
D=Rn–×R+=(x,y)∈Rn:x=x> ,y= (x, . . . ,xn)∈Rn–
, (.)
S=∂D=Rn–× {}=(x,y)∈Rn:x= ,y∈Rn–. (.)
The aim of this paper is to study the following Problem . with a non-classical heat-flow
feedback problem in the domainDwith nonlocal source, for which the internal energy
supply depends on the integraltux(,y,s)dson the boundaryS.
Problem . Find the temperatureuat (x,y,t) such that it satisfies the following condi-tions:
ut–u= –F
t
ux(,y,s)ds
, x=x> ,y∈Rn–,t> ,
u(,y,t) = , y∈Rn–,t> ,
u(x,y, ) =h(x,y), x> ,y∈Rn–,
wheredenotes the Laplacian inRn. This problem is motivated by the modeling of
tem-perature regulation in an isotropic medium, with non-uniform and nonlocal sources that provide cooling or heating system; this could represent a feedback air-conditioning
sys-tem in a macro scale installation. According to the properties of the function F with
respect to the heat flow, V(y,s) =ux(,y,s) at the boundaryS. For example, assuming
that
VF(V) > , ∀V= ,F() = , (.)
with
FV(y,t) =F
t
V(y,s)ds
(.)
then, see [, ], the cooling source occurs whenV(y,t) > and the heating source occurs whenV(y,t) < .
Some references on the subject are [] whereF(V) =F(V) and [–] where the following semi-dimension of this nonlinear problem was considered. The non-classical one-dimensional heat equation in a slab with fixed or moving boundaries was studied in [– ]. More references on the subject can be found in [–]. To our knowledge, it is the first time that the solution to a non-classical heat conduction of the type of Problem . is given. Other non-classical problems can be found in [].
The goal of this paper is to obtain in Section the existence and uniqueness of the global solution of the non-classical heat conduction Problem ., which is given through a Volterra integral equation. In Section we obtain the explicit solution of the one-dimensional case of Problem ., with some simplifications, which is obtained by using the Adomian method through a double induction principle.
We recall here the Green’s function for then-dimensional heat equation with
homoge-neous Dirichlet’s boundary conditions, given the following expression [, ]
G(x,y,t;ξ,η,τ) = exp[–
y–η
(t–τ)]
(√π(t–τ))n–G(x,t,ξ,τ), (.)
whereGis the Green’s function for the one-dimensional case given by
G(x,t,ξ,τ) = e –((x–t–ξ)τ)
–e–((x+tξ–)τ)
√π(t–τ) , t>τ.
2 Existence results
In this section, we give first in Theorem . the integral representation (.) of the solution
of considered Problem ., but it depends on the heat flowVon the boundaryS, which
Theorem . The integral representation of a solution of considered Problem.is given by the following expression:
u(x,y,t) =u(x,y,t)
–
t
erf( x √t–τ)
(√π(t–τ))n–
Rn– exp
–y–η
(t–τ)
FV(η,τ) dη
dτ, (.)
where
erf(ζ) =
√
π
ζ
e–XdX
is the error function,with
u(x,y,t) =
D
G(x,y,t;ξ,η, )h(ξ,η)dξdη, (.)
and the heat flux V(y,t) =ux(,y,t)on the surface x= satisfies the following Volterra
integral equation:
V(y,t) =V(y,t)
–
t
(√π(t–τ))n
Rn– exp
–y–η
(t–τ)
FV(η,τ) dη
dτ (.)
in the variable t> ,with y∈Rn–is a parameter and
V(y,t) =
D
G,x(,y,t;ξ,η, )h(ξ,η)dξdη, (.)
where the function(y,t)→F(V(y,t))is defined by(.)for y∈Rn–and t> .
Proof As the boundary condition in Problem . is homogeneous, we have from []
u(x,y,t) =
D
G(x,y,t;ξ,η, )h(ξ,η)dξdη
+
t
D
G(x,y,t;ξ,η,τ)–FV(η,τ) dξdηdτ, (.)
and therefore
ux(x,y,t) =
D
G,x(x,y,t;ξ,η, )h(ξ,η)dξdη
+
t
D
G,x(x,y,t;ξ,η,τ)
–FV(η,τ) dξdηdτ. (.)
From (.) (the definition ofG) by derivation with respect tox, then takingx= , we obtain
D
G,x(,y,t;ξ,η,τ)F
=
Rn–
F(V(η,τ))e–
y–η (t–τ)
(t–τ)n+ (√π)n
+∞
ξe– ξ (t–τ)dξ
dη
=
(√π(t–τ))n
Rn–F
V(η,τ) e–
y–η
(t–τ) dη, (.)
as
+∞
ξe–
ξ
(t–τ)dξ= (t–τ).
Thus, takingx= in (.) with (.), we get (.). Also by (.) we obtain
D
G(x,y,t;ξ,η,τ)FV(η,τ) dξdη
=
(√π(t–τ))n
D
e–
y–η (t–τ) e–
(x–ξ) (t–τ) –e–
(x+ξ)
(t–τ)FV(η,τ) dξdη
=
(√π(t–τ))n
R+
e–((x–t–ξτ)) –e– (x+ξ) (t–τ)dξ
Rn– e–y–η
(t–τ) FV(η,τ) dη,
and by using
+∞
e–((xt––ξτ)) dξ = √t–τ
–∞
e–XdX+
x
√t–τ
e–XdX
=π(t–τ)
+erf
x
√t–τ
and
+∞
e –(x+ξ)
(t–τ) dξ = √t–τ
+∞
e–XdX–
x
√t–τ
e–XdX
=π(t–τ)
–erf
x
√t–τ
,
we get
D
G(x,y,t;ξ,η,τ)FV(η,τ) dξdη=
erf(√x
t–τ)
(√π(t–τ))n–
Rn–e
–(y–t–ητ)F
V(η,τ) dη.
Taking this formula in (.), we obtain (.).
To solve Volterra integral equation (.), we rewrite it in a suitable form.
Lemma . Volterra integral equation(.)can be rewritten in the following form:
V(y,t) =
t(√πt)n
R+ ξe–ξ
t
Rn–e
–y–ηth(ξ,η)dη
dξ
–
(√π)n
t
(t–τ)n/
Rn–F
V(η,τ)e–
y–η
Proof Using the derivative, with respect tox, of (.), then takingx= andτ = , then taking the new expression ofV(y,t) in Volterra integral equation (.), we obtain (.).
Theorem . Assume that h∈C(D),F∈C(R)and locally Lipschitz inR,then there exists a unique solution of Problem.locally in times which can be extended globally in times.
Proof We know from Theorem . that, to prove the existence and uniqueness of the so-lution (.) of Problem ., it is enough to solve Volterra integral equation (.). So we rewrite it again as follows:
V(y,t) =f(y,t) +
t
gy,τ,V(y,τ) dτ (.)
with
f(y,t) =
t(√πt)n
R+ξe –ξt
Rn–e
–y–ηth(ξ,η)dη
dξ (.)
and
gt,τ,y,V(y,τ) = –(t–τ) –n/
(√π)n
Rn–F
(V(η,τ))e–y–η
(t–τ) dη. (.)
We have to check conditions H to H in Theorem . page , and H and H in The-orem . page in [].
• The functionf is defined and continuous for all(y,t)∈Rn–×R+, so H holds. • The functiongis measurable in(t,τ,y,x)for≤τ≤t< +∞,x∈R,y∈Rn–, and
continuous inxfor all(y,t,τ)∈Rn–×R+×R+,g(y,t,τ,x) = ifτ >t, so here we need the continuity of
V(η,τ)→FV(η,τ) =F
τ
V(η,s)ds
,
which follows from the hypothesis thatF∈C(R). So H holds. • For allk> and all bounded setsBinR, we have
g(y,t,τ,X)≤ (√π)nsup
X∈B
F(X)(t–τ)–n/
Rn–e –y–η
(t–τ)
dη
≤
(√π)nsup X∈B
F(X)(t–τ)–n/π(t–τ) n–
= √ πsupX∈B
F(X)√ (t–τ),
thus there exists a measurable functionmgiven by
m(t,τ) =√ πsupX∈B
F(X)√
(t–τ) (.)
such that
and it satisfies
sup
t∈[,K]
t
m(t,τ)dτ=√ πsupX∈B
F(X) sup
t∈[,k]
t
√
t–τ dτ
=
πsupX∈B
F(X) sup
t∈[,k]
–(t–τ)|t
=
πsupX∈B
F(X) sup
t∈[,k]
√
t≤
√
k π supX∈B
F(X)<∞,
so H holds.
• Moreover, we have also
lim
t→+
t
m(t,τ)dτ=√ πsupX∈B
F(X)lim
t→+
t
dτ
√
t–τ
=√ πsupX∈B
F(X)lim
t→+(
√
t) = , (.)
and
lim
t→+
T+t T
m(t,τ)dτ=√ πsupX∈B
F(X)lim
t→+(
√
t) = . (.)
• For each compact subintervalJofR+, each bounded setBinRn–, and eacht ∈R+,
we set
At,y,V(η) =gt,τ;y,V(η,τ) –gt,τ;y,V(η,τ) ,
At,y,V(η) = (√π)n
J
Rn–
e–y –η
(t–τ) F(V(η,τ)) (t–τ)–n/ –e
–(yt–η
–τ)F(V(η,τ)) (t–τ)–n/
dηdτ
as the functionτ→V(η,τ)is continuous, then
τ →
τ
V(η,s)ds
isC(R)and is in the compactB⊂Rfor allη∈Rn–, so by the continuity ofFwe get
F(V(η,τ))⊂F(B). That is, there existsM> such that|F(V(η,τ))| ≤Mfor all
(η,τ)∈Rn–×R+. So
sup
V(η)∈C(J,B)
At,y,V(η)
≤ M
(√π)n sup V(η)∈C(J,B)
Rn–
e–
y–η
(t–τ)
√
(t–τ)ndη–
Rn– e–
y–η
(t–τ)
√
(t–τ)
ndη
.
Using that
Rn– exp
–y–η
(t–τ)
we obtain
sup
V(η)∈C(J,B)
At,y,V(η) ≤ M
(√π)n sup V(η)∈C(J,B)
(√(√π(t–τ))n–
t–τ)n –
(√π(t–τ))n– (√t–τ)n
thus
sup
V(η)∈C(J,B)
At,y,V(η) ≤√M
πV(ηsup)∈C(J,B)
√t–τ–
√
t–τ
√
(t–τ)(t–τ)
.
Thus we deduce that
lim
t→t
J
sup
V(η)∈C(J,B)
At,y,V(η) dτ= .
So H holds.
• For all compactI⊂R+, for all functionsψ∈C(I,Rn), and allt
> ,
gt,τ;ψ(τ) –gt,τ,ψ(τ)
=
(√π)n
Rn–F
ψ(τ) e –(y–tη–τ)
(t–τ)n/–
e– y–η (t–τ)
(t–τ)n/
dτ
asF∈C(R)andψ∈C(I,Rn), then there exists a constantM> such that
|F(ψ(τ))| ≤Mfor allτ∈I. Then we obtain, as for H, that
lim
t→t
I
gt,τ;ψ(τ) –gt,τ,ψ(τ) dτ= .
So H holds.
• Now, for each constantK> and each bounded setB⊂Rn–, there exists a
measurable functionϕsuch that
g(y,t,τ,x) –g(y,t,τ,X)≤ϕ(t,τ)|x–X|
whenever≤τ≤t≤Kand bothxandXare inB. Indeed, asFis assumed to be a locally Lipschitz function inR, there exists a constantL> such that
F(x) –F(X)≤L(τ)|x–X| ∀(x,X)∈B
withL(τ) =Lτ. Then we have
g(y,t,τ,x) –g(y,t,τ,X)= (√π)n
Rn–
(t–τ)–n/e–(y–t–ητ)F
(x) –F(X) dη
≤
(√π)n
Rn–e –(y–t–ητ)
dη
(t–τ)–n/Lτ|x–X|
≤√ Lτ
thenϕ(t,τ) =√Lτ
π(t–τ). We have also for eacht∈[,k]the functionϕ∈L
(,t)as a
function ofτ and
t+l t
ϕ(t+l,τ)τdτ =√L π
t+l t
τdτ
√
t+l–τ = L
√
π
l
u–t–l du
=√Ll π
l+t–
→ withl→,
whereu=√t+l–τ.
So H holds. All the conditions H to H are satisfied with (.) and (.). Thus from [] (Theorem . p., Theorem . p. and Theorem . p.) there exists a unique local times solution of Volterra integral equation (.) which can be extended globally in times. 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 . for the temperature defined by
Problem . Find the temperatureuat (x,t) such that it satisfies the following conditions:
ut–uxx= –F
t
ux(,s)ds
, x> ,t> ,
u(,t) = , t> ,
u(x, ) =h(x), x> .
Taking into account that
t
G(x,t,ξ,τ)dξ=erf
x
√t–τ
, (.)
thus the solution of Problem . is given by
u(x,t) =u(x,t) –
t
erf
x
√t–τ
F
τ
W(σ)dσ
dτ (.)
with
u(x,t) =
t
G(x,t,ξ, )h(ξ)dξ, (.)
andW(t) =ux(,t) is the solution of the following Volterra integral equation:
W(t) =V(t) –
t
F(τW(σ)dσ)
√
π(t–τ) dτ, (.)
where
V(t) = √πt/
+∞
ξe–ξ/th(ξ)dξ=√
πt
+∞
For the particular case
h(x) =h> pourx> , andF(W) =λWforλ∈R, (.)
we have
u(t,x) =herf
x
√t
, (.)
and integral equation (.) becomes
W(t) =√h πt–λ
t
τ
W(σ)dσ
√
π(t–τ) dτ. (.)
Lemma . Assume(.)holds.The solution of Problem.is given by
u(x,t) =herf
x
√t
–λ
t
erf
x
√t–τ
U(τ)dτ, (.)
where U is given by
U(t) = √h π
t
g(τ)
√
t–τ dτ, (.)
and g is the solution of the Volterra integral equation
g(t) = –√λ π
t
g(τ)√t–τdτ. (.)
Moreover,the heat flux on x= is given by
ux(,t) =U(t) =
h
√
πt–hλ
t
g(τ)dτ, t> . (.)
Proof We set
U(t) =
t
W(τ)dτ, (.)
thus the functionUsatisfies the following new Volterra integral equation:
U(t) = h
t π –
λ
√
π
t
τ
U(σ)
√
τ–σ dσdτ
= h
t π –
λ
√
π
t
U(τ)√t–τdτ, t> (.)
by using the following equality:
t σ
dτ
√
τ–σ =
√
From [], p., the solutiont→U(t) of integral equation (.) is given by (.), where
gis the solution of Volterra equation (.). From (.) we obtain that
t
g(τ)
√
t–τ dτ =
√
t–λ√π
t
g(τ)√t–τdτ, (.)
using the following equality:
t σ
√
τ–σ
√
t–τ dτ = (t–σ)
√
ξ
√
–ξ dξ= (t–σ)B
,
= (t–σ)( )(
) () =
π
(t–σ), (.)
whereBandare the classical beta and gamma functions defined below.
Therefore, we have that
U(t) = h
t π –λh
t
g(τ)(t–τ)dτ, (.)
and then the heat flux onx= is given byux(,t) =W(t) =U(t), that is, (.) holds.
We recall here the well-known beta and gamma functions defined respectively by
B(x,y) =
tx–( –t)y–dt, x> ,y> ,
(x) =
+∞
tx–e–tdt, x> .
We will use in the next theorem the well-known relations
B(x,y) =(x)(y)
(x+y), (x+ ) =x(x) ∀x> ,
=√π, (n+ ) =n! ∀n∈N,
and in particular the following one.
Lemma . For all integers n≥,we have
n+
=(n– )!! n
√
π,
and we use the definition
(n– )!! = (n– )(n– )(n– )· · ···
Proof Forn= , we get() =
√
π
, which is true. By induction we obtain that
n+ +
=
n+ + =
n+
n+
=
n+
(n– )!! n
√
π=(n+ )!! n+
√
π,
thus the lemma is true.
Corollary . For all integers n≥,we have also
n+ +
=(n+ )!! n+
√
π,
B
, n+
=(
)((n+ ) + ) (n+ +) =
((n+ ))!(n+)+
(n+ )!! ,
B
, n+
=(
)(n+
) ((n+ ) + )=
π(n+ )!! ((n+ ))!(n+),
which will be useful in the next lemma.
First, we need some preliminary simple results in order to obtain the solution of integral equation (.).
t
√
t–τdτ= t
/,
t
τ/√t–τdτ= π t
, (.)
t
τ√t–τdτ= !
!! t /,
t
τ/√t–τdτ=π!! !t
, (.)
t
τ√t–τdτ = !
!!t /,
t
τ/√t–τdτ=π!! !t
, (.)
which can be generalized by the following ones.
Lemma . For all integers n≥,we have
t
τn+√t–τdτ=
n+((n+ ))!
(n+ )!! t
(n+)/, (.)
t
τ(n+)√t–τdτ= π(n+ )!! (n+)((n+ ))!t
(n+). (.)
Proof Taking the change of variableτ=tξin (.) and using Corollary ., we get
t
τn+√t–τdτ =t(n+)
ξn+( –ξ)dξ
=t(n+)
ξ(n+)–( –ξ)–dξ
=t(n+)B
, n+
=((n+ ))! (n+)+
and
t
τ(n+)√t–τdτ=t(n+)
ξ(n+)–( –ξ)–dξ
=t(n+)B
, n+
= π(n+ )!! ((n+ ))!(n+)t
(n+),
thus (.)-(.) hold.
Now, we will obtain the explicit solution of the integral equation
y(t) = –√λ π
t
y(τ)√t–τdτ, t> , (.)
by using the Adomian decomposition method [–] through a series expansion.
Theorem . The solution of integral equation(.)is given by the following expression:
y(t) =I(t) –
πJ(t), t> , (.)
with
I(t) = +∞
n=
(λ/t)n
(n)! (.)
and
J(t) = +∞
n=
(λ/t)(n+)
((n+ ))!! (.)
are series with infinite radii of convergence.
Proof Following the idea of [–], we propose, for the solution of integral equation (.), the series of expansion functions given by
y(t) = +∞
n=
yn(t), (.)
and we obtain the following recurrence expressions:
y(t) = , yn(t) = –
λ
√
π
t
yn–(τ)
√
t–τdτ, ∀n≥. (.)
Then we get
y(t) = –√λ π
t
√
t–τdτ= – λ √πt
/= –
π
(λ/t)/
!! , (.)
y(t) = –√λ π
t
– λ
√πτ /
√
t–τdτ =λ
π
t
τ/√t–τdτ=λ t
The first step of the double induction principle is just verified by (.) taking into ac-count (.), (.). For the second step, we suppose by induction hypothesis that we have
Jn(t) =
λn (n)!t
n, J
n+(t) = –
n+ ((n+ ))!!
λn+
√
π t (n+)
. (.)
Therefore, we obtain
Jn+(t) = –
λ
√
π
t
yn+(τ) √
t–τdτ=λ n+
π
n+ (n+ )!!
t
τ(n+)√t–τdτ
=λ
n+
π
(n+) (n+ )!!
π (n+)
(n+ )!! ((n+ ))!t
(n+)
= λ
n+
((n+ ))!t
(n+) (.)
and
Yn+(t) = –
λ
√
π
t
yn+(τ) √
t–τdτ= – λ n+
((n+ ))!√π
t
τn+√t–τdτ
= – λ
n+
((n+ ))!√π
n+((n+ ))! (n+ )!! t
(n+)
= –
(n+)+λ(n+)+
√
π((n+ ) + )!!t (n+)+
. (.)
This ends the proof.
Remark . Takingt→+in (.), (.), and (.), we obtain
W+ = +∞, W+ = –∞,
U+ = , U+ = +∞,
g+ = , g+ = .
So we deduce that the heat fluxWand the total heat fluxU, and alsog, are positive func-tions in a neighborhood oft= .
4 Conclusion
We have obtained the global solution of a non-classical heat conduction problem in a semi-n-dimensional space. Moreover, for the one-dimensional case, we have obtained the explicit solution by using the Adomian method with a double induction principle.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Author details
1Institut Camille Jordan CNRS UMR 5208, Lyon University, 23 rue Paul Michelon 42023 Saint-Etienne Cedex 2,
Saint-Etienne, 42023, France.2Departamento de Matemática-CONICET, FCE, Universidad Austral, Paraguay 1950, Rosario,
Acknowledgements
This paper was partially sponsored by the Institut Camille Jordan St-Etienne University for first author, and the projects PIP # 0534 from CONICET-Austral (Rosario, Argentina) and Grant AFOSR-SOARD FA 9550-14-1-0122 for the second author.
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Received: 5 October 2016 Accepted: 31 March 2017 References
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