A Cauchy problem in nonlinear heat conduction

5 Pages·2006·0.082 MB·English

by  De Lillo S., Lupo G., Sanchini G.

INSTITUTEOFPHYSICSPUBLISHING JOURNALOFPHYSICSA:MATHEMATICALANDGENERAL
J.Phys.A:Math.Gen.39(2006)7299–7303 doi:10.1088/0305-4470/39/23/008
A Cauchy problem in nonlinear heat conduction
SDeLillo1,GLupo2andGSanchini1
1IstitutoNazionalediFisicaNucleare,SezionediPerugia,Perugia,Italy
2DipartimentodiMatematicaeInformatica,Universita`degliStudidiPerugia,
ViaVanvitelli,1,06123Perugia,Italy
E-mail:[email protected]
Received14March2006
Published23May2006
Onlineatstacks.iop.org/JPhysA/39/7299
Abstract
A Cauchy problem on the semiline for a nonlinear diffusion equation is
considered, withaboundaryconditioncorrespondingtoaprescribedthermal
conductivity at the origin. The problem is mapped into a moving boundary
problem for the linear heat equation with a Robin-type boundary condition.
SuchaproblemisthenreducedtoalinearintegralVolterraequationofIItype
whichadmitsauniquesolution.
PACSnumbers: 02.30.Ik,02.60.Lj
Thenonlineardiffusionequation
(cid:1) (cid:2)
u
u = x u=u(x, t) (1)
t u2 x
is a well-known mathematical model for heat conduction in high polymer systems [1] and
in simple monoatomic metals of Storm type [2]. Fixed and moving boundary problems for
equation (1)have beensolvedinthepastthroughalinearizingtransformationwhichallows
us to reduce equation (1) to the linear heat equation [3, 4]. In the following we limit our
consideration to materials of Storm type and consider for equation (1) an initial/boundary
valueproblemonthesemilinewithaprescribedthermalconductivityattheorigin. Weshow
that the corresponding problem for the linear heat equation is a semiline problem with a
moving boundary and a Robin-type boundary condition. Such problem is then solved, i.e.
reducedtoalinearintegralequationofVolterraIItypewhichadmitsauniquesolution. An
explicitexampleisalsodiscussed.
Westartouranalysisbyobservingthatthethermalvariableuinequation(1)isrelatedto
thetemperaturedistributionofthesystemthroughtherelation[3]
(cid:3)
T
u= ρc (T(cid:1))dT(cid:1), (2)
p
T0
where ρ and c (T) represent in turn the density (assumed to be constant) and the specific
p
heatofthesystem. Inourmodelthethermalvariableurepresentsthereforetheheatenergy
0305-4470/06/237299+05$30.00 ©2006IOPPublishingLtd PrintedintheUK 7299
7300 SDeLilloetal
(quantity of heat for unitary length) propagating through a semi-infinite one-dimensional
metallicrod. MoreoverweobservethatformaterialsofStormtypewecanwrite
(cid:1) (cid:2)
u
x =k(T), (3)
u2 x
k(T)beingthethermalconductivityofthematerial.
Letusnowanalyseforequation(1)theinitial/boundaryvalueproblemonthesemiline
0(cid:1)x <∞,characterizedbythefollowinginitialandboundarydata:
u(x,0)=u (x), 0(cid:1)x <∞ (4a)
0
u(∞,t)=γ >0, u (∞,t)=0, t (cid:2)0 (4b)
x
u (0,t)
x =α >0, (4c)
u2(0,t)
whereα andγ arepositiveconstants. Dueto(3), theboundarycondition(4c)corresponds,
fromthephysicalpointofview,toaprescribedconstantthermalconductivityattheorigin.
Nextweintroducethehodographtransform
u(x,t)=[v(z,t)]−1 (5a)
with
∂z
=u(x,t) (5b)
∂x
(cid:4) (cid:5)
∂z 1
=− (5c)
∂t u(x,t)
x
whosecompatibility, ∂2z = ∂2z ,isguaranteedby(1).
∂x∂t ∂t∂x
Undertheabovetransformationequation(1)ismappedinto
v =v (6)
t zz
overthedomainαt (cid:1)z<∞,withtheinitialdatum
v(z,0)≡v (z )=[u (x)]−1, (7a)
0 0 0
where
(cid:3)
x
z ≡z (x)= dx(cid:1)u (x(cid:1)). (7b)
0 0 0
0
Moreover,theboundaryconditions(4b),(4c)become
1
v(∞, t)= , v (∞, t)=0 (7c)
z
γ
αv(αt, t)+v (αt, t)=0. (7d)
z
Theinitial/boundaryvalueproblemforthenonlineardiffusionequation(1),withtheinitial
datum (4a) and the boundary conditions (4b), (4c) is then mapped into an initial/boundary
valueproblemforthelinearheatequation(6)overadomainwithalinearlymovingboundary,
characterized by the initial datum (7a) and boundary conditions (7c), (7d). We observe
that(7d)isaRobin-typeboundaryconditionatthemovingboundary. Inordertosolvethis
problemweintroducethefundamentalkerneloftheheatequation
(cid:6) (cid:7)
1 1 1(z−ξ)2
K(z−ξ,t −t(cid:1))= √ √ exp − (8)
2 π t −t(cid:1) 4 (t −t(cid:1))
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andintegrateGreen’sidentityfortheheatequation
(cid:4) (cid:5)
∂ ∂v ∂K ∂
K −v − (Kv)=0 (9)
∂ξ ∂ξ ∂ξ ∂t(cid:1)
over the domain αt(cid:1) < ξ < ∞, ε < t(cid:1) < t − ε and let ε → 0. Using (7d) and
K(z−ξ,0)=δ(z−ξ),weobtain
(cid:3) (cid:3)
+∞ t
v(z,t)= dξK(z−ξ,t)v (ξ)+ dt(cid:1)K (z−αt,t −t(cid:1))v(αt(cid:1),t(cid:1)). (10)
0 ξ
0 0
From (10) it follows that v(z, t) has to be determined in terms of the boundary value
v(αt,t)whichisunknown;itisthereforeconvenienttoevaluate(10)atz = αt. Byputting
v(αt,t)=w(t),weobtain
(cid:3)
t
w(t)=G(t)+ dt(cid:1)R(t −t(cid:1))w(t(cid:1)), (11a)
0
with
(cid:3) (cid:6) (cid:7)
1 +∞ (αt −ξ)2
G(t)= √ dξexp − v (ξ) (11b)
0
2 πt 4t
0
and
α 1 α2
R(t)= √ √ e−βt, β = . (11c)
4 π t 4
Equation (11a) is a linear Volterra integral equation of the convolution type with a mildly
singular kernel; it admits a unique solution under the assumption that G(t) is an integrable,
boundedfunctionofitsargument[5]. Thesolutionof(11a)canbewrittenas
(cid:3)
t
w(t)=G(t)+ dt(cid:1)S(t −t(cid:1))G(t(cid:1)), (12a)
0
whereS(t)istheresolventkernelof(11a)givenby
(cid:8) √ (cid:6) (cid:4)√ (cid:5)(cid:7)(cid:9)
1 β βt
S(t)=e−βt √ + eβt/4 1+Erf , (12b)
πt 2 2
withG(t)givenby(11b)and
(cid:3)
2 y
Erf(y)= √ dτe−τ2. (12c)
π
0
Having established existence and uniqueness of the boundary datum w(t), it then follows,
via (10), existence and uniqueness of the solution of the linear problem v(z, t). We can
thereforeconcludethat,dueto(5a),theinitial/boundaryvalueproblem(1),(4a)–(4c)forthe
nonlineardiffusiveequationadmitsauniquesolutionu(x,t).
Asanexample,letusnowconsideraninitialdatumu (x),compatiblewiththeasymptotic
0
conditions(4b),givenby
u (x)=γ tanh(x). (13)
0
(11)implies,via(5a)and(5b),
z (x)=γ lncosh(x) (14a)
0
and
v (z )= ez0/γ(e2z0/γ −1)−1/2 (14b)
0 0
γ
whichistheinitialdatumforthelinearproblem(6),(7a)–(7d).
7302 SDeLilloetal
When (14b) and (11b) are used, the function G(t) on the right-hand side of (12a) takes
theform
(cid:3) √
1 1 +∞ e( β+1/γ)ξ
G(t)= √ e−βt dξe−ξ2/4t√ . (15)
2γ πt 0 e2ξ/γ −1
Inthefollowingweconcentrateourattentiononthecasewhentheparameterγ issmall
(0<γ <1)andanalyseforthiscasetheasymptotic,largetbehaviourofv(z,t). Itiseasyto
checkthatforsmallγ weobtaintheapproximateexpression
(cid:10)
1
G(t)∼= [1+Erf( βt)]. (16)
2γ
In the same approximation, via (10), (8) and (14b), we can write the solution of the linear
problemv(z,t)as
(cid:3) (cid:6) (cid:7)
1 1 +∞ (z−ξ)2
v(z,t)∼= √ dξexp −
2γ πt 4t
(cid:3)0 (cid:6) (cid:7)
1 t (z−αt(cid:1)) 1(z−αt(cid:1))2
+ √ dt(cid:1) exp − w(t(cid:1)). (17)
4 π (t −t(cid:1))3/2 4 (t −t(cid:1))
0
Thetwotermsontheright-handsideof(17)canbeevaluatedinthelargetimelimitt → ∞
(seetheappendix). Wedenotebyv∞(z,t)theasymptoticvalueofv(z,t)andobtainfrom(17)
(cid:11) (cid:4) (cid:5)(cid:12)
1 e−z2/4t 2
v∞(z,t) ≈ 1+ √ z− √ +O(t−1/2) . (18)
tlarge2γ πt β
Finally, the solution of equation (1) with initial datum (13) and boundary data (4b), (4c) is
obtained(forsmallγ)inthelargetimelimitas
(cid:4) (cid:5)
∂z
u∞(x,t)= (19)
∂x
where,invirtueof(3a),(3b),z(x,t)solves
(cid:3)
z
x = dz(cid:1)v∞(z(cid:1),t) (20)
0
withv∞(z,t)givenby(18).
Appendix
Wewrite(17)intheform
v(z,t)=I (z,t)+I (z,t), (A.1)
1 2
whereI (z,t)andI (z,t)denote,respectively,thefirstandthesecondtermontheright-hand
1 2
sideof(17).
Wethenget
(cid:3) (cid:6) (cid:7)
1 1 +∞ (z−ξ)2
I (z,t)= √ dξexp −
1
2γ πt 4t
(cid:6) 0 (cid:4) (cid:5)(cid:7)
1 z
= 1+Erf √ . (A.2)
2γ 2 t
Inthelargetimetlimitweobtainfrom(A.2)
(cid:11) (cid:13) (cid:14)(cid:12)
1 z e−z2/4t e−z2/4t
I (z,t) ≈ 1+ √ √ +O . (A.3)
1 tlarge2γ π t t3/2
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Wenowturnourattentiontotheasymptotic,larget,evaluationofI (z,t). From(17)we
2
canwrite
(cid:3) (cid:6) (cid:7) (cid:4) (cid:5)
1 1 1 1(z−αtu)2 3
I (z,t)∼= √ √ du(z−αtu)exp − w(tu) 1+ u+··· . (A.4)
2 4 π t 4 t(1−u) 2
0
Inthelargetlimit,byusingtheLaplacemethod,weobtainfrom(A.4)
(cid:13) (cid:14)
2 e−z2/4t z e−z2/4t e−z2/4t
I (z,t) ≈ −√ √ w(0)+ √ w(0)+O , (A.5)
2 tlarge πβ t 4 β t t3/2
where,via(12a)and(16),itisw(0)∼=1/2γ.
When (A.3) and (A.5) are used in (A.1), there immediately follows the result reported
in(18).
References
[1] BlumanGandKumeiS1980J.Math.Phys.211019
[2] StormML1951J.Appl.Phys.22940
[3] RogersC1985J.Phys.A:Math.Gen.18L105
RogersC1986J.NonlinearMech.21249
[4] NataleM.FandTarziaDA2000J.Phys.A:Math.Gen.33395
[5] TricomiFG1985IntegralEquations(NewYork:Dover)