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Invariant solutions and wave features related to a non-linear hyperbolic model for heat conduction
Int. 1. Non-Lineor Mccknics. Primed in Chat Britain.
Vol. 25. No. 4, pp. 99-106. 1990 C
ooze-746fW 53.00 + .Jo 1990 Pergamon Press plc
INVARIANT SOLUTIONS AND WAVE FEATURES RELATED TO A NON-LINEAR HYPERBOLIC MODEL FOR HEAT CONDUCTION N. MANGANARO Dipartimentodi Matematica dell’Univcrsita’di Messina, Contrada Papardo, Salita Spcrone 31, 98166 Sant’Agata, Messina, Italy (Received for publication
18 May
1989)
non-linear hyperbolic model for heat conduction is considered in connection with the group analysis approach. Once the groups of transformations leaving invariant the governing
Abstract-A
system are characterized, then a set of possible exact similarity solutions are determined. The conditions which make a similarity line a weak discontinuity line are investigated. 1. INTRODUCTION
Several hyperbolic mathematical models to describe heat conduction have been proposed to avoid the celebrated paradox of infinite wave speeds for thermal disturbances caused by the well-known Fourier law. An exhaustive list of references can be found in [l].
Among others, in [2], within the framework either of the extended thermodynamics or of the symmetrizable hyperbolic conservative systems, the following system of quasilinear first-order equations have been considered in connection with rigid conductors:
U, + A(U)U, = B(U),
(1.1)
where
U=[
;];
A=[i
;];
B=
[
;O)qj.
(1.2)
In (1.1) and (1.2) e = e(O)is the specific internal energy, 8 is the absolute temperature, 4 is the heat flux, ~(0) is the thermal conductivity, T(B) = aX02, a is a constant, x and t are respectively space and time variables. Moreover, in (1.2) and in what follows the subscripts denote the derivatives with respect to the indicated variables. The pair of equations (1.1) is obtained from the most general hyperbolic conservative system [Z] compatible with the entropy balance, which requires that the internal energy depends only on the absolute temperature, and the evolution equation for the heat flux is a Maxwell-Cattaneo-like equation [3] with temperaturedependent relaxation time. Furthermore, it is possible to show that, by means of a field variables transformation, the governing system admits a special symmetric conservative form. Consequently, the Cauchy problem is well posed in suitable functional spaces, the study of the weak solutions is permitted and all the wave speeds are real and finite. In this paper we investigate the system of governing equations (1.1) within the theoretical group analysis [4-6]. Making use of a well-established procedure, in Section 2 we determine the groups of transformations which leave invariant the model (1.1) as well as a class of functional forms to the response functions e = e(O)and x(e) which allows for the existence of the characterized groups. Furthermore, in Section 3, we present some possible exact (similarity) solutions to the governing equations. These solutions, apart from their own theoretical value, cannot in general fit in with prescribed initial or boundary value problems. Nevertheless, they can be used for testing numerical solutions to the non-linear system (1.1). Finally, in Section 4, along the lines suggested in [7], the possible existence of singularities in the similarity solutions across a similarity line is investigated in a special case. Usually this kind of analysis is carried out when the governing system is invariant with Contributed by W. F. Ames. 99
N. MANGANARO
100
respect to a dilatation group. Here we develop a similar analysis for a group of transformations different from the stretching group.
2. GROUPS
OF TRANSFORMATIONS
We consider the infinitesimal one-parameter
(E) Lie group of transformations:
u* = u + &V(U, x, c) + O(.?) x* = x + &IYl(X,I) + 0(&Z) t* = t + &l-(X,
t) f O(2),
where
v=
V, [
(2.1)
1
v* *
Actually I?, (a = 0, l), and V are not the most general generators of the non-extended Lie group. However, it is easy to ascertain that (2.1) represents an extended form of the group transformations considered by Morgan [6,8]. Usually these transformations include all the cases of physical interest in which group analysis can be applied. An exhaustive treatment of the theory of the Lie group and algebra may be found in [4-6]. We introduce the generator L of the group (2.1) together with the first extension defined respectively by L(*) = I-=8=(.) + V(*)V, (2.2) where
ht.1 = Lt.1 + $-)A.
(2.3)
with
u,=a,u;
DV
DTs
Aa=Dx"-Dx"uB;
+p(*)
= a=(.) + V(*)U=.
Furthermore, in (2.2) and (2.3) a, /.?= 0, 1 and x0 = C, x1 = x. Set II = U, + A U, - B = 0.The invariance condition for (1.1) is given by L,(C) = 0.
(2.4)
Substituting U, = B - AU, into (2.4) we obtain
{(VA)V+A(VV)-(VV)A +a,r" A=A - d,rlA”}U, + A=d,V + (VV)B + -(VB)V - 8,r'A'B= 0, (2.5) where A0 = I (I is the unit matrix) and A’ = A. As the derivatives U, must be treated as independent quantities, from (2.5) we deduce an overdetermined set of linear partial differential equations involving the coefficients of the Lie-generator P, V as well as the functions e(0), x(0). After some algebra, from (2.9, the following expressions for P and V are obtained: (I)
r"=5. r1 =h
(2.6)
v, = 0 v, = 0, with T and h constant. Moreover, no hypothesis is needed for the functions e(6) and x(0). Relations (2.6) give the Galilean group of transformations.
Non-linear hyperbolic model for heat conduction
(2)
101
I-O = T lY’=h
(2.7)
v, = 0 v2 = cl
( >
exp -t
,
with T, h, cl, (r constant and the constitutive restriction
x(e)= 5.
f2.8)
A constitutive equation such as (2.8) has been considered in [9] by completely different arguments. As a consequence of (2.8) the relaxation time of the Maxwell-Cattaneo equation is constant: T = ua = constant 3 0. (2.9) (3) r” = T r’=yx+h
(2.IO) VI = - W
+ f)e e6
V2 = - Hyq +
cl
,
exp
where r, y, h, H, c, are constants and y # 0. Moreover, x(e) assumes the form (2.8) and e(B) is given by
de) =
& j--t*- I)(b,_;)>‘““‘i(H-“_2b,),
(2.11)
with b,, b,, b3 constant and H # k 1. If H = + 1 we obtain respectively: e(e) = -b,+b,exxp
(
-%
)
e,,=h,Ib,-log(~(b,+~)>>.
(2.12) (2.13)
The expressions (2.12) and (2.13) are not acceptable from a physical point of view, whereas (2.11) contains, with suitable constants, the usual forms of the internal energy in a rigid conductor. Indeed, in many cases, the specific heat at constant volume C,(e) = (&/a@, is constant or proportional to e3 [lo], and both cases can be obtained from (2.11) if b, = b, = 0 and H = 0 or H = 3. (4)
r” +-t-r ri=yx+h VI = (9 - Y)W +
(2.14)
1)-f@
Vz = W - ~14, with f, 7, y, h constant and y #f. Here e(e) is given by (2.11) if H # + 1, or by (2.12) and (2.13) if H = If: 1. Furthermore, x(e) assumes the following expression: ~(0) = c1 e(2y-i)/(J-r) e,i/Yi-71, where cI is constant. In passing, we note that the group characterized stretching and the translation group.
by (2.14) is a combination
(2.15) of the
102
N.
MANGANARO
3. SIMILARITY
SOLUTIONS
Along the lines of a well-established procedure [4-6], once the generator of the Lie group has been determined, the invariant surfaces are obtained by integrating: Pu,
+ rru,
= V,
(3.1)
whose characteristic curves are given by dt dx ro=r’=v,
dU,
(3.2)
with i = 1, 2. Furthermore, the insertion of the expression of U(x, t) obtained from (3.1) into (1.1) leads to a system of ordinary differential equations, whose integration provides a class of invariant solutions to the governing model. In what follows, to be as close as possible to the cases of physical interest, we assume that the internal energy has the form e(6) = be”,
(3.3) where 2, n are positive constants. In fact, the usual forms of the internal energy in a rigid conductor are recovered [lo] if n=lorn = 4. Moreover, comparing (3.3) with (2.11) we obtain b, = 6, = 0 and H =
(n- 1)/b+ 1). (1)First we consider the group of transformations
characterized by (2.10). In this case,
the invariant surfaces for the system (1.1) are
0=
eo(~)exp( -&c)
9=40(w)exp(
-Tt)+Cexp(
(3.4)
-b),
where C = c1 aa/(Haya - z) and w = ( yx + h)‘lY exp ( - c/z) is the similarity variable. If we introduce (3.4) into (l.l), we obtain the following system of ordinary differential equations for the functions do and qo: @-"&)'-Yq;) = o&-1.ne*
T&-y oqb - --0; a e,2
2y8O
I__
n-t-l
= - 2qo*
(3.5)
We may integrate equation (3.9, if n = 1:
qoye, A
40 =
-Iq*,
(3.6)
where q* is constant. Thus, substituting (3.6) in (3.5),, a further integration leads to the solution of (3.5) in the implicit form: (3.7) provided that q* = 0 and 3t = aay. Apart from (3.6) and (3.7), if we look for solutions of (3.5) in the form B. = 8* = constant, taking into account (3.4), we obtain 0 = 8*exp
---& (
q = exp
S(yx
-h (
with the condition y = t(n + 1)/2naa.
) >I
f h) + c
, 1
(3.8)
Non-&war hyperbolic model for heat conduction
103
(2) Here we shah be concerned with the generator determined by (2.14). The invariant surfaces give (j = (j*(@(y”r + r)W-vMfi~+n
4 = q&)(jb
+ r)H@+,
(3.9)
with 0 = (yx + h)“Y($ + r)-I/y’.
(3.10)
In this case the system of ordinary differential equations for t$, and q. is
(3.11) where e* = (Pn)~~2tY-9! One possible solution of (3.11) is given by 8, = 8* = constant 1
q”=
24y -
*
Y) e*wy,
(3.12)
y(n+l)
with the condition (3.13) Thus, bearing in mind (3.9) and (3.12) we obtain 0 = e*($r + 4
Tpi-~)/i(n+
1)
24Y - ?I e*yyx + h)(ft + Tp-2~ln+l
=
r(n + 1)
(3.14)
As remarked in the previous section the group of transformations (2.14) includes the diIatation group. In particular, if T = h = 0, y = 1 and p = 2, then (3.10) specializes to the well-known similarity variable o = ~t-r’~, connected to the celebrated Fourier equation. Moreover, in the parabolic case (Fourier law), which we recover if a = 0, from (3.11) we may easily obtain the same equation studied in [4] for non-linear parabolic heat conduction. (3) Finally, we consider the group of transfo~atio~s characterized by (2.7). The invariant surfaces of (1.1) are
8 = e,(w)
q=qO(w)-yexp
-f (
,
(3.15)
>
where w = x - UCand v = h/r. The system of ordinary differential equations for e. and q. gives
1 f 4b-ze2
-et 0 0
(3.16)
From (3.16), we obtain 40 = &IV~“,+ q*,
q* = constant.
(3.17)
Further insertion of (3.17) into (3.16), yields loge0
+
(n+1)8,~-1 =-LO, anev
am
(3.18)
where we have taken q* = 0. The relation (3.18) in connection with (3.17) gives the solution of (3.16) in implicit form.
N. MANGANARO
104 4. SINGULARITY
AND WEAK DISCONTINUITY
Recently, the problem of the limiting characteristic [l l] for a hyperboiic system of partial differential equations has been investigated [73 by means of the group analysis approach. In particular, requiring the invariance of the governing system with respect to the dilatation group, in [73, the conditions which make a similarity line a weak discontinuity line are studied. Here we develop such an analysis for the group of transformations (2.10). First, the pair of equations (3.5) are invariant with respect to the dilatation group t3,*= &- 2v/fl+(1,~* & = w*
~Ytl-R)/(i+~)~*
(4.1)
= EW.
In passing, we note that when the group of transformations leaving invariant the governing system of partial differential equations is the stretching group, then (4.1) is known in the literature as an “associated group” [12]. As a consequence of (4.1), by means of the transformation f& = w-2Yi(l +qo 40
logo
=
wY(I
=
0,
-*MI
+n)40
(4.2)
the system (3.5) can be reduced to the autonomous
form:
_ (I +
2,g =ii,
(4.3)
fI. I.
where 0
7
0
--
vu -
4
ne^(l + n)
fj,tp
jj=
[
2Y7 ------x-+-40 a(1 + n)@,
r aa
Solving (4.3) with respect to go, Go we obtain A& = A,
A& = AZ,
(4.4)
where A = det (I-i- 2) and
(4.5) In (4.4) there is a possible singularity if A = 0 and this would represent a singularity through the curve where w is constant. Indeed, bearing in mind that w = (yx + /I)“~ exp( -t/r), if o is constant the characteristic velocities of (1.1) are given by 0.9.
105
Non-linear hyperbolic model for heat conduction
Denoting by J+(L)the c~~r~~~~~t~~polynomial associated with f1.Q and taking account (3.4)and (4.2)we obtain
P(4 =$exp
(
27; A,
into
(4.71
)
which leads to the expected result. To characte~ze the value wf of the limiting chara~e~sti~, following [7], we suppose that the similarity solutions (3.4) have a jump in their first derivatives across of, Then it is necessary to require A = 0 or, equivalently, A@,) = - 1,
(4.8)
where A is an eigenvalue of the matrix A^. Moreover, as we assume that the field (3.4) is continuous across or we must have A, = 0.
(4.91
It is easiiy verified that the condition (4.9)implies A2 = 0 and conversely. In the present case, bearing in mind (4.2) and (4.5),the relations (4.8) and (4-9)give
getting a value ~5~of or from (4.10)we det~~ine & = 80(@ff and & = q*~~~~. Then, assuming as boundary conditions the values go and &, we may integrate nume~cally the system (3.5),and we obtain B,(O)and PO(O),Thus, assigning the boundary problem q&O) = 4 * = constant and bearing in mind &JO), &,(O)and (4.1) we are able to find the value of os The requirement qOfO) = q* originates from the boundary value problem 4t4 t1 = W,
vtao,
which is compatible with the group of transformations (2.10) provided that h = 0 and $(t)=Cexp(
--f-)+q*exp(
-Tt).
(4.11)
As far as the weak discontinuity propagation is concerned, denoting by g _ lim da II = fim w+,:+’ da @.+, da it is known that [13] II = xdO,
(4.12)
where d, is the right eigenvector of A^corresponding to A&ur) and the subscript 0 means that the quantities are evaluated at the unperturbed state. It is easily found that the amplitude x of the discontinuity may be calculated from the relation [7] (4.13) In (4.13),I is the left eigenvector of A”corresponding to the eigenvalue A(#,), Here from (4.13)we obtain
Finally, setting x the amplitude of the discontinuity related to the jump of d&,/dw and dq,/dw we obtain x = @Pi-Wfl Wrr. (4.15)
106
N. MANGANARO
Of course, such an analysis can be carried out to investigate when a similarity line may be a shock line [7, 123. Acknowledgement-This
work was supported by the C.N.R. through G.N.F.M.
REFERENCES 1. A. Morro and T. Ruggeri, Propagazione de1 calore ed equazioni costitutive. Quademi de1 C.N.R. (G.N.F.M), Ravello (1983). 2. T. Ruggeri, Struttura dei sistemi alle derivate parziali compatibili con un principio di entropia e termodinamica estesa. Suppl. B.V.M.I. IV-5. l(l985). 3. C. Cattaneo, Sulla conduzione del calore. Atti Sem. Mot. Fis. Modena 3, 83-101 (1948). 4. G. W. Bluman and J. D. Cole, Similarity Methodsfir Difirential Equations. Springer, Berlin (1974). 5. L. V. Ovsiannikov. Group Annlysis of Diferenbal Equations (in Russian). Nauka, Moscow (1978); English translation, Academic Press, New York (1982). 6. W. F. Ames, Non-lineor Partial Difirentiul Equations in Engineering, Vol. II. Academic Press, New York (1972). I. A. Donato, Similarity analysis and non-linear wave propagation. Int. J. Non-linear Mech. 22(4), 307-314 (1987). 8. D. Fusco, Group analysis and constitutive laws for fluid filled elastic tubes. Int. J. Non-linear Mech. 19, 565-574 (1984). 9. C. Rogers and T. Ruggeri, A reciprocal Backlund transformation: application to a non-linear hyperbolic model in heat conduction. Lett. Nuouo Cimento 2(44), 289-296 (1985). 10. L. Landau and E. Lifchitz, Physique stotistique. Mir, Moscow (1984). 11. G. B. Whitham, Linear and Non-linear Wuoes. John Wiley, New York (1974). 12. L. Dresner, Similarity solutions of non-linear partial differential equations, Research Notes in Mathematics, 88. Pitman, London (1983). 13. G. Boillat, La Propagation des Ondes. Gauthier-Villars, Paris (1965).
Vol. 25. No. 4, pp. 99-106. 1990 C
ooze-746fW 53.00 + .Jo 1990 Pergamon Press plc
INVARIANT SOLUTIONS AND WAVE FEATURES RELATED TO A NON-LINEAR HYPERBOLIC MODEL FOR HEAT CONDUCTION N. MANGANARO Dipartimentodi Matematica dell’Univcrsita’di Messina, Contrada Papardo, Salita Spcrone 31, 98166 Sant’Agata, Messina, Italy (Received for publication
18 May
1989)
non-linear hyperbolic model for heat conduction is considered in connection with the group analysis approach. Once the groups of transformations leaving invariant the governing
Abstract-A
system are characterized, then a set of possible exact similarity solutions are determined. The conditions which make a similarity line a weak discontinuity line are investigated. 1. INTRODUCTION
Several hyperbolic mathematical models to describe heat conduction have been proposed to avoid the celebrated paradox of infinite wave speeds for thermal disturbances caused by the well-known Fourier law. An exhaustive list of references can be found in [l].
Among others, in [2], within the framework either of the extended thermodynamics or of the symmetrizable hyperbolic conservative systems, the following system of quasilinear first-order equations have been considered in connection with rigid conductors:
U, + A(U)U, = B(U),
(1.1)
where
U=[
;];
A=[i
;];
B=
[
;O)qj.
(1.2)
In (1.1) and (1.2) e = e(O)is the specific internal energy, 8 is the absolute temperature, 4 is the heat flux, ~(0) is the thermal conductivity, T(B) = aX02, a is a constant, x and t are respectively space and time variables. Moreover, in (1.2) and in what follows the subscripts denote the derivatives with respect to the indicated variables. The pair of equations (1.1) is obtained from the most general hyperbolic conservative system [Z] compatible with the entropy balance, which requires that the internal energy depends only on the absolute temperature, and the evolution equation for the heat flux is a Maxwell-Cattaneo-like equation [3] with temperaturedependent relaxation time. Furthermore, it is possible to show that, by means of a field variables transformation, the governing system admits a special symmetric conservative form. Consequently, the Cauchy problem is well posed in suitable functional spaces, the study of the weak solutions is permitted and all the wave speeds are real and finite. In this paper we investigate the system of governing equations (1.1) within the theoretical group analysis [4-6]. Making use of a well-established procedure, in Section 2 we determine the groups of transformations which leave invariant the model (1.1) as well as a class of functional forms to the response functions e = e(O)and x(e) which allows for the existence of the characterized groups. Furthermore, in Section 3, we present some possible exact (similarity) solutions to the governing equations. These solutions, apart from their own theoretical value, cannot in general fit in with prescribed initial or boundary value problems. Nevertheless, they can be used for testing numerical solutions to the non-linear system (1.1). Finally, in Section 4, along the lines suggested in [7], the possible existence of singularities in the similarity solutions across a similarity line is investigated in a special case. Usually this kind of analysis is carried out when the governing system is invariant with Contributed by W. F. Ames. 99
N. MANGANARO
100
respect to a dilatation group. Here we develop a similar analysis for a group of transformations different from the stretching group.
2. GROUPS
OF TRANSFORMATIONS
We consider the infinitesimal one-parameter
(E) Lie group of transformations:
u* = u + &V(U, x, c) + O(.?) x* = x + &IYl(X,I) + 0(&Z) t* = t + &l-(X,
t) f O(2),
where
v=
V, [
(2.1)
1
v* *
Actually I?, (a = 0, l), and V are not the most general generators of the non-extended Lie group. However, it is easy to ascertain that (2.1) represents an extended form of the group transformations considered by Morgan [6,8]. Usually these transformations include all the cases of physical interest in which group analysis can be applied. An exhaustive treatment of the theory of the Lie group and algebra may be found in [4-6]. We introduce the generator L of the group (2.1) together with the first extension defined respectively by L(*) = I-=8=(.) + V(*)V, (2.2) where
ht.1 = Lt.1 + $-)A.
(2.3)
with
u,=a,u;
DV
DTs
Aa=Dx"-Dx"uB;
+p(*)
= a=(.) + V(*)U=.
Furthermore, in (2.2) and (2.3) a, /.?= 0, 1 and x0 = C, x1 = x. Set II = U, + A U, - B = 0.The invariance condition for (1.1) is given by L,(C) = 0.
(2.4)
Substituting U, = B - AU, into (2.4) we obtain
{(VA)V+A(VV)-(VV)A +a,r" A=A - d,rlA”}U, + A=d,V + (VV)B + -(VB)V - 8,r'A'B= 0, (2.5) where A0 = I (I is the unit matrix) and A’ = A. As the derivatives U, must be treated as independent quantities, from (2.5) we deduce an overdetermined set of linear partial differential equations involving the coefficients of the Lie-generator P, V as well as the functions e(0), x(0). After some algebra, from (2.9, the following expressions for P and V are obtained: (I)
r"=5. r1 =h
(2.6)
v, = 0 v, = 0, with T and h constant. Moreover, no hypothesis is needed for the functions e(6) and x(0). Relations (2.6) give the Galilean group of transformations.
Non-linear hyperbolic model for heat conduction
(2)
101
I-O = T lY’=h
(2.7)
v, = 0 v2 = cl
( >
exp -t
,
with T, h, cl, (r constant and the constitutive restriction
x(e)= 5.
f2.8)
A constitutive equation such as (2.8) has been considered in [9] by completely different arguments. As a consequence of (2.8) the relaxation time of the Maxwell-Cattaneo equation is constant: T = ua = constant 3 0. (2.9) (3) r” = T r’=yx+h
(2.IO) VI = - W
+ f)e e6
V2 = - Hyq +
cl
,
exp
where r, y, h, H, c, are constants and y # 0. Moreover, x(e) assumes the form (2.8) and e(B) is given by
de) =
& j--t*- I)(b,_;)>‘““‘i(H-“_2b,),
(2.11)
with b,, b,, b3 constant and H # k 1. If H = + 1 we obtain respectively: e(e) = -b,+b,exxp
(
-%
)
e,,=h,Ib,-log(~(b,+~)>>.
(2.12) (2.13)
The expressions (2.12) and (2.13) are not acceptable from a physical point of view, whereas (2.11) contains, with suitable constants, the usual forms of the internal energy in a rigid conductor. Indeed, in many cases, the specific heat at constant volume C,(e) = (&/a@, is constant or proportional to e3 [lo], and both cases can be obtained from (2.11) if b, = b, = 0 and H = 0 or H = 3. (4)
r” +-t-r ri=yx+h VI = (9 - Y)W +
(2.14)
1)-f@
Vz = W - ~14, with f, 7, y, h constant and y #f. Here e(e) is given by (2.11) if H # + 1, or by (2.12) and (2.13) if H = If: 1. Furthermore, x(e) assumes the following expression: ~(0) = c1 e(2y-i)/(J-r) e,i/Yi-71, where cI is constant. In passing, we note that the group characterized stretching and the translation group.
by (2.14) is a combination
(2.15) of the
102
N.
MANGANARO
3. SIMILARITY
SOLUTIONS
Along the lines of a well-established procedure [4-6], once the generator of the Lie group has been determined, the invariant surfaces are obtained by integrating: Pu,
+ rru,
= V,
(3.1)
whose characteristic curves are given by dt dx ro=r’=v,
dU,
(3.2)
with i = 1, 2. Furthermore, the insertion of the expression of U(x, t) obtained from (3.1) into (1.1) leads to a system of ordinary differential equations, whose integration provides a class of invariant solutions to the governing model. In what follows, to be as close as possible to the cases of physical interest, we assume that the internal energy has the form e(6) = be”,
(3.3) where 2, n are positive constants. In fact, the usual forms of the internal energy in a rigid conductor are recovered [lo] if n=lorn = 4. Moreover, comparing (3.3) with (2.11) we obtain b, = 6, = 0 and H =
(n- 1)/b+ 1). (1)First we consider the group of transformations
characterized by (2.10). In this case,
the invariant surfaces for the system (1.1) are
0=
eo(~)exp( -&c)
9=40(w)exp(
-Tt)+Cexp(
(3.4)
-b),
where C = c1 aa/(Haya - z) and w = ( yx + h)‘lY exp ( - c/z) is the similarity variable. If we introduce (3.4) into (l.l), we obtain the following system of ordinary differential equations for the functions do and qo: @-"&)'-Yq;) = o&-1.ne*
T&-y oqb - --0; a e,2
2y8O
I__
n-t-l
= - 2qo*
(3.5)
We may integrate equation (3.9, if n = 1:
qoye, A
40 =
-Iq*,
(3.6)
where q* is constant. Thus, substituting (3.6) in (3.5),, a further integration leads to the solution of (3.5) in the implicit form: (3.7) provided that q* = 0 and 3t = aay. Apart from (3.6) and (3.7), if we look for solutions of (3.5) in the form B. = 8* = constant, taking into account (3.4), we obtain 0 = 8*exp
---& (
q = exp
S(yx
-h (
with the condition y = t(n + 1)/2naa.
) >I
f h) + c
, 1
(3.8)
Non-&war hyperbolic model for heat conduction
103
(2) Here we shah be concerned with the generator determined by (2.14). The invariant surfaces give (j = (j*(@(y”r + r)W-vMfi~+n
4 = q&)(jb
+ r)H@+,
(3.9)
with 0 = (yx + h)“Y($ + r)-I/y’.
(3.10)
In this case the system of ordinary differential equations for t$, and q. is
(3.11) where e* = (Pn)~~2tY-9! One possible solution of (3.11) is given by 8, = 8* = constant 1
q”=
24y -
*
Y) e*wy,
(3.12)
y(n+l)
with the condition (3.13) Thus, bearing in mind (3.9) and (3.12) we obtain 0 = e*($r + 4
Tpi-~)/i(n+
1)
24Y - ?I e*yyx + h)(ft + Tp-2~ln+l
=
r(n + 1)
(3.14)
As remarked in the previous section the group of transformations (2.14) includes the diIatation group. In particular, if T = h = 0, y = 1 and p = 2, then (3.10) specializes to the well-known similarity variable o = ~t-r’~, connected to the celebrated Fourier equation. Moreover, in the parabolic case (Fourier law), which we recover if a = 0, from (3.11) we may easily obtain the same equation studied in [4] for non-linear parabolic heat conduction. (3) Finally, we consider the group of transfo~atio~s characterized by (2.7). The invariant surfaces of (1.1) are
8 = e,(w)
q=qO(w)-yexp
-f (
,
(3.15)
>
where w = x - UCand v = h/r. The system of ordinary differential equations for e. and q. gives
1 f 4b-ze2
-et 0 0
(3.16)
From (3.16), we obtain 40 = &IV~“,+ q*,
q* = constant.
(3.17)
Further insertion of (3.17) into (3.16), yields loge0
+
(n+1)8,~-1 =-LO, anev
am
(3.18)
where we have taken q* = 0. The relation (3.18) in connection with (3.17) gives the solution of (3.16) in implicit form.
N. MANGANARO
104 4. SINGULARITY
AND WEAK DISCONTINUITY
Recently, the problem of the limiting characteristic [l l] for a hyperboiic system of partial differential equations has been investigated [73 by means of the group analysis approach. In particular, requiring the invariance of the governing system with respect to the dilatation group, in [73, the conditions which make a similarity line a weak discontinuity line are studied. Here we develop such an analysis for the group of transformations (2.10). First, the pair of equations (3.5) are invariant with respect to the dilatation group t3,*= &- 2v/fl+(1,~* & = w*
~Ytl-R)/(i+~)~*
(4.1)
= EW.
In passing, we note that when the group of transformations leaving invariant the governing system of partial differential equations is the stretching group, then (4.1) is known in the literature as an “associated group” [12]. As a consequence of (4.1), by means of the transformation f& = w-2Yi(l +qo 40
logo
=
wY(I
=
0,
-*MI
+n)40
(4.2)
the system (3.5) can be reduced to the autonomous
form:
_ (I +
2,g =ii,
(4.3)
fI. I.
where 0
7
0
--
vu -
4
ne^(l + n)
fj,tp
jj=
[
2Y7 ------x-+-40 a(1 + n)@,
r aa
Solving (4.3) with respect to go, Go we obtain A& = A,
A& = AZ,
(4.4)
where A = det (I-i- 2) and
(4.5) In (4.4) there is a possible singularity if A = 0 and this would represent a singularity through the curve where w is constant. Indeed, bearing in mind that w = (yx + /I)“~ exp( -t/r), if o is constant the characteristic velocities of (1.1) are given by 0.9.
105
Non-linear hyperbolic model for heat conduction
Denoting by J+(L)the c~~r~~~~~t~~polynomial associated with f1.Q and taking account (3.4)and (4.2)we obtain
P(4 =$exp
(
27; A,
into
(4.71
)
which leads to the expected result. To characte~ze the value wf of the limiting chara~e~sti~, following [7], we suppose that the similarity solutions (3.4) have a jump in their first derivatives across of, Then it is necessary to require A = 0 or, equivalently, A@,) = - 1,
(4.8)
where A is an eigenvalue of the matrix A^. Moreover, as we assume that the field (3.4) is continuous across or we must have A, = 0.
(4.91
It is easiiy verified that the condition (4.9)implies A2 = 0 and conversely. In the present case, bearing in mind (4.2) and (4.5),the relations (4.8) and (4-9)give
getting a value ~5~of or from (4.10)we det~~ine & = 80(@ff and & = q*~~~~. Then, assuming as boundary conditions the values go and &, we may integrate nume~cally the system (3.5),and we obtain B,(O)and PO(O),Thus, assigning the boundary problem q&O) = 4 * = constant and bearing in mind &JO), &,(O)and (4.1) we are able to find the value of os The requirement qOfO) = q* originates from the boundary value problem 4t4 t1 = W,
vtao,
which is compatible with the group of transformations (2.10) provided that h = 0 and $(t)=Cexp(
--f-)+q*exp(
-Tt).
(4.11)
As far as the weak discontinuity propagation is concerned, denoting by g _ lim da II = fim w+,:+’ da @.+, da it is known that [13] II = xdO,
(4.12)
where d, is the right eigenvector of A^corresponding to A&ur) and the subscript 0 means that the quantities are evaluated at the unperturbed state. It is easily found that the amplitude x of the discontinuity may be calculated from the relation [7] (4.13) In (4.13),I is the left eigenvector of A”corresponding to the eigenvalue A(#,), Here from (4.13)we obtain
Finally, setting x the amplitude of the discontinuity related to the jump of d&,/dw and dq,/dw we obtain x = @Pi-Wfl Wrr. (4.15)
106
N. MANGANARO
Of course, such an analysis can be carried out to investigate when a similarity line may be a shock line [7, 123. Acknowledgement-This
work was supported by the C.N.R. through G.N.F.M.
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