Symmetries and reduction techniques for dissipative models

Transcription

1 Symmetries and reduction techniques for dissipative models M. Ruggieri and A. Valenti Dipartimento di Matematica e Informatica Università di Catania viale A. Doria 6, Catania, Italy Fourth Workshop on Group Analysis of Differential Equations and Integrable Systems Protaras, Cyprus, October 26-30, 2008

2 OUTLINE We consider the third order partial differential equation w tt = f (w x )w xx + [λ(w x )w tx ] x, (1) where, f and λ are smooth functions, w(t,x) is the dependent variable and subscripts denote partial derivative with respect to the independent variables t and x.

3 Some mathematical questions as the global existence, uniqueness and stability of solutions can be found in C. M. Dafermos, J. of Differential Equations 6 (1969), 71 and in R. C. MacCamy,Indiana Univ. Math. J. 20 (1970), 231. Moreover, shear wave solutions are found in K. R. Rajagopal and G. Saccomandi, Q. Jl Mech. Appl. Math. 56 (2003), 311. While a symmetry analysis of the equation (1) was performed in M. Ruggieri and A. Valenti, Proceedings of WASCOM 2007, N. Manganaro et al. Eds., World Sc. Pub., (2008) 516.

4 When λ(w x ) = λ 0, with λ 0 a positive constant, a symmetry analysis can be found in M. Ruggieri and A. Valenti, Proceedings of MOGRAN X, N. H. Ibragimov et al. Eds., (2005) 175 and in M. Ruggieri and A. Valenti, Proceedings of WASCOM 2005, R. Monaco, G. Mulone, S. Rionero and T. Ruggeri Eds., World Sc. Pub., Singapore, (2006) 481. Moreover, for λ 0 = ε 1 approximate symmetries were studied in A. Valenti, Proceedings of MOGRAN X, N. H. Ibragimov et al. Eds., (2005) 236.

5 By setting w x = u and w t = v equation (1) can be written as a 2x2 system of the form u t v x = 0, (2) v t f (u)u x = [λ(u)v x ] x, (3) where, u corresponds to the specific volume, p(u) = u f (s)ds is the pressure and v is the velocity. As it is well known, the system (2)-(3) is equivalent to the equation (1). Consequently, a symmetry of any one of them defines a symmetry of the other. More specifically, because of the nonlocal transformation connecting (1) and (2)-(3), it is possible for a point symmetry of (2)-(3) to yield a contact symmetry of (1) (for details see Ovsiannikov [1]).

6 AIM - Symmetry classifications - Construction of the Optimal Systems - Reductions to ODEs - Exact solutions

7 SYMMETRY CLASSIFICATIONS In order to investigate on the symmetry classification of equation (1), we apply the classical Lie method and look for the one-parameter Lie group of infinitesimal transformations in (t, x, w)-space given by ˆt = t + a ξ 1 (t,x,w) + O(a 2 ), (4) ˆx = x + a ξ 2 (t,x,w) + O(a 2 ), (5) ŵ = w + a η(t,x,w) + O(a 2 ), (6) where a is the group parameter and the associated Lie algebra L is the set of vector fields of the form X = ξ 1 t + ξ 2 x + η w. (7)

8 We then require that the transformation (4)-(6) leaves invariant the set of solutions of the equation (1). In others words, we require that the transformed equation has the same form as the original one. The invariance condition reads X (3) (w tt f (w x )w xx [λ(w x )w tx ] x ) = 0, (8) under the constraint that the variable w tt has to satisfy the equation (1) and X (3) is the third prolongation of the operator X. (See the well known monographs on this argument [1]-[6]).

9 The determining system arising from the invariance condition (8), leads to the following result: and ξ 1 = a 8 t + a 1, (9) ξ 2 = a 5 x + a 2, (10) η = a 6 w + a 7 x + a 4 t + a 3, (11) [(a 6 a 5 )w x + a 7 ]f + 2(a 8 a 5 )f = 0, (12) [(a 6 a 5 )w x + a 7 ]λ + (a 8 2a 5 )λ = 0, (13) where a i (i = 1,2,...,8) are arbitrary constants.

10 The determining system allows us to find the infinitesimal generator of the symmetry transformations (i.e. (9)-(11)) and, at the same time, gives the functional dependencies of the constitutive functions f and λ (i.e. (12) and (13)) for which (1) does admit symmetries. So that, for arbitrary f and λ, we obtain the Principal Lie Algebra L P of equation (1), which is four-dimensional and spanned by the operators X 1 = t, X 2 = x, X 3 = w, X 4 = t w, (14) otherwise we obtain the results summarized in Table: 1.

11 case Forms of f and λ Extensions of L P I f = f 0 e wx p, λ = λ 0 e wx s X 5 = 2(s p)t t + (s 2p) x x + [(s 2p) w 2p s x] w II f = f 0 (w x + q) 1 p, X 5 = 2r t t + (1 + 2r) x x λ = λ 0 (w x + q) 1+r p + {[1 + 2(p + r)]w + 2p q x} w Table: 1. Group classification of the equation (1). f 0, λ 0, p, q, r and s are constitutive constants with f 0, λ 0 > 0, and p, s 0.

12 Now, we look for the one-parameter Lie group of infinitesimal transformations of the system (2)-(3) in the (t,x,u,v)-space and the associated Lie algebra L is the set of vector fields of the form X = ξ 1 t + ξ 2 x + η 1 u + η 2 v, (15) where the coordinates ξ 1, ξ 2,η 1,η 2 are functions of t,x,u and v. In this case, the invariance condition reads X (2) (u t v x ) = 0, (16) X (2) (v t f (u)u x [λ(u)v x ] x ) = 0, (17) under the constraint that the variables u t and v t have to satisfy the system (2)-(3) and X (2) is the second prolongation of the operator X.

13 In this case, the determining system gives to the following result: ξ 1 = a 8 t + a 1, (18) ξ 2 = a 5 x + a 2, (19) η 1 = (a 6 a 5 )u + a 7, (20) η 2 = (a 6 a 8 )v + a 4, (21) [(a 6 a 5 )u + a 7 ]f + 2(a 8 a 5 )f = 0, (22) [(a 6 a 5 )u + a 7 ]λ + (a 8 2a 5 )λ = 0. (23) For arbitrary f and λ, the Principal Lie Algebra L P of the system (2)-(3) is three-dimensional and spanned by the operators X 1 = t, X2 = x, X3 = v, (24) otherwise we obtain the results summarized in Table: 2.

14 case Forms of f and λ Extensions of L P I f = f 0 e u p, X4 = 2(s 2p) t t + (s 2p)x x λ = λ 0 e u s 2p s u s v v II f = f 0 (u + q) 1 p, X4 = 2r t t + (1 + 2r) x x λ = λ 0 (u + q) 1+r p + 2p (u + q) u + (2p + 1) v v Table: 2. Group classification of the system (2)-(3). f 0, λ 0, p, q, r and s are constitutive constants with f 0, λ 0 > 0, and p, s 0.

15 DISCUSSION OF THE GROUP CLASSIFICATIONS By comparing the two classifications we can deduce easily that the point symmetries of the system (2)-(3) do not produce any contact symmetry of the equation (1). Moreover, after observing that the following relations ξ 1 = ξ 1 = a 8 t + a 1, (25) ξ 2 = ξ 2 = a 5 x + a 2, (26) hold, we will demonstrate the following statement:

16 Theorem For any f and λ, a point symmetry admitted by (1) induces a point symmetry admitted by (2)-(3) and viceversa.

17 In order to demonstrate the theorem, we introduce the infinitesimal operator X = η w + ζ 1 wt + ζ 2 wx + η 1 u + η 2 v, (27) where ζ 1 and ζ 2 are the coordinates of the first prolongation of the operator X, given by ζ 1 = D t (η) w t D t (ξ 1 ) w x D t (ξ 2 ), (28) ζ 2 = D x (η) w t D x (ξ 1 ) w x D x (ξ 2 ) (29) (D t and D x denote total derivatives with respect to t and x) and require the invariance of the transformation w x = u, w t = v, i.e. X (w x u) (wx u=0, w t v=0) = 0, (30) X (w t v) (wx u=0, w t v=0) = 0. (31)

18 Starting from the classification of (1) (i.e. (9)-(11)), taking (28) and (29) into account, through (30) and (31) we obtain immediately the other two coordinates of the operator X, namely η 1 = ζ 2 wx=u = (a 6 a 5 )u + a 7, (32) η 2 = ζ 1 wt=v = (a 6 a 8 )v + a 4. (33) So, we have demonstrated that: For any f and λ, a point symmetry admitted by (1) induces a point symmetry admitted by (2)-(3).

19 Viceversa, starting from the classification of the system (2)-(3) (i.e. (18)-(21)), relations (30) and (31) can be written as ζ 2 = η 1 u=wx = (a 6 a 5 )w x + a 7, (34) ζ 1 = η 2 v=wt = (a 6 a 8 )w t + a 4 (35) and taking (28) and (29) into account, we obtain η x + η w w x a 5 w x = (a 6 a 5 )w x + a 7, (36) η t + η w w t a 8 w t = (a 6 a 8 )w t + a 4. (37) Relations (36) and (37) must be verified identically, so that the following conditions hold: η w = a 6, η t = a 4, η x = a 7. (38) Then the expression (11) of η follows in a simple way from (38).

20 OPTIMAL SYSTEM In general, when a differential equation admits a Lie group G r and its Lie algebra L r is of dimension r > 1, one desires to minimize the search for invariant solutions by finding the nonequivalent branches of solutions. This leads to the concept of optimal system. It is well known that for one-dimensional subalgebras, the problem of finding an optimal system of subalgebras is essentially the same as the problem of classifying the orbits of the adjoint transformations.

21 In Ovsiannikov [1], the global matrix of the adjoint transformations is used in constructing the one-dimensional optimal system. In Olver [4], a slightly different technique is employed: it consists in constructing a table, named the adjoint table, showing the separate adjoint actions of each element in L r as it acts on all the other elements. Here, we made use of the procedure reported in Ruggieri and Valenti [7], which is a mixed of the above procedures and consists in constructing the global matrix of the adjoint transformations by means of the adjoint table.

22 The non trivial operator of the optimal system of L p is X o1 = X 1 + c 2 X 2 + c 4 X 4 = t + c 2 x + c 4 t w, (39) where c 2 and c 4 are real parameters. Because of the optimal systems of the Lie algebras of (1), are an extension of the optimal system of L p, we show, in Tables: 3 and 4, the extensions with respect to (39) only.

23 Case Generators of the optimal systems I a s = p X o2 = c 1 X 1 + X 5 = c 1 t x x (w + 2p x) w I b s = 2p X o2 = c 2 X 2 + X 5 = t t + c 2 x 2p x w I c s p, 2p X o2 = X 5 = 2 (s p)t t + (s 2p) x x + [(s 2 p)w 2 p s x] w Table: 3. Case I, extensions of the optimal system of L p

24 Case Generators of the optimal systems X o2 = X 1 + c 4 X 4 + X 5 = t + x x + (c 4 t q x) w II a II b II c II d II e II f II g X o3 = c 1 X 1 + X 4 + X 5 = c 1 t + x x + (t q x) w p 1, r = 0 2 X o2 = c 1X 1 + X 5 = c 1 t + x x + [(1 + 2p)w + 2 p q x] w p = 1, r = X o2 = c 2 X 2 + c 4 X 4 + X 5 = c 2 x t t + (c 4 t w q x) w p 1, r = X o2 = c 2 X 2 + X 5 = t t + c 2 x + 2p(w + q x) w p = 1, r 1, X o2 = c 4 X 4 + X 5 = 2r t t + (1 + 2 r) x x + (c 4 t + 2r w q x) w p 1, r = 1+2p 2 2 X o2 = c 3X 3 + X 5 = (1 + 2p)t t 2 p x x + (c 3 + 2p qx) w p 1, r 1, X o2 = X 5 = 2 r t t + (1 + 2r) x x + {[1 + 2(p + r)] w + 2p qx} w Table: 4. Case II, extensions of the optimal system of L p

25 REDUCTIONS TO ODEs AND EXACT SOLUTIONS One of the advantages of the symmetry analysis is the possibility to find solutions of the original PDEs by solving ODEs. These ODEs, called reduced equations, are obtained by introducing suitable new variables, determined as invariant functions with respect to the infinitesimal generators. On the basis of the infinitesimal generators of the optimal systems of the Lie algebras of (1), we can construct the reduced ODEs of (1) and find exact solutions. We show the details of the analysis in an application.

26 AN APPLICATION As an application, we consider the following equation w tt = (w x + q)w xx + [(w x + q) 1 w tx ] x, (40) so that we belong in the Case II of the Table 1 with p = 1, r = 2, f 0 = λ 0 = 1 and the generator of the optimal system X o2 reads as (see Case II g of the Table 4): X o2 = 4t t + 3x x + (w 2q x) w. (41)

27 By applying the invariant surface condition, through (41), we obtain dt 4t = dx 3x = dw w 2q x, (42) which gives the following similarity variable and solution, respectively z = x, w = t 1 4ψ(z) q x, (43) t 3 4 where ψ(z) must satisfy the ODE to which (40) is reduced by means of (41), namely 3(3z 2 ψ + 5z ψ ψ)ψ 2 16ψ (ψ ) 3 + 4[(3z ψ + 5ψ )ψ (3z ψ + 2ψ )ψ ] = 0. (44)

28 An exact solution of (44) is ψ(z) = z3 3 (45) and coming back to (43), the solution of (40) can be written as w = x3 q x, (46) 3t2 So that, the solution of the system equivalent to (40) is u = x2 q, (47) t2 v = 2x3 3t 3. (48)

29 FINAL REMARK These solutions are an example of exact solutions in the field of nonlinear dissipative media. Apart their own theoretical value, they can be used as benchmark test for numerical schemes and codes. Moreover, because of known exact solutions in viscoelasticity are very few, these ones found in the present paper give a contribution to the literature in this field.

30 [1] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York (1982). [2] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New-York (1989). [3] N. H. Ibragimov, CRC Hanbook of Lie Group Analysis of Differential Equations, CRC Press, Boca Raton, FL (1994). [4] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1986). [5] W. F. Ames and C. Rogers, Nonlinear Equations in the Applied Sciences, Academic Press, Boston (1992). [6] W. I. Fushchych and W. M. Shtelen, Symmetry Analysis and Exact Solutions of Nonlinear Equations of Mathematical Physics, Kluwer, Dordrecht (1993). [7] M. Ruggieri and A. Valenti, Proc. WASCOM 2005, R. Monaco, G. Mulone, S. Rionero and T. Ruggeri eds., World Sc. Pub., Singapore, (2006), 481.