NONCLASSICAL EQUIVALENCE TRANSFORMATIONS ASSOCIATED WITH A PARAMETER IDENTIFICATION PROBLEM

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1 NONCLASSICAL EQUIVALENCE TRANSFORMATIONS ASSOCIATED WITH A PARAMETER IDENTIFICATION PROBLEM NICOLETA BÎLĂ AND JITSE NIESEN Abstract. A special class of smmetr reductions called nonclassical equivalence transformations is discussed in connection to a class of parameter identification problems represented b partial differential equations. These smmetr reductions relate the forward and inverse problems, reduce the dimension of the equation, ield special tpes of solutions, and ma be incorporated into the boundar conditions as well. As an eample, we discuss the nonlinear stationar heat conduction equation and show that this approach permits the stud of the model on new tpes of domains. Our MAPLE routine GENDEFNC which uses the package DESOLV (authors Carminati and Vu) has been updated for this propose and its output is the nonlinear partial differential equation sstem of the determining equations of the nonclassical equivalence transformations. Ke words. smmetr reductions, parameter identification problems AMS subject classifications. 58J7, 7G65, 35R3, 35G3. Introduction. One of the fastest developing research fields in the last few ears is the area of inverse problems. These problems arise frequentl in engineering, mathematics, and phsics. In particular, parameter identification problems deal with the identification of phsical parameters from observations of the evolution of a sstem and especiall arise when the phsical laws governing the processes are known, but the information about the parameters occurring in equations is needed. In general, these are ill-posed problems, in the sense that the do not fulfill Hadamard s postulates for all admissible data: a solution eists, the solution is unique, and the solution depends continuousl on the given data. Arbitrar small changes in data ma lead to arbitrar large changes in the solution. The iterative approach of studing parameter identification problems is a functional-analtic setup with a special emphasis on iterative regularization methods []. Consider the following class of parameter identification problems modeled b partial differential equations (PDEs) of the form F(, w (m), E (n) ) =, (.) where the unknown function E = E() is called parameter, and, respectivel, the arbitrar function w = w() is called data, with = (,..., p ) Ω R p a given domain, where w (m) denotes the function w together with its partial derivatives up to order m. Suppose that the parameters and data are analtical functions. The PDE (.) sometimes augmented with certain boundar conditions is called the inverse problem associated with a direct (or forward) problem. While the direct problem is the same equation but the unknown function is the data (for which certain boundar conditions are imposed), for inverse problems, onl positive solutions ma be of phsical interest. This work was partiall supported b the Austrian Science Foundation FWF, Project SFB 38 Large Scale Inverse Problems. Department of Mathematics and Computer Science, Faetteville State Universit, Faetteville, NC 283, USA. (nbila@uncfsu.edu). Department of Applied Mathematics, School of Mathematics, Universit of Leeds, Leeds LS2 9JT, United Kingdom. (jitse@maths.leeds.ac.uk).

2 2 N.Bîlă, J. Niesen On the other hand, smmetr analsis theor has been widel used to stud nonlinear PDEs. A remarkable number of mathematical phsics models have been successfull analzed from this point of view. There is a considerable bod of literature on this topic (see, for eample, [], [4] [6], [9], [], [3] [5], [8] [2], [22] [24] and references from there). Sophus Lie [7] introduced the notion of continuous groups of transformations (known toda as Lie groups of transformations) and, subsequentl, a method for finding the smmetr group associated with a PDE. This (local) group of transformations acting on the space of the independent variables and the space of the dependent variables of the equation has the propert that it leaves the set of all analtical solutions of the PDE invariant. Moreover, the form or the equation remains unchanged and group-invariant solutions can be found. Lie s method has been applied etensivel to various mathematical models represented b PDEs. Consequentl, new methods for seeking eplicit solutions that cannot be obtained b Lie s method have been proposed over the ears. The nonclassical method introduced b Bluman and Cole [4] is a technique for determining the nonclassical smmetries (or conditional smmetries) related to a PDE. These are transformations that act on the space of the independent variables and the space of the dependent variables with the propert that the leave onl a subset of the set of all analtical solutions invariant. Knowledge of these classical and nonclassical smmetries allows one to reduce the order of the studied PDE and to determine solutions of special tpe. It ma happen that certain group-invariant solutions cannot be found eplicitl, but at least additional information about the studied model can be obtained in this case. An classical smmetr is a nonclassical smmetr but not conversel. Ovsiannikov [2] introduced the notion of equivalence transformations for a PDE depending on an arbitrar function and these transformations have been recentl generalized b Meleshko in [8]. The (generalized) equivalence transformations are groups of transformations acting on the space of the independent variables, the space of the dependent variables, and the space of the arbitrar functions with the propert that the leave the differential structure of the equation unchanged. B dropping the auiliar conditions that characterize the functional dependence, Torrisi and Tracina introduced the notion of weak equivalence transformations [24] and applied them to certain mathematical phsics models (see for eample [23]). Notice that all these methods do not take into account the boundar conditions attached to a PDE. Moreover, in the equivalence transformations approach the arbitrar functions ma depend on the dependent variable onl. In our case, for parameter identification problems of the form (.), the data does not depend on the parameter both data and parameter act on the space of the independent variables. The aim of this paper is to continue the stud of the parameter identification problems (.) from the point of view of smmetr analsis theor. This is a second paper in which we analze smmetr reductions suitable for (.). It is known that investigating special Lie groups of transformations related to a model, its dimension can be reduced and further information about its solutions can be obtained. As discussed in [3], finding the classical Lie smmetries related to the inverse problem (.) might be a ver difficult task and the success of the method depends on the nonlinearit of the equation. Therefore, considering both the parameter and data unknown functions in (.), the classical Lie method can be applied with less difficulties (in [3] a mathematical model arising in car windshield design was analzed using this assumption). We shall refer to these transformations as equivalence transformations, taking into account that w was initiall an arbitrar function. Furthermore, once the equivalence transformations related to (.) have been discussed, it is nat-

3 NONCLASSICAL SYMMETRIES 3 ural to investigate the nonclassical smmetries associated with this equation as well. These new smmetr reductions also relate the forward and inverse problems as the equivalence transformations do. To the best of our knowledge, these tpe of transformations have not been studied so far for parameter identification problems. Since the differential structure of the equation is preserved and w is an arbitrar function, we shall call these transformations nonclassical equivalence transformations. While the equivalence transformations are found b integrating a linear PDE sstem, the nonclassical equivalence transformations are solutions of the nonlinear PDE sstem. Tpicall, a large amount of calculations are required and, hence, in both cases, the use of a smbolic manipulation program would represent a great advantage. Therefore, for this purpose, we have updated our Maple package GENDEFNC [2] (which uses the package DESOLV b Carminati and Vu [7]). The GENDEFNC output is the nonlinear partial differential equation sstem of the determining equations of the nonclassical equivalence transformations. Notice that GENDEFNC is based on a different algorithm for finding the nonclassical smmetries (see [2] for further details). Recentl, Bruzón and Gandarias have etended this algorithm to a different case (see [6] for more information). As an eample, we shall consider a mathematical model arising in heat conduction, namel, the nonlinear stationar heat conduction equation given b div(e(, ) w(, )) = in Ω, (.2) where the unknown function E = E(, ) is the parameter, the arbitrar function w = w(, ) is the data, and (, ) Ω with Ω R 2 a bounded domain (here w = (w, w ) denotes the gradient of w). The data function must also satisf the Dirichlet boundar condition w Ω =. (.3) In 3D, the above problem is related to the heat conduction in a material occuping a domain Ω whose temperature is kept zero at the boundar []. After sufficientl long time, the temperature distribution w can be modeled b div(e(,, z) w(,, z)) = f(,, z) in Ω, (.4) where E is the heat conductivit and f represents the heat sources. For given E and f, the forward problem is to find the temperature distribution w satisfing (.4) and (.3). Conversel, the inverse problem is to determine E from (.4) and (.3) when w is known. While the direct problem is an elliptic PDE for w, the inverse problem is a linear PDE (with variable coefficients) for E. On the other hand, in the inverse problems approach, if the solution of the forward problem is unique for each parameter E, the parameter-to-solution map associates with each parameter E the forward problem solution. Since for the above problem the parameter-to-output map is nonlinear, (.4) and (.3) is a nonlinear problem from this point of view. In addition, when w = w(e), a new dependent variable can be introduced b using the Kirchoff transformation [25, p. 3]; if the heat conductivit E =, then the equation (.4) becomes Poisson s equation [2, p. 36]. For simplicit, in this paper, we discuss the 2D case with the heat sources f =. Since (.2) can be written as it can be easil seen that the vector field w E + w E + E(w + w ) =, (.5) X = w + w [ + E(w + w )] E,

4 4 N.Bîlă, J. Niesen generates the classical Lie smmetries associated with (.5). The paper is organized as follows. The equivalence transformations related to (.5) are presented in 2 and the nonclassical equivalence transformations are discussed in 3. We show that the nonclassical smmetries are related to the Monge equation (3.9), the Monge-Ampère equation (3.22), and the Abel ordinar differential equations (ODEs) of second kind (3.29) and (3.37). The new smmetr reductions related to (.5) are given b (3.27), (3.3), (3.35), (3.38), and, respectivel, (3.3) and (3.39) ecepting the case when A(, ) = (k k 3 + k 4 )/(k 2 + k 3 + k 4 ). Section 4 contains a few eamples for data defined on oval and rounded corner domains. 2. Equivalence transformations related to (.5). Let us consider a oneparameter Lie group of transformations acting on an open set D Ω W E, where W is the space of the data functions, and E is the space of the parameter functions, given b = + εγ(,, w, E) + O(ε 2 ), ỹ = + ελ(,, w, E) + O(ε 2 ), (2.) w = w + εφ(,, w, E) + O(ε 2 ), Ẽ = E + εψ(,, w, E) + O(ε 2 ), where ε is the group parameter. Let V = Γ(,, w, E) + Λ(,, w, E) + Φ(,, w, E) w + Ψ(,, w, E) E (2.2) be its associated general infinitesimal generator. Assume that E = E(, ) and w = w(, ) are both dependent variables in (.5). The transformation (2.) is called an equivalence transformation related to the PDE (.5) if this leaves the equation invariant, i.e., w Ẽ + wỹẽỹ +Ẽ( w + wỹỹ ) =. Note that the set of all analtical solutions of (.5) will also be invariant. For this case, the criterion for infinitesimal invariance is given b pr (2) V(F) F= =, with F (,, w (2), E ()) = w E + w E + E(w + w ) +, where pr (2) V denotes the second order prolongation of the vector field V [9]. Notice that this prolongation is determined b taking into account that E and w are both dependent variables, eactl as one would proceed in finding the classical Lie smmetries for a PDE without arbitrar functions. The order of the prolongation of the vector field V is given b the highest leading derivative of the dependent variables. Appling the classical Lie method, we obtain the following infinitesimals Γ(,, w, E) = k k 3 + k 4 Λ(,, w, E) = k 2 + k 3 + k 4 Φ(,, w, E) = µ(w) Ψ(,, w, E) = E(2k 4 µ (w)), (2.3) where k i, i =,..., 4 are real constants and µ = µ(w) is an arbitrar function. Hence, the infinitesimal generator (2.2) becomes 4 V = k i V i + V µ, (2.4a) i=

5 NONCLASSICAL SYMMETRIES 5 where V =, V 2 =, V 3 = +, V 4 = + + 2E E, (2.4b) V µ = µ(w) w Eµ (w) E. We obtain the following result Proposition 2.. There is an infinite dimensional Lie algebra of the equivalence transformations related to (.5) spanned b the infinitesimal generators (2.4). Therefore, the PDE (.5) is invariant under translations in the -space, -space, rotations in the (, )-space, and, respectivel, scaling transformations in the (,, E)-space. Since µ = µ(w) is an arbitrar function, we can obtain other Lie groups of transformations that leave the equation (.5) invariant as well. For instance, if µ = const., (.5) is invariant under translations in the w-space and, when µ(w) = w, the equation remains unchanged under scaling transformations in the (w, E)-space. Furthermore, the dimension of (.5) can be reduced if this is augmented with { Γ(,, w, E)w + Λ(,, w, E)w Φ(,, w, E) =, Γ(,, w, E)E + Λ(,, w, E)E Ψ(,, w, E) =, which is a first order PDE sstem defining the characteristics of the vector field (2.2). These relations are also called invariance surface conditions. 3. Nonclassical equivalence transformations related to (.5). Consider a one-parameter Lie group of transformations acting on an open set D Ω W E, where W is the space of the data functions, and E is the space of the parameter functions, given b = + εξ(,, w, E) + O(ε 2 ), ỹ = + εη(,, w, E) + O(ε 2 ), (3.) w = w + εφ(,, w, E) + O(ε 2 ), Ẽ = E + εψ(,, w, E) + O(ε 2 ), where ε is the group parameter. Let the following vector field U = ξ(,, w, E) + η(,, w, E) + φ(,, w, E) w + ψ(,, w, E) E (3.2) be the general infinitesimal generator related to (3.). The transformation (3.) is called a nonclassical equivalence transformation (or a conditional smmetr) of the PDE (.5) if this leaves the subset S F,φ,φ 2 = {F(,, w (2), E (2) ) =, φ (,, w (), E () ) =, φ 2 (,, w (), E () ) = } of the set of all analtical solutions invariant, where { φ := ξ(,, w, E)w + η(,, w, E)w φ(,, w, E) =, (3.3) φ 2 := ξ(,, w, E)E + η(,, w, E)E ψ(,, w, E) = represents the characteristics of the vector field U (or the invariant surface conditions). Here the criterion for infinitesimal invariance is the following pr (2) U(F) F=,φ=,φ 2= =, pr () U(φ ) F=,φ=,φ 2= =, pr () U(φ 2 ) F=,φ=,φ 2= =.

6 6 N.Bîlă, J. Niesen If η, one can assume without loss of generalit that η = (the case η = is not discussed in this paper), and, hence, (3.3) turns into { w = φ(,, w, E) ξ(,, w, E)w, (3.4) E = ψ(,, w, E) ξ(,, w, E)E. At the first step, we augment the original PDE with (3.4) and eliminate all the partial derivatives of w and E with respect to occurring in (.5). Hence, b using (3.4) and its differential consequences, we obtain A w + A 2 w 2 + A 3w E + A 4 w + A 5 E + A 6 =, (3.5a) where the coefficients A i = A i (,, w, E), i =...6, are the following A = E(ξ 2 + ), A 2 = 2Eξξ w, A 3 = ξ 2 + 2Eξξ E +, A 4 = ξψ + E (ξξ ξ φξ w ψξ E 2ξφ w ), A 5 = ξ (φ + 2Eφ E ), A 6 = φψ + E (ξφ φ φφ w ψφ E ). (3.5b) The equation (3.5) has been obtained b using the GENDEFNC command gendefnc(pde, [w,e], [,],, 3). Since A, (3.5) ma be regarded as an ODE in the unknown functions w and E (with as a parameter). At the second step, b using the GENDEFNC command gendefnc(pde, [w,e], [,], ) we obtain the determining equations of the nonclassical smmetries. This is an overdetermined nonlinear PDE sstem for the infinitesimals ξ = ξ(,, w, E) and φ = φ(,, w, E). Among these equations, we get which implies ξ w =, ξ E =, φ E =, ξ(,, w, E) = A(, ), (3.6) and φ(,, w, E) = G(,, w). (3.7) The substitution of the above functions into the remaining equations ields ψ(,, w, E) = EF(,, w). (3.8) B using the above relations, the determining sstem is reduced to F = G w + 2(A AA ) A 2, (3.9) + G AG 2AA A (A 2 ) A 2 G =, (3.) +

7 NONCLASSICAL SYMMETRIES 7 and G + G + F G + FG + 2GG w + 2(A AA) A 2 + (GG w + G + FG) 2(AA+A) A 2 + G =, (A 4 )A + 4A(A 2 + )A (A 4 )A 2A(A 2 3)A 2 4(3A 2 )A A + 2A(A 2 3)A 2 =, (3.) (3.2) where A = A(, ), G = G(,, w), and F = F(,, w) are the unknown functions. Remark. The nonclassical smmetries do not leave the form of the equation (.5) invariant. Moreover, the nonclassical operators (3.2) do not form a vector space, still less a Lie algebra, as the smmetr operators do. Since ever classical smmetr is a nonclassical smmetr but not conversel, there eists a set of common solutions of the determining sstem of the nonclassical smmetries (nonclassical equivalence smmetries, in this contet) and the determining sstem of the classical smmetries (equivalence transformations, respectivel). This common solution is given b ξ(,, w, E) = Γ(,,w,E) Λ(,,w,E) = k k3+k4 k, 2+k 3+k 4 φ(,, w, E) = Φ(,,w,E) Λ(,,w,E) = µ(w) k 2+k 3+k 4, ψ(,, w, E) = Ψ(,,w,E) Λ(,,w,E) = E(2k4 µ (w)) k 2+k 3+k 4, (3.3) where at least one of the constants k 2, k 3 or k 4 is nonzero. To solve the determining equations (3.9) (3.2), we proceed as follows: the equations (3.) and (3.) are analzed in 3., the equation (3.2) is studied in 3.2, and the solutions of the determining equations (3.9) (3.2) are given in The equations (3.) and (3.) for φ = G(,, w). First observe that (3.) can be written in the following conservation form ( ) ( ) G AG A 2 + A 2 =. + If A, then two cases might occur: Case G. G. In this case, (3.) and (3.) are both satisfied. Case G2. G. There eists a potential function K = K(,, w) such that K = AG A 2 + K = G A 2 +. The above sstem ields K = AK whose general solution is K(,, w) = P(u, w), where u = u(, ) is a solution of the equation From these relations we obtain u = A(, )u. (3.4) G = u (A 2 + )S, (3.5)

8 8 N.Bîlă, J. Niesen where S(u, w) denotes the partial derivative P u (u, w). Thus, the solution of (3.) is given b (3.5), where u satisfies (3.4). The substitution of (3.5) into (3.) implies q (S uw S S u S w ) + q S uu + q 2 S u + q 3 S =, (3.6) where the coefficients q i are epressed in terms of A and u as follows q = u 3 (A2 + ) 3, q 2 = u (A 2 + ) [ 3u (A 2 + ) 2 + u (5A + 3A 3 A + 3A 2 A + AA ) ], q 3 = u (A 2 + ) 3 + u (A 2 + )[(3A 2 + 5)A + (3A 2 + )AA ] +u [2A(A 2 + )A + (A 2 + )(A 2 + 3)A + A(A 2 + ) 2 A 2(A 2 3)A 2 + 2A(A 2 3)A A + (A 4 )A 2 ]. In particular, the method of separation of variables applied to (3.6) implies solutions of the form S(u, w) = p(u)µ(w), where p = p(u) satisfies the equation q p uu + q 2 p u + q 3 p = (3.7) and µ = µ(w) is an arbitrar function of its argument. To summarize, the equation (3.) has been reduced to (3.6), where A = A(, ) satisfies (3.2) and u = u(, ) is a solution of (3.4) The equation (3.2) for ξ = A(, ). In the following, we show that (3.2) can be reduced to a Monge-Ampère equation. Case A. A The equation (3.2) admits the trivial solution A. Case A2. A = k, with k The constant function A = k, with k is also a solution of (3.2). Case A3. A is a nonconstant function If A is a nonconstant function, then an alternative formulation of (3.2) is B(B 2 + )B + (B 2 )(B 2 + )B B(B 2 + )B (3B 2 )B 2 2B(B 2 3)B B + (3B 2 )B 2 = where B = (A + )/(A ). The conservation form of the above PDE is ( ) ( ) B(B + BB ) B + BB (B 2 + ) 2 (B 2 + ) 2 =. (3.8) We distinguish the following two cases: Case A3.. B = B(, ) is a solution of (3.9). If B = B(, ) satisfies the Monge equation B + BB =, (3.9)

9 NONCLASSICAL SYMMETRIES 9 then (3.8) holds. Since the general solution of (3.9) is given implicitl b B = ν(b), where ν is an arbitrar function, the corresponding solution of (3.2) is A + A = ν Case A3.2. B = B(, ) does not satisf (3.9) ( ) A +. (3.2) A It follows from (3.8) that there eists a potential function T = T(, ) such that T = B+BB (B 2 +) 2, T = B(B+BB) (B 2 +) 2. (3.2) Since T cannot be identicall zero, the above equations ield B = T /T and, b substituting it into the first equation of (3.2) we have T T T ( T 2 T 2 ) T T T T + ( T 2 + T 2 ) 2 =. B using the following Legendre transformation [25, p. 353] H(a, b) + T(, ) = a + b, where T = a, = H a, T = b, and = H b, the above PDE turns into the following Monge-Ampère equation H aa H bb H 2 ab ab (a 2 + b 2 ) 2 H aa + a2 b 2 (a 2 + b 2 ) 2 H ab ab + (a 2 + b 2 ) 2 H bb =. Furthermore, this can be reduced to the Monge-Ampère equation V aa V bb Vab 2 = (a 2 + b 2 ) 2, (3.22) where V (a, b) = H(a, b) 2 arctan ( a b ). (3.23) Eact solutions for particular Monge-Ampère equations have been etensivel analzed in [22]. The Monge-Ampère equation (3.22) ma be included in Case 7 [22, p. 458] or Case 2 [22, p. 46] Solutions of the determining equations (3.9) (3.2). We distinguish the following four cases: Case. A and G Proposition 3.. If A and G, the infinitesimal generator (3.2) becomes U =, which implies the invariance of the equation (.5) with respect to translations in the -space.

10 N.Bîlă, J. Niesen This nonclassical smmetr reduction is an equivalence transformation that can be obtained from (3.3) for µ =, k 2 =, and k i =, where i =, 3, 4. Case 2. A and G The PDE (3.) takes the form G = and it follows that G = H(, w). After substituting it into (3.9) and (3.), we get F = H w and, respectivel, H + HH w H H w =. (3.24) Notice that H in the above equation. We distinguish the following cases: Case 2.. H = µ(w), where µ is an arbitrar function Indeed, H = µ(w) is a particular solution of (3.24). Proposition 3.2. If A and G = µ(w), where µ is an arbitrar function, the infinitesimal generator (3.2) turns into U = + µ(w) w µ (w)e E. (3.25) The above nonclassical equivalence transformation is in fact an equivalence transformation and corresponds to the case k 2 =, and k i =, where i =, 3, 4 in (3.3). Case 2.2. H The PDE (3.24) can be written in the following conservation form ( ) ( ) H + =. After introducing the potential function g = g(, w), we get { g = H H H H w g w = H. Clearl, g w. After eliminating g in the above sstem, we obtain H = g /g w. Net, substituting it into the second equation, the following PDE results The following two cases occur: Case g = g (g ) w g w (g ) =. (3.26) It results g(, w) = 2 /2 + h(w), where h is an arbitrar nonconstant function (otherwise, g w ). Since H = g /g w, we get H = µ(w), where µ = /h. Proposition 3.3. If A and G = µ(w), where µ is an arbitrar function, then the infinitesimal generator (3.2) becomes U = + µ(w) w µ (w)e E, (3.27) The nonclassical smmetr generated b (3.27) is a new smmetr reduction for (.5) which cannot be obtained from (3.3).

11 Case g NONCLASSICAL SYMMETRIES Since the equation (3.26) is the Jacobian of the functions g and g, there eists a function α such that g = + α(g). (3.28) In the above relation, w is viewed as a parameter. The equation (3.28) can be written as the following Abel ODE of second kind ( + α(g)) d dg =, for = (g). The canonical substitutions z = α(g) and v = +α(g) reduce the above ODE to its canonical form vv v = β(z), (3.29) where v = v(z) and β = /(α α ). A collection of the known cases of solvable Abel ODEs of the form (3.37) is presented in [2, pp. 7 2] and new results can be found, for eample, in [8]. Each of these ODEs corresponds to a nonclassical equivalence transformation related to (.5). Proposition 3.4. If A and G = g /g w, where g is a solution of (3.28), then the infinitesimal generator (3.2) turns into U = g /g w w + E(g /g w ) w E, (3.3) Since g satisfies (3.26), the above vector field generates new smmetr reductions related to (.5) that are not equivalence transformations. Case 3. A and G B (3.9), we obtain F = 2(A AA )/(A 2 + ). Proposition 3.5. If A is a solution of the equation (3.2) and G, then the nonclassical infinitesimal generator (3.2) becomes U = A(, ) + + 2(A AA ) A 2 E E. (3.3) + Case 3.. A = k, where k Proposition 3.6. If A = k, where k is a constant, the infinitesimal generator (3.3) rewrites as U = k +. Replacing µ =, k 2 =, and k i = with i =, 3, 4 in (3.3), we obtain the smmetr reduction generated b the above vector field, and, therefore, this is an equivalence transformation. Case 3.2. A is a nonconstant function If A is a nonconstant function, then we distinguish two subcases: Case A is given implicitl b (3.2). Case The equation for A is reduced to the Monge-Ampère equation (3.22).

12 2 N.Bîlă, J. Niesen In the above two cases, the nonclassical operator (3.3) generates new smmetr reductions for (.5) ecepting the case when A(, ) = (k k 3 +k 4 )/(k 2 +k 3 +k 4 ). Case 4. A and G Case 4.. A = k, where k Without loss of generalit, we consider the particular solution u(, ) = k + of the equation (3.4). Since u =, the relation (3.5) ields G = (k 2 + )S, where S is a nontrivial solution of the equation (3.6) which rewrites as The following cases might occur: Case 4... S = µ(w), where µ is an arbitrar function S uu + SS uw S u S w =. (3.32) In this case, (3.32) is satisfied and (3.9) implies F = µ (w). Proposition 3.7. If A = k with k and G = (k 2 + )µ(w), where µ is an arbitrar function, then the nonclassical infinitesimal generator (3.2) becomes U = k + + (k 2 + )µ(w) w (k 2 + )µ (w)e E. (3.33) The above nonclassical operator generates an equivalence transformation that can be obtained from (3.3) for k = k/(k 2 + ), k 2 = /(k 2 + ), k 3 =, and k 4 =. Case S u Since the PDE (3.32) can be written in the conservation form ( ) ( ) S + =, there eists a potential function Q = Q(u, w) such that { Qu = S S u, S u u S u w Q w = S u. Eliminating S in the above sstem, we obtain S = Q u /Q w, where We distinguish the following two cases: Case 4..2.a. Q u = u Q u (Q u u) w Q w (Q u u) u =. (3.34) In this case, Q(u, w) = u 2 /2 + p(w), where p is an arbitrar nonconstant function (otherwise Q w ). With the aid of S = Q u /Q w, we get S(u, w) = uµ(w), where µ = /p and u(, ) = k +. Proposition 3.8. For A(, ) = k (k ) and G(,, w) = (k 2 +)(k+)µ(w), where µ is an arbitrar nonconstant function, the nonclassical infinitesimal generator (3.2) rewrites as follows U = k + + (k 2 + )(k + )µ(w) w (k 2 + )(k + )µ (w)e E. (3.35)

13 NONCLASSICAL SYMMETRIES 3 The nonclassical operator (3.35) generates new transformations that are not equivalence transformations related to (.5). Case 4..2.b. Q u u Observe that (3.34) is the Jacobian of Q and Q u u. Thus, there eists a function γ such that Q u = u + γ(q). (3.36) In the above ODE, w is viewed as a parameter. Similar to Case 2.2.2, the equation (3.36) can be reduced to (u + γ(q)) du dq =, which is an Abel ODE of second kind for u = u(q). Moreover, after using the substitutions s = γ(q) and V = u + γ(q), the above ODE can be reduced to the canonical form V V V = θ(s), (3.37) where θ = /(γ γ ). This is an Abel equation of second kind for the unknown function V = V (s). For each solution of (3.37), we obtain a nonclassical smmetr for (.5). The solvable Abel ODEs of the form (3.37) that are known so far are listed in [2, pp. 7 2]. More recent results can be found, for instance, in [8]. Proposition 3.9. If A(, ) = k with k and G(,, w) = (k 2 + )Q u /Q w, where u(, ) = k +, and Q = Q(u, w) satisfies (3.34), then the nonclassical infinitesimal generator (3.2) is given b U = k + (k 2 + )Q u /Q w w + (k 2 + )E(Q u /Q w ) w E. (3.38) Since Q satisfies (3.34), the above vector field generates new smmetr reductions related to (.5) that are not equivalence transformations. Case 4.2. Suppose A is a nonconstant function and G. Proposition 3.. If A is a nonconstant function satisfing the equation (3.2) and G = u (A 2 + )S, where u satisfies (3.4) and S is a nonzero solution of (3.6), the nonclassical infinitesimal generator (3.2) is written as ( U = A + + u (A 2 + )S w + E u (A 2 + )S w + 2(A ) AA ) A 2 E. (3.39) + Two cases might occur: Case A is given implicitl b (3.2). Case The equation for A is reduced to the Monge-Ampère equation (3.22). In the above two cases, the vector (3.39) generates new smmetr reductions that cannot be obtained from (3.3) ecepting the case when A(, ) = (k k 3 + k 4 )/(k 2 + k 3 + k 4 ).

14 4 N.Bîlă, J. Niesen 4. Eamples. The nonclassical smmetries related to (.5) ield classes of data and suitable domains for which the dimension of the problem can be reduced. Since the homogeneous Dirichlet boundar condition (.3) is imposed, the data w must satisf (3.4) and the boundar Ω must be compatible with the smmetr reduction as well. To illustrate this approach, we discuss a few eamples of data modeled b the functions w(, ) = 2 + W() (4.) which are invariant with respect to the nonclassical infinitesimal generator U = + 2 w. (4.2) Indeed, substituting (4.2) in (3.4), we get w = 2 and E =. This implies E(, ) = E () (i.e., the heat conductivit is constant along the lines = const.) and w is given b (4.). Note that (4.2) is obtained from (3.27) for µ(w) = 2. Boundar curves compatible with (4.2) will have the form 2 = P() where P is a polnomial. Therefore, we can be considered circles, ellipses, generalized Lamé curves 2p + 2 = (p > 2), Granville s egg curve 2 2 = ( b)( ), where b,, or elliptical curves in particular, Newton s egg curve 2 = ( 2 )( a), where a ±. In this case, as it is shown below, the parameter cannot be determined at the points (, ) for which W ( ) =. Despite this fact, additional information about the parameter on the specified domain can be obtained. Eample. (Newton s egg curve) Suppose Ω = {(, ) : 2 = ( 2 )( 3), [, ]} (see Figure 4.). For the data w(, ) = 2 + ( 2 )( 3), the equation (.5) is reduced to the ODE (3 2 6 )E () + (6 8)E () =, whose general solution is [ E () = C E ( )] () ep 3 arctanh ( ), 2 for 2/ 3, where E () = ep[ 3/3 arctanh ( ( ) 3/2 )]. Figure 4.2 shows the parameters satisfing the conditions E( ) =.2, and, respectivel, E() =.2. The parameter E cannot be determined for = 2/ 3. Eample 2. (Granville s egg curve) Consider Ω = {(, ) : 2 2 = ( 3)( ), [, 3]} (see Figure 4.3) and the data given b w(, ) = 2 ( 3)( ) 2. The reduced ODE 2(2 3)E () + 2( )E () = 4 implies E(, ) = E () = [C + E ()] 3 [ ] (2 3) ep 43/6 24 ( ), which is defined for.5, where E () =.5(2 3) 27/6 ep [ 24 ( ) ]. Similarl, the parameter cannot be determined for =.5. See Figure 4.4 for E() =.2 and, respectivel, E(3) =.6.

15 NONCLASSICAL SYMMETRIES w Fig. 4.. The boundar Ω = {(, ) : 2 ( 2 )( 3) =, [,]} and the data w(, ) = 2 + ( 2 )( 3) discussed in Eample E E Fig The parameter E(, ) = E () corresponding to data discussed in Eample. Eample 3. (Generalized Lamé curve) For Ω = {(, ) : =, [, ]} (see Figure 4.5) and the following data w(, ) = 4 2, the PDE (.5) becomes 4 3 E () + 2(62 + ) =. It results E () = [ ( ) π erf + C ] ( ) 2 3 ep 4 2 that is defined for (here erf denote is the error function). Therefore, the parameter cannot be estimated at =. See Figure 4.6 for the cases E ( ) =.25, and, respectivel, E () =.25.

16 6 N.Bîlă, J. Niesen w Fig The boundar Ω = {(, ) : 2 2 = ( 3)( ), [,3]} and the data w(, ) = 2 ( 3)( ) 2 considered in Eample E 5 E Fig The parameter E(, ) = E () corresponding to data considered in Eample 2. Eample 4. (Rounded corner rectangular domain) Suppose Ω = {(, ) : 2 = , [ a, a]}, where a = + 65/4 (see Figure 4.7). For the data w(, ) = , we obtain the following reduced equation 2(6 2 )E () E () =. Thus, E () = C arctan ( 6 2 ) /2 2(6 2 ) 3/2 for [ a,.25) (.25, a], and E () = C arctan ( 6 2) /2 2( 6 2 ) 3/2

17 NONCLASSICAL SYMMETRIES w Fig The boundar Ω = {(, ) : =, [, ]} and the data w(, ) = 4 2 discussed in Eample E 8 E Fig The parameter E(, ) = E () corresponding to data discussed in Eample 3. for (.25,.25). E cannot be determined for = ±.25. Figure 4.3 shows the graph of the parameter for C = C 2 =. 5. Conclusion. In this paper we point out another sstematic wa of finding classes of smmetr reductions related to parameter identification problems of the form (.). Similar to the stud in [3], we emphasize that the geometrical significance of the nonlinearit occurring between the data and parameter in (.) can be reflected b the group analsis tool. Seeking different shapes for domains on which the dimension of the problem can be reduced is not an eas task. Therefore, in this paper, we discuss the nonclassical equivalence transformations related to (.5). Briefl, to determine these transformations, the data is considered as a dependent variable as well as the parameter, and the nonclassical method (due to Bluman and Cole) is applied to the studied equation. These smmetr reductions provide us with additional information about the parameter on different parts of the domain (as it has been shown in 4). In general, for a known data w, one should check the invariance of this function in (3.4), where ξ, φ, and ψ are discussed in 3. Net, from the second equation in

18 8 N.Bîlă, J. Niesen 2.5 w Fig The boundar Ω = {(, ) : 2 = , [ a, a]}, where a = p + 65/4 and the data w(, ) = considered in Eample 4. (3.4), the form of the parameter should be obtained in terms of the invariants of the smmetr reduction. At the end, substituting E and w into (.5), the dimension of the model should be reduced b one. Acknowledgements. One of the authors, N. Bîlă would like to thank Prof. Dr. Heinz W. Engl at Johann Radon Institute for Computational and Applied Mathematics (RICAM) who encouraged her to appl the smmetr analsis to parameter identification problems. She is also grateful to Dr. Philipp Kügler at RICAM and Dr. Antonio Leitão at Federal Universit of Santa Catarina, for man stimulating and rewarding discussions during her sta at RICAM. REFERENCES [] W. F. Ames, Nonlinear Partial Differential Equations, Academic Press, New York, 967. [2] N. Bîlă and J. Niesen, On a new procedure for finding nonclassical smmetries, J. Smbolic Comput., 38, 6(24), [3] N. Bîlă, Application of smmetr analsis to a PDE arising in the car windshield design, SIAM Journal on Applied Mathematics, 65, (24), pp [4] G. W. Bluman and J. D. Cole, The general similarit solutions of the heat equation, J. Math. Mech., 8 (969), pp [5] G. W. Bluman and S. Kumei, Smmetries and Differential Equations, Appl. Math. Sci. 8, Springer-Verlag, New York, 989. [6] M. S. Bruzóna and M. L. Gandarias, Appling a new algorithm to derive nonclassical smmetries, Commun. in Nonlinear Sci. Numer. Simul., 3, 3(28), pp [7] J. Carminati and K. Vu, Smbolic computation and differential equations: Lie smmetries, J. Smbolic Comput., 29 (2), pp [8] E. S. Cheb-Terraba and A. D. Roched, Abel ODEs: Equivalence and Integrable Classes, Comput. Phs. Comm., 3, (2), pp [9] P. A. Clarkson and M. Kruskal, New similarit reductions of the Boussinesq equation, J. Math. Phs., 3 (989), pp [] P. A. Clarkson and E. L. Mansfield, Algorithms for the nonclassical method of smmetr reductions, SIAM J. Appl. Math. 54(994), pp [] H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, The Netherlands, 996. [2] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations,

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