Abstract
Nonlinear boundary value problems (BVPs) by means of the classical Lie symmetry method are studied. A new definition of Lie invariance for BVPs is proposed by the generalization of existing those on much wider class of BVPs. A class of twodimensional nonlinear boundary value problems, modeling the process of melting and evaporation of metals, is studied in details. Using the definition proposed, all possible Lie symmetries and the relevant reductions (with physical meaning) to BVPs for ordinary differential equations are constructed. An example how to construct exact solution of the problem with correctlyspecified coefficients is presented and compared with the results of numerical simulations published earlier.
2010 Mathematics Subject Classification : 22E70, 35K61, 80A22.
Lie symmetries of nonlinear boundary value problems
Roman Cherniha and Sergii Kovalenko
Institute of Mathematics, Ukrainian National Academy of Sciences,
3 Tereshchenkivs’ka Street, Kyiv 01601, Ukraine
Department of Mathematics, National University ‘Kyiv Mohyla Academy’,
2 Skovoroda Street, Kyiv 04070, Ukraine
Email: cherniha@imath.kiev.ua and kovalenko@imath.kiev.ua
1 Introduction
It is well known that principle of linear superposition cannot be applied to generate new exact solutions to nonlinear partial differential equations (PDEs). Thus, the classical methods (the Fourier method, the method of the Laplace transformations, and so forth) are not applicable for solving nonlinear PDEs. While there is no existing general theory for integrating nonlinear PDEs, construction of particular exact solutions for these equations is a nontrivial and important problem. Now the most popular methods for construction of exact solutions to nonintegrable nonlinear PDEs are the Lie, LieBäcklund and conditional symmetry methods [1, 2, 6, 3, 4, 5]. Although these methods are very powerful provided the relevant symmetry is known, several other approaches for solving nonintegrable nonlinear PDEs were independently suggested during the last decades. Among them the method of compatible differential constraints [7, 8], the method of linear invariant subspaces [9], method and its various modifications [10, 11, 12, 13, 14], the method of additional generating conditions [15, 16], and the transformed rational function method [17] should be marked out (see, e.g., Supplements in [18] about other methods).
The Lie symmetries are widely applied to study nonlinear differential equations (including multicomponent systems of PDEs) since 60s of the last century, notably, for constructing their exact solutions. Nevertheless there are a huge number of papers and many excellent books (see, e.g., [1, 2, 3, 4, 5] and papers cited therein) devoted to such applications, one may note that a very small number of them involve Lie symmetries to solve boundary value problems for the given PDEs. To the best of our knowledge, the first papers in this directions were published in the beginning of 1970s [19] and [20] (the extended versions of these papers are presented in books [21] and [2], respectively). The books, which highlight essential role of Lie symmetries in solving boundary value problems (BVPs) and present several examples, were published only in 1989 [2, 22].
The main object of this paper is a nonlinear BVP of Stefan type, which belongs to the class of BVPs with free (moving) boundaries. Boundary value problems of Stefan type are widely used in mathematical modeling a huge number of processes, which arise in physics, biology and industry [23, 24, 25, 26, 27]. Nevertheless these processes can be very different from formal point of view, they have the common peculiarity, unknown moving boundaries. Movement of unknown boundaries are described by famous Stefan boundary conditions [27, 28]. It is wellknown that exact solutions of BVPs of Stefan type can be derived only in exceptional cases and the relevant list is rather short at the present time (see [23, 29, 30, 31, 32, 33, 34, 35] and papers cited therein).
Nevertheless BVPs with free boundaries are more complicated objects than the standard BVPs with the fixed boundaries, it can be noted that the Lie symmetry method should be more applicable just for solving problems with moving boundaries. In fact, the structure of such boundaries may depend on invariant variable(s) and this gives a possibility to reduce the given BVP to that of lover dimensionality. This is the reason why different authors applied the Lie symmetry method to BVPs with free boundaries ignoring BVPs with fixed boundaries [36, 20, 37, 19, 38].
The paper is organized as follows. In Section 2, we discuss the existing definitions of Lie invariance for BVPs and propose their generalization on much wider class of BVPs. As an example the direct application of the definition for the wellknown BVP with the fixed boundaries is presented. In Section 3, we apply the definition derived to the class of (1+1)–dimensional BVPs of Stefan type used to describe melting and evaporation of materials in the case when their surface is exposed to a powerful flux of energy [31, 39]. The result obtained is an essential generalization of paper [37]. In Section 4, we reduce the problem to BVPs for ordinary differential equations, using Lie symmetry operators obtained in the previous section. An example how to construct exact solution of the problem with correctlyspecified coefficients is also presented. Finally, we present conclusions in the last section.
2 Definition of Lie invariance for BVPs
We start from a definition of invariance of a BVP under the given infinitesimal operator presented in [5, 2] and restrict ourselves on the case when the basic equation of BVP is an (1+1)–dimensional evolution PDE of th–order (). In this case the relevant BVP may be formulated as follows:
(1) 
(2) 
where and are smooth functions in the corresponding domains, and are a domain with smooth boundaries and smooth curves, respectively. Hereafter the subscripts and denote differentiation with respect to these variables, . We assume that BVP (1) and (2) has a classical solution (in a usual sense).
Consider the infinitesimal generator
(3) 
(hereafter and are known smooth functions), which defines a Lie symmetry acting on both –space as well as on its projection to –space. Let be the th–prolongation of the generator calculated by the wellknown prolongation formulae (see, e.g. [3, 1]).
Definition 1
The definition can straightforwardly be extended on BVPs for a system of PDEs. However, one easily notes that this definition cannot be applied to BVPs with free boundaries, because such problems contain moving surfaces, say where are unknown functions. Obviously, these functions should be interpreted as additional variables. In [36] (see Appendix 2), a criteria of invariance for BVP with a free boundary was formulated. Another deficiency of Definition 1 appears if one consider BVPs in the unbounded domain when the boundary conditions for arise. In fact, item (b) has no sense in this case and cannot be replaced by the natural passage to the limit, i.e., Probably this deficiency for the first time was noted in [40] (see Section 4.3) where the transformation was suggested to avoid the nonregular manifold generated by .
Now we present a definition which takes into account all possible boundary conditions and is applicable to a wide range of BVPs. Consider a BVP for a system of evolution equations () with independent and dependent variables. Let us assume that the th–order () basic equations of evolution type
(4) 
are defined on a domain with smooth boundaries. Consider three types of boundary and initial conditions, which can arise in applications:
(5) 
(6) 
and
(7) 
Here and are the given numbers, and are the known functions, while the functions defining free boundary surfaces must be found. We assume that all functions arising in (4)–(7) are sufficiently smooth so that a classical solution exists for this BVP.
Consider an –parameter (local) Lie group of point transformations of variables in the Euclidean space (open subset of ), which is given by equations
(8) 
where are the group parameters. According to the general Lie group theory, one may construct the corresponding –dimensional Lie algebra with the basic generators
(9) 
where .
Consider the Lie algebra in the extended space of the variables , where are new dependent variables with respect to and . In the extended space , the Lie group corresponding to this algebra is given by transformations
(10) 
Now we propose a new definition, which is based on the standard definition of differential equation invariance as an invariant manifold in the relevant space of variables and on the prolongation theory [3].
Definition 2
A boundary value problem (4)–(7) is called to be invariant with respect to the Lie group (10) if:

the manifold determined by Eqs. (4) in the space of variables is invariant with respect to the th–order prolongation of the group ;

each manifold determined by conditions (5) with any fixed number is invariant with respect to the th–order prolongation of the group in the space of variables , where ;

each manifold determined by conditions (6) with any fixed number is invariant with respect to the th–order prolongation of the group in the space of variables , where ;

each manifold determined by conditions (7) with any fixed number is invariant with respect to the th–order prolongation of the group in the space of variables , where .
Definition 3
Remark 1
Definition 2 can be generalized on more general systems (including hyperbolic and elliptic those) and boundary conditions containing highorder derivatives for .
Remark 2
If free boundaries are given in the form , where then we simply take . On the other hand, one can formulate a definition of Lie invariance for BVPs with such form of the free boundaries (see, e.g., [36]). However, the form used in Definition 2 is more convenient for generalization on multidimensional BVPs.
Now we present a nontrivial result to illustrate Definition 2. Let us consider the nonlinear BVP modeling the heat transfer in semiinfinite solid rod assuming that thermal diffusivity depends on temperature and the rod is exposed to a periodical flux of energy at the left endpoint. It should be noted that we neglect the initial distribution of the temperature in the rod, i.e., consider the process on the stage when the heat transfer already started. Thus the nonlinear BVP reads as
(11)  
(12)  
(13) 
where is an unknown temperature field, is a thermal diffusivity coefficient, is an energy flux. We assume that all functions arising in (11)–(13) are sufficiently smooth, so that a classical solution exists for this BVP.
Here we restrict ourselves to the case when the thermal diffusivity coefficient depends on the temperature as a power low, i.e. , where (in the case , the problem is liner and can be solved by classical methods, see, e.g., [41]). Notably, equation (11) with presents the most interesting cases of Lie symmetry invariance [1]. In the case ( ), it admits a fourdimensional Lie group. The corresponding algebra possesses the basic operators . These operators generate the oneparameter Lie groups
(14)  
(15)  
(16)  
(17) 
respectively (hereafter are arbitrary group parameters). If , then the additional conformal generator occurs, which extends to the fivedimensional Lie algebra . Thus, the case should be examined separately.
Theorem 1
Proof. On the first step of the proof we will consider BVP (11)–(13) with the constant energy flux , i.e. . Let us study the case of arbitrary power . First of all, we consider the oneparameter Lie groups (14)–(17) generated by the basic operators of . One easily notes that BVP (11)–(13) is invariant with respect to the Lie group and isn’t invariant under the Lie group since the boundary curve isn’t invariant with respect to the transformations (15).
According to item (b) of Definition 2, the boundary condition (12) is invariant with respect to the oneparameter group , if the manifold = satisfies the conditions
(18) 
The first equation of (18) is an identity, while the second equation leads to the expression , which immediately gives
(19) 
The invariance of (13) under the oneparameter group is obvious. Thus, BVP (11)–(13) is invariant with respect to the Lie group if and only if the restriction (19) takes place.
Dealing in a similar way with the Lie group , we obtain that BVP (11)–(13) is invariant with respect to only in two cases: , and , . Indeed, according to item (b) of Definition 2, the boundary condition (12) is invariant with respect to , if conditions (18) are satisfied on the manifold . Hence, we arrive at the restriction
(20) 
which immediately leads to provided , and if . The invariance of (13) under the oneparameter Lie group is evident.
Taking into account the restrictions considered above on and , one concludes that BVP (11)–(13) is invariant with respect to the twoparameter Lie group iff and with respect to the threeparameter Lie group iff (it is exactly case 1 of Table 1).
To find other Lie groups of invariance, one needs to consider a linear combination of the basic operators of excepting the operator (we remind that the BVP is invariant under the Lie group for arbitrary and )
(21) 
where are arbitrary parameters and at least two of them are nonzero. If , then one arrives only at the results obtained above for the Lie group , if then the result obtained above for the Lie group is recovered. If then two possibilities occur: and . Consider the case when operator (21) generates the Lie group
(22) 
Clearly, the boundary condition (13) is invariant with respect to . Boundary conditions (12) is invariant under , if and only if conditions (18) are satisfied. Now we realize that the first equation of (18) leads to the requirement while the second equation of (18) gives
(23) 
Since , one immediately obtains provided and . If then we immediately arrive at case 1 from Table 1. On the other hand, the Lie group transforms into the group , when . Thus, we can conclude that the BVP under study is invariant with respect to the twoparameter Lie group if and only if
(24) 
It is exactly case 2 of Table 1. The examination of the case leads to case 2 with . Thus, the invariance of BVP (11)–(13) with is completely examined.
Now we examine the special case . One easily checks that the oneparameter groups (with ) listed in cases 1–2 of Table 1 are the groups of invariance of BVP (11)–(13) (with ) under the same restrictions on the constant .
Thus, we need to examine whether the BVP in question can be invariant with respect to a Lie group corresponding to any liner combination of the basic operators of
(25) 
To avoid cumbersome formulae, we consider the oneparameter Lie group corresponding to the pure conformal operator
(26) 
Let us study the invariance of the boundary condition (13). According to item (d) of Definition 2, the following equalities should take place
(27) 
where = . However, . Thus, the contradiction is obtained and we conclude that BVP (11)–(13) with isn’t invariant under .
In a quite similar way, one may show that the boundary condition (13) isn’t invariant under any Lie group corresponding to operator (25).
Finally, to complete the proof, we must consider the case, when the flux of energy has periodical form, i.e. . Obviously, must be nonzero, otherwise we obtain the case examined above. Since calculations are quite similar to the case (an analog of formula (20) plays a crucial role to derive the special power ), we present the result: BVP (11)–(13) with the periodic energy flux is invariant only with respect to the oneparameter Lie group with (case 3 from Table 1).
The proof is now completed.
3 Lie invariance of a class of (1+1)–dimensional nonlinear BVPs of Stefan type
In this section we consider a class of (1+1)–dimensional BVPs of Stefan type used to describe melting and evaporation of materials in the case that their surface is exposed to a powerful flux of energy. Such problems also arise in mathematical modeling of other processes in biology (tumor growth) and physics (crystal growth). The class of BVPs after some simplifications (like using the Goodman substitution to transform the basic equations to the standard heat equations) can be written as follows
(28)  
(29)  
(30)  
(31)  
(32) 
where and are the unknown temperature fields; are the unknown functions, which determine the phase division boundaries (they can be also presented in the form ); are the phase division boundary velocities; is the known strictly positive function presenting the energy flux being absorbed by the material; is the known nonnegative function describing dynamics of evaporation process; are the known strictly positive function presenting specific heat values per unit volume of liquid and solid phases. The parameters and are assumed to known, moreover, .
Here Eqs. (28) and (29) describe the heat transfer process in liquid and solid phases, the boundary conditions (30) present evaporation dynamics on the surface , and the boundary conditions (31) are the famous Stefan conditions on the surface dividing the liquid and solid phases. Assuming that the liquid phase thickness is considerably less than the solid phase thickness, one may use the Dirichlet condition (32). It should be stressed that we neglect the initial distribution of the temperature in the solid phase and consider the process on the stage when two phases take already place.
One may claim that formulae (28)–(32) present a class of BVPs with moving boundaries and take into account a number of different situations, which occur in the melting and evaporation processes. Setting and , where is a correctlyspecified function, one obtains the problem, which is the most typical, see, e.g., [30]. In the case of a process when surfaces are exposed to very powerful periodic laser pulses these functions take complicated forms [39].
The BVP obtained is based on the standard nonlinear heat equations. Lie symmetry of noncoupled system (28)–(29) can be easily derived using the determining equations from paper [42], where reactiondiffusion systems of more general form have been investigated. Now we formulate a theorem, which gives complete information on Lie symmetry of this system.
Theorem 2
All possible maximal algebras of invariance (up to equivalent representations generated by transformations of the form (33)) of the system (28) and (29) for any fixed vectors with strictly positive functions and are presented in Table 2. Any other system of the form (28) and (29) is reduced to one of those with diffusivities from Table 2 by an equivalence transformation of the form
(33) 
where , and are arbitrary parameters.
no  Basic operators of MAI  

1.  
2.  
3.  
4.  
5.  
6.  
7.  
8.  
9.  
Remark 4
In the case of linear system (28)–(29) with (see case 9 of Table 2), the Lie algebra extension occurs by the operators and . However, BVP (28)–(32) with is rather artificial from physical point because diffusivities of solid and liquid phases must be different. Thus, we don’t consider this case below.
Remark 5
If one takes into account the trivial discrete transformations , and , then cases 2 and 3, 5 and 6 arising in Table 2 are equivalent. However, the class of BVPs (28)–(32) isn’t invariant under these transformations because of boundary conditions (30) and (32). Thus, we don’t take into account them in what follows.
Using the set of equivalence transformations (33), we can straightforwardly extend one to the relevant set for BVP (28)–(32) by adding the identical transformations for the variables . Direct calculations show that the most general form of those is
(34) 
where , and are arbitrary parameters ().
Now we formulate the main result of this section.
Theorem 3
BVP (28)–(32) with any smooth functions , and is invariant under the oneparameter Lie group presented in case 1 of Table 3. All possible extensions of this Lie group invariance (up to equivalent representations generated by equivalence transformations of the form (34)) depend only on the form of the functions and , and are presented in cases 2 and 3 of Table 3. Any other BVP of the form (28)–(32) is invariant under twoparameter Lie group is reduced by transformations (34) to one of those with the functions and from Table 3.
Proof. According to Definition 2 and Theorem 2 we need to examine the nine different cases listed in Table 2. It turns out that the examination of the first case, when the functions and are arbitrary, leads to the main result of the theorem presented in Table 3.
Let us consider the oneparameter Lie groups corresponding to the basic operators of algebra . Obviously, BVP (28)–(32) with arbitrary given functions is invariant under the group of space translations generated by the operator and this is listed in the first case of Table 3. Since any linear combination of other two operators is equivalent (up to transformations (34)) either to (if ) or to (if ), we should separately examine these two operators.
Now we apply Definition 2 to . Taking into account that BVP (28)–(32) has two free boundaries, we construct the extended Lie group corresponding to the operator :
(35) 
According to item (c), the boundary conditions (30) are invariant with respect to the group , if the manifold satisfies the conditions
(36) 
Taking into account (35), one finds
(37) 
so that the second and third equations of (36) produce the equations
(38) 
to find the functions and . Solving (38) one obtains
(39) 
where and are arbitrary smooth functions. The invariance criterium of the boundary conditions (31) for is fulfilled for arbitrary parameters arising in (31), while the invariance of condition (32) under is obvious. Thus, BVP (28)–(32) is invariant with respect to the Lie group if and only if restrictions (39) take place. This is exactly listed in case 3 of Table 3.
In a quite similar way one can show, that the BVP under study is invariant with respect to the extended Lie group corresponding to the operator if and only if the restrictions on
(40) 
take place, and this is what exactly listed in case 2 of Table 3.
Much more cumbersome calculations are needed to show that there are no any new Lie group invariance for BVP (28)–(32) nevertheless there are eight special cases listed in Table 2, which lead to the extensions of MAI of the basic equations (28).
Let us consider case 2 of Table 2. Firstly, we check the invariance of BVP (28)–(32) with respect to the oneparameter extended Lie groups corresponding to the operators and :
(41) 
and
(42) 
According to item (c) of Definition 2, the boundary conditions (31) are invariant with respect to the group , if the conditions
(43) 
are satisfied, where the manifold
Taking into account (41) and the second equation of (43), we arrive at the requirement
(44) 
Similarly, one easily checks that the boundary conditions (31) isn’t invariant with respect to the Lie group , too. Indeed, to satisfy the third equation of (43), one obtains the requirement
(45) 
Let us now examine the invariance of BVP (28)–(32) with respect to an extended Lie group corresponding to a liner combination of operators , , , and , i.e.
(46) 
where are arbitrary parameters and (otherwise the operator is obtained). Having transformations (34), we can put and in (46) so that the operator takes the form
(47) 
The corresponding Lie group is