Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

The method of conditional Lie-Bäcklund symmetry is applied to solve a class of reaction-diffusion equations ut + uxx + Q(x)u2 x + P(x)u + R(x) = 0, which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamical systems. The exact solutions obtained in concrete examples possess the extended forms of the separation of variables.


Introduction
In this paper, we analyze a class of reaction-diffusion equations (RDEs) which admit certain conditional Lie-Bäcklund symmetries (CLBSs). Equation (1) can be simplified from the following RDEs: which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory [1][2][3][4][5]. Here the coefficient functions depend upon the variable which typically represents the value of the underlying asset, such as the price of a stock upon which an option is placed. There exist several special cases of (2), say, the Black-Scholes-Merton equation, the Longstaff equation [6], the Vasicek equation [7], and the Cox-Ingersoll-Ross equation [8], in which ( ) is zero. In order to keep the computations as simple as and consistent with this requirement, authors in [4] transform (2) into (1) by using an equivalence point transformation, namely, by means of ( , ) = ( , ) + ( ) , = ℎ ( ) , with ℎ ( ) 2 = 1 ( ) , Equation (2) can be transformed into

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where the primes denote differentiation with respect to x. Then, with the inverse transformation = ℎ −1 ( ), this equation can be performed as (1) on reversion to lower case variables and redefinition of the coefficient functions as appropriate. When ( ) = and ( ) = 0, (1) becomes which can simplify the linear form of under the transformation = log(V)/ + ∫ ( ) /2 + with 4 ( ) = −(2 + 2 + 4 ).
It is known that symmetry reductions and exact solutions play important roles in the study of RDEs. The conditional Lie-Bäcklund symmetry (CLBS) method introduced by Zhdanov [9] and Fokas and Liu [10,11] firstly has been proved to be very powerful to classify equations or specify the functions appeared in the equations and construct the corresponding group invariant solutions. Furthermore, authors have shown that CLBS is closely related to the invariant subspace; namely, exact solutions defined on invariant subspaces for equations or their variant forms can be obtained by using the CLBS method [12][13][14][15][16][17][18][19][20][21][22][23][24].
Motivated by the form of (1), we set the following secondorder nonlinear CLBSs: which are very powerful to specify the functions appeared in (1) and construct the corresponding exact solutions. The remainder of this paper is organized as follows. In Section 2, some equations of the form (1) admitting CLBSs generated by (8) are obtained. CLBS reductions and exact solutions of two concrete examples are used to illustrate the results. Section 3 is devoted to conclusions and discussions.

Equations Admitting CLBSs and Two Examples
To consider further, we need the following proposition derived in [9,10]. (8) if and only if = 0 whenever satisfies (1) and = 0, where the prime denotes the Gateaux derivative, that is,

Proposition 1. RDEs (1) admit CLBSs
A direct computation from the above proposition yields To vanish all the coefficients of (9), we have the following overdetermined system: Solving this system, we can obtain the unknown functions in (1) and the corresponding CLBSs (8). From the first and seventh equations of system (10), it is apparent that the solutions can be divided into two cases including 1 ( ) = ( ) and 1 ( ) = 3 ( ) = 0, 1 ( ) ̸ = ( ).

Example 1. Equation
admits the CLBS The corresponding solutions are given by Advances in Mathematical Physics 5 where 1 ( ) and 2 ( ) satisfy the finite-dimensional dynamical system Exact solutions can be obtained as with two arbitrary constants 1 and 2 .

Example 2. Equation
The corresponding solutions are given by where 1 ( ) and 2 ( ) satisfy the system This dynamical system can be solved and exact solutions are with two arbitrary constants 1 and 2 . Here the solutions ( , ) preserve the forms of the separation of variables = 1 ( ) + 2 ( ) ( ), which are associated with the invariant subspace L{1, 2 } mentioned in [3] and the references therein.

Conclusions and Discussions
In this paper, we have discussed RDEs (1) by means of CLBS with characteristic (8). The key for this method is to determine presumably the form of the CLBS. For (1), we found that nonlinear CLBS (8) is very effective, which can yield some interesting symmetry reductions and exact solutions. Two examples are considered to illustrate this method in terms of the compatibility of CLBSs and the governing equations. Generally speaking, the obtained solutions cannot be derived within the framework of Lie's classical method and nonclassical method.
In addition, it must be pointed out that, for the corresponding equations with certain fractional derivative [25], we can do similar work including CLBS classification, reductions, and exact solutions.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.