#### Abstract

We study the new conservation forms of the nonlinear fin equation in mathematical physics. In this study, first, Lie point symmetries of the fin equation are identified and classified. Then by using the relationship of Lie symmetry and -symmetry, new -functions are investigated. In addition, the Jacobi Last Multiplier method and the approach, which is based on the fact -functions are assumed to be of linear form, are considered as different procedures for lambda symmetry analysis. Finally, the corresponding new conservation laws and invariant solutions of the equation are presented.

#### 1. Introduction

Fins are used in a large number of applications to increase the heat transfer from surfaces. Typically, the fin material has a high thermal conductivity. Due to having this property, it is very important in terms of technology. Generally, nonlinear heat conduction equation with fins is a mathematical model which can be represented by nonlinear differential equation. Pakdemirli and Sahin [1] obtained scaling, translational, and spiral group symmetries of the fin equation considered as a partial differential equation in which heat conduction coefficient is assumed to be function of temperature but the heat transfer coefficient is assumed to be only function of spatial variable. Bokhari et al. [2] investigated group theoretic analysis that provides different exact solutions or reduced equations specifically on traveling wave solutions and steady state type solutions. Vaneeva et al. [3] analyzed equivalence transformations and conditional equivalence groups and nonclassical symmetries of the fin equations are discussed in their study.

Lie point transformations are used for analyzing of differential equations mostly. These transformations leave the equation invariant which acts on the space of the dependent and independent variables. Revealing the symmetries of the equation by Lie group method perhaps enables us to obtain new solutions directly or from the known ones or via similarity reductions. In addition, the group classification of the differential equation based on the Lie point symmetries can be important in understanding the possible solutions of equations [4–12]. A part of this paper deals with Lie group analysis (symmetries, classification, and invariant solutions) of fin equation. Here, we analyze the special forms of thermal conductivity coeffcient and the heat transfer coefficient .

For any second order ordinary differential equation by using different approaches -symmetries can be obtained directly. First studies based on this idea have been introduced by Muriel and Romero [13–16]. They have proved that under the invariance criteria obtained Lie symmetries enable deriving -symmetries in a direct way [13]. Moreover, they have demonstrated that integrating factors and first integrals can be determined algorithmically by making use of -symmetries. Another way obtaining -symmetry, for the sake of simplicity -function can be assumed in a linear form. As a result of this assumption the determining equation can be solved easily. In addition, in this study, we present the connection between Lie point symmetry and -symmetry to find nontrival -functions, corresponding integrating factors and first integrals.

The last approach to find -symmetry is based on the Jacobi last multiplier method which is presented by Nucci and Levi [17]. -symmetries and corresponding invariant solutions can be obtained by using the Jacobi last multiplier directly. This new method admits the new determining equation which includes -function that can be obtained from the divergence of the ordinary differential equation. -symmetries can be found from a new form of the prolongation formula which includes three unknown variables; therefore, the determining equations cannot be reduced to a simpler form. Despite this difficulty, we can reduce -function to two; the number of unknown functions by using the Jacobi last multiplier approach and the obtaining new functions called as infinitesimals functions can be evaluated simply. When all these reasons are taken into consideration, we examine -symmetries of the fin equation for different cases.

This study is organized as follows. In Section 2 we give some preliminaries on relationship for Lie symmetries and -symmetries. In Section 3 we introduce nonlinear fin equation and the corresponding determining equations. In Section 4, we present Lie symmetries of fin equation for different heat transfer coefficient and thermal conductivity. -symmetries, conservation laws and new reduced form of fin equation are obtained by using these Lie symmetries. In Section 5 -symmetries in linear form are obtained and -symmetries based on Jacobi Last Multiplier method are considered as an alternative approach. Finally, In Section 6 we discuss some important results in the study.

#### 2. Relationship between Lie Point Symmetries and -Symmetries

Let us consider the second order differential equation of the form with , for some open subset . We denote by the corresponding space , for . Their elements are , where, for , denotes the derivative of order of with respect to . We assume that the implicit function theorem can be applied to (1), and, as a consequence, that this equation can locally be written in the explicit form and let vector field of (2) be in the form of

*Definition 1 (generalized prolongation formula). *Let be a vector field defined on , and let be an arbitrary function. The -prolongation of second order of denoted by is the vector field defined on by
where and
for where denotes the total derivative operator with respect to .

The relationship between -symmetries, integrating factors and first integrals of second order differential equations is important from the mathematical point of view [13–16]. In terms of a first integral of (2) is any function in the form of providing equality of . An integrating factor of (2) is any function satisfying the following equation:
Thus, -symmetries of second order differential equation (2) can be obtained directly by using Lie symmetries of this same equation. Secondly, let
be a Lie point symmetry of (2) and then the characteristic of is
and thus the vector field is called -symmetry of (2) if the following equality:
is satisfied.

If is assumed to be a -symmetry of (2) and is a first order invariant of , namely, any particular solution of the equation
then a first order invariant reduced equation of the form is obtained by using the reduction process associated to the -symmetry. Thus the general solution is found such as an equation of the implicit form:
It is clear that is an equivalent form of (2). Consequently,
is an integrating factor of (2).

Theorem 2. *Let be a second order ordinary differential equation, where is an analytic function of its arguments. There exists a function , for some , such that the vector field is a -symmetry of the equation [15].*

#### 3. Determining Equations for the Infinitesimal Symmetries

The differential equation describing the nonlinear fin problem has been derived as follows: where and are thermal conductivity and heat transfer coefficient, respectively, which are considered as functions of temperature and is the temperature function and is dimensional spatial variable [12].

If we consider an operator in the following form the nonlinear fin equation (13): where and are infinitesimal functions. Here we consider second prolongation operator of (14) as in the following form: since the highest derivative in (13) is second order in which and are defined: The application of (13) to (15) yields the invariance condition or equivalently In order to obtain the determining equations, the equation (19) can be separated with respect to and its powers:

#### 4. Lie Symmetries and Corresponding -Symmetries

In this section, the aim of our study is to get Lie point symmetries, corresponding lambda symmetries, conserved forms and invariant solutions by using different and functions. To find the infinitesimals, (20)–(23) should be solved together. First the equation (20) is integrated: and integrating (21) with respect to and solving for we have The infinitesimals and are inserted into (22), and the heat transfer coefficient is calculated in the following form: in which the equation (26) leads to that heat transfer coefficient can be obtained by examining different cases of thermal conductivity .

##### 4.1. Constant Thermal Conductivity:

Firstly we consider a constant thermal conductivity. Substituting in (26) we can write and in (16) we see that the heat transfer coefficient is in the linear form. By defining and such that then, rewriting (16) with (17) the relation is obtained in which and are constants. To classify the results systematically, we consider the following subcases: , , , and .

*Case 1 (if ). *The fin equation (13) is
For this case (20)–(23) finally yield
where , are constants. The algebra consists of an eight-parameter finite Lie group of transformations [4].

If parameter is selected and the remaining ones are set to be zero, the infinitesimals and are
Therefore, the characteristic is written:
By using (9), we obtain the -symmetry

A solution of (10) for this case is
and we can write ; then to obtain fin equation in terms of one can write
By using these equalities (36) we find the following equation:
in which the general solution is
and then the integrating factor becomes
Conservation law is
and the invariant solution is
where is a constant.

*Case 2 (). *The fin equation becomes
and the solutions of (20)–(23) yield an eight-parameter Lie group of transformations:
where , are constants.

For the selection of parameter , the infinitesimals are

By using (9), the -symmetry yields
A solution of (10) is equal to
and reduced form becomes
in which the general solution is
and the integrating factor is

Conservation law is found as
and the invariant solution is
where and are constants.

*Case 3 (if ). *Equation (13) is equal to
and the infinitesimals for this equation are
where , are constants.

For selection of parameter infinitesimals are found:
By using (9) we obtain the -symmetry
A solution of (10) for this case is
and reduced form is
in which the general solution is
and then the integrating factor becomes

Conservation law is
and the invariant solution is
where and are constants.

*Case 4 (). *In this case we obtain (13)

Infinitesimals are found as follows:
where , are constants.

For selection of parameter infinitesimals are found:
By using (9) we obtain the -symmetry
A solution of (10) for this case is
and reduced form is equal to
in which the general solution is
and the integrating factor yields
Conservation law is
and the invariant solution is
where and are constants.

##### 4.2. Linear Thermal Conductivity

If inserted into (26), and then the heat transfer coefficient is obtained in the following form: Let us consider the form of and such as is Then, To classify the results systematically, we consider the following subcases: , , , , , and , .

*Case 1 (if ). *Then the equation is
and the solution of (20)–(23) gives to the infinitesimals
where , are constants.

If parameter is selected and the remaining ones are set to be zero, the infinitesimals and are
By using (9) we obtain the -symmetry
A solution of (10) for this case is
and reduced form is found as below:
in which the general solution is
and the integrating factor becomes as the form
Conservation law is
and the invariant solution is
where and are constants.

*Case 2 (). *For this case we obtain
then the infinitesimals (24) and (25) are
where , are constants.

If parameter is selected and the remaining ones are set to be zero, the infinitesimals and are
By using (9) we obtain the -symmetry
A solution of (10) for this case is
and reduced form is
in which the general solution is
and then the integrating factor becomes

Conservation law is
and the invariant solution is
where and are constants.

*Case 3. *For the choice of the fin equation (13) becomes
After some manipulations equations (20)–(23) yield
where , are constants.

For selection of parameter infinitesimals are found:
By using (9) we obtain the -symmetry
A solution of (10) for this case is
and reduced form is
in which the general solution is
and the integrating factor becomes

Conservation law is
and the invariant solution is
where and are constants.

*Case 4 (). *For this case, (13) is
and infinitesimals are
where is constant. This result corresponds to principal Lie algebra which is a one-parameter finite Lie group of transformations.

By applying (9) we derive the -symmetry

A solution of (10) can be obtained:
and the reduced form is
in which the general solution is found
and the integrating factor can be derived as below:
Conservation law is

##### 4.3. ,

If is inserted into (26), then we have By defining then, is rewritten in the form To classify the results systematically, we consider the following subcases: , , , , , and , .

*Case 1 (). *The fin equation is
The solution of (20)–(23) gives to an eight-parameter Lie group of transformations:
where , are constants.

For selection of parameter infinitesimals are found:
By using (9) we obtain the -symmetry
A solution of (10) for this case is
and reduced form is equal to
in which the general solution yields
and then the integrating factor becomes
Conservation law is
and the invariant solution is
where and are constants.

*Case 2 (). *For this case (13) equals
where , are constants.

For selection of parameter infinitesimals are found:
By using (9) we obtain the -symmetry
A solution of (10) for this case is
and reduced form is
in which the general solution is
and then the integrating factor becomes
Conservation law is

*Case 3 (). *In this case fin equation becomes
and infinitesimals functions are found as below
where , are constants.

For selection of parameter infinitesimals are found
equation (9) gives the -symmetry
For this case the solution of (10) is
and reduced form can be written as
in which the general solution is
and the integrating factor is found as below:
Conservation law is
and the invariant solution is

*Case 4 (). *For the last case, (13) can be rewritten:
and infinitesimals are
where is a constant.

For selection of parameter infinitesimals are found:
By using (9) we obtain the -symmetry
A solution of (10) is equal to
and reduced form is found as
in which the general solution is
and the integrating factor can be written as
Conservation law is

#### 5. Alternative Approaches for -Symmetries

##### 5.1. Assuming Linear Form of

Let us consider an th-order ODE: Thus the invariance criteria [14] of (153) is The expansion of relation (154) for gives the determining equation related to fin equation, which are the system of partial differential equations. This system is difficult to solve in terms of lambda function because it is highly nonlinear. For the sake of simplicity can be assumed to be in a linear form [13–15] such that When , is -symmetry of (1), if and only if is satisfied. Applying (2) and (155) to (156) gives and are defined in equation (1). If the equation (155) is substituted in to equation (157), then we obtain determining equation and we find the functions and from the coefficients of this determining equation. We emphasize that is a particular solution of (157). If we consider in terms of fin equation and it is substituted into (157), the prolongation formula for the fin equation can be written as By analyzing (158) the coefficient of gives first determining equation such that

A particular solution of this equation gives like this, Then if we consider coefficient of , we obtain second determining equation as the form by applying (160) to (161), the general solution of this equation gives such that The last determining equation becomes To obtain a simpler form of (163) one can rewrite this equation in the form: In (164), we assume in which is a constant and we write our assumption in (164); we obtain the ordinary differential equation with respect to and so solution of this equation is equal to From the above relation, we find and if we solve this equation we obtain equation which defines relationship between and Now we investigate -symmetries of fin equation for different cases of .

*Case 1 (). *It is easy to see that from (166) thermal conductivity yields
For this case if and are substituted into (160) and (162), respectively, then we find
where is a constant. And is obtained by (155)
In order to obtain an integrating factor associated to we must find a first order invariant of . And so it is clear that the solution of (10) gives
If we take