On New Conservation Laws of Fin Equation

We study the new conservation forms of the nonlinear fin equation inmathematical physics. In this study, first, Lie point symmetries of the fin equation are identified and classified.Thenby using the relationship of Lie symmetry andλ-symmetry, newλ-functions are investigated. In addition, the Jacobi LastMultiplier method and the approach, which is based on the fact λ-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.


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][5][6][7][8][9][10][11][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][14][15][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 2 Advances in Mathematical Physics 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.
The relationship between -symmetries, integrating factors and first integrals of second order differential equations is important from the mathematical point of view [13][14][15][16].In terms of  a first integral of ( 2) is any function in the form of (, ,   ) providing equality of () = 0.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  [, (1)] , namely, any particular solution of the equation then a first order invariant reduced equation of the form Δ  (, ,   ) = 0 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   ((, (, ,   ))) = 0 is an equivalent form of (2).Consequently, is an integrating factor of (2).
Theorem 2. Let  (2) = (,  (1) ) be a second order ordinary differential equation, where  is an analytic function of its arguments.There exists a function (,  () ), for some  < 2, such that the vector field   is a -symmetry of the equation [15].
Advances in Mathematical Physics 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  1 and  2 are defined: The application of (13) to (15) yields the invariance condition  (2) In order to obtain the determining equations, the equation ( 19) can be separated with respect to  1 and its powers: and in (16) we see that the heat transfer coefficient () is in the linear form.By defining  1 and  2 such that then, rewriting ( 16) with ( 17) the relation is obtained in which  1 and  2 are constants.To classify the results systematically, we consider the following subcases: Case 1 (if  1 = 0,  2 = 0).The fin equation ( 13) is For this case (20)-( 23) finally yield where   ,  = 1, . . ., 8 are constants.The algebra consists of an eight-parameter finite Lie group of transformations [4].
If parameter  5 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.
For the selection of parameter  5 , 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.
For selection of parameter  1 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.

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For selection of parameter  1 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.26), and then the heat transfer coefficient is obtained in the following form:

Linear Thermal Conductivity
Let us consider the form of  1 and  2 such as is Then, To classify the results systematically, we consider the following subcases: Case 1 (if  1 = 0,  2 = 0).Then the equation is and the solution of ( 20)-( 23) gives to the infinitesimals where   ,  = 1, . . ., 8 are constants.
If parameter  5 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.
If parameter  3 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.
For selection of parameter  3 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 Advances in Mathematical Physics and the invariant solution is where  and  are constants.
Case 4 ( 1 ̸ = 0,  2 ̸ = 0).For this case, ( 13) is and infinitesimals are where  1 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 By defining then, () is rewritten in the form To classify the results systematically, we consider the following subcases: (i)  1 =  2 = 0, (ii)  1 = 0,  2 ̸ = 0, (iii)  1 ̸ = 0,  2 = 0, and (iv)  1 ̸ = 0,  2 ̸ = 0.
For selection of parameter  1 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.
For selection of parameter  3 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 ( 1 ̸ = 0,  2 = 0).In this case fin equation becomes and infinitesimals functions are found as below where   ,  = 1, . . ., 2 are constants.
For selection of parameter  2 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 ( 1 ̸ = 0,  2 ̸ = 0).For the last case, ( 13) can be rewritten: and infinitesimals are where  1 is a constant.For selection of parameter  1 infinitesimals are found: By using ( 9) we obtain the -symmetry A solution of ( 10) is equal to and reduced form is found as Advances in Mathematical Physics in which the general solution is and the integrating factor can be written as Conservation law is (152)

Alternative Approaches for 𝜆-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  = 2 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][14][15] such that  (, ,   ) =  1 (, )   +  2 (, ) .
When V =   , V 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  1 and  2 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   2 gives first determining equation such that A particular solution of this equation gives  1 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  2 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   ()/() −  = 0 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  1 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 V  1 .And so it is clear that the solution of (10) gives If we take  1 () = 1 and to write (13) in terms of {, ,   } we can express the following equality using (170): Taking derivative of (171) with respect to  gives and by using   and   (13) becomes and, solving this equation we have To find the integrating factor one can write above equation in terms of : It is clear that from (12),  is obtained in the following form: The conserved form of this equation is given by which gives the original fin equation.Thus the reduced equation is where  3 is a constant.Integrating above equation we obtain the solution that satisfies the original equation: where  2 is a constant.
Case 2 (() =   ).From (166) thermal conductivity () can be written as By inserting () and () to ( 161) and (162), respectively, we have where  1 is a constant.From (155)  is equal to The general solution of (10) for this case is given by Now we again use the assumption  1 () = 1 and to express (13) in terms of {, ,   } we can write the following equality using (183): Differentiating (184), we obtain If we write   ,   , (), () in original fin equation ( 13) then we obtain the same equation ( 173) such as and the solution of this equation is given by To obtain the integrating factor one can write above equation in terms of : It is clear that from (12) that  can be derived as The conserved form of fin equation for this case is which gives the original fin equation.Thus the reduced equation is where  3 is a constant.Solution of this reduced equation which can be called a invariant solution of ( 13) is where  2 is a constant.

𝜆-Symmetries and Jacobi Last Multiplier Approach.
The -prolongation of  (4) is pr =  (, )   +  (, )   +  (1) (, ,   ,   , . . .,  (−1) ) ×    +  (2) (, ,   ,   , . . .,  (−1) )    , with where   is total derivative operator with respect to  such that In this section we analyze -symmetries of fin equation by using Jacobi Last Multiplier as another approach.First (153) can be written by using system of first order equations, which is equivalent to the expression the Jacobi Last Multiplier of (207)  is found where, namely,  is The nonlocal approach [17] to -symmetries is analyzed to seek -symmetries such that With this approach the function is considered in the form such as  = log(1/).However it is a fact that the relation (207) cannot be used if the divergence of (207) Div ≡ ∑  =1 (  /  ) is equal to zero.Therefore  has to be chosen in this form because any Jacobi Last Multiplier is a first integral of (207).In this subsection we consider some cases, which we discussed in Section 3 for functions () and () in order to compare -symmetries in terms of different approaches: For this case the divergence of the fin equation ( 13) yields Substituting   into (204) then from the solution of the determining equations (154) we obtain eight-parameter infinitesimals: where   ,  = 1, . . ., 8 are constants.The generators can be written as which corresponds to the classical Lie point symmetries since   is equal to zero.
For this case we obtain the divergence of (13) in the form and, by substituting   into the prolongation formula, the infinitesimals can be found as follows: In this section we present some invariant solutions based on Jacobi multiplier approach.
Case 1.For the case () =  0 , () = and the Lagrange equations are which give the first-order invariants: Then one can write the first-order equation in the form and the solution of this equation yields which is equal to the original fin equation (13).The new reduced form is and the solution of (234) is where  1 and  are constants.

Concluding Remarks
This study focuses on applications of different symmetry approaches such as classical Lie point and -symmetries for the one-dimensional steady-state fin problem.We consider both the thermal conductivity and the heat transfer coefficient to be arbitrary functions of temperature.Firstly we concern with determining equations of fin problem to determine infinitesimals functions.The solutions of these determining equations enable us determine various forms of the heat transfer coefficient functions () corresponding to different cases of thermal conductivity function ().
And so we obtain a lot of cases to investigate in detail and then Lie point symmetries of fin equation are examined.By making use of these Lie point symmetries we determine symmetries, conservation laws, and new reduced form of fin equation algorithmically.
Another part of this study consists of investigation of symmetries, in which -function is assumed to be in linear form for simplicity since the corresponding determining equations are highly nonlinear and difficult to solve.This approach is also provided to illustrate new forms of () and () functions for new conservation forms of nonlinear fin equation.As a new approach to -symmetry concept the Jacobi Last Multiplier method is discussed and applied in this study.This method has some differences from previous methods mentioned here.In this approach we can use divergence of the equation to construct the prolongation of the same equation and as a result new -symmetries are achieved.So these symmetries are compared with each other.In particular, the comparison is made between Lie point and -symmetry approach with Jacobi last multiplier method.One can say that if Lie point symmetries of given equation are trivial and then this method has advantage to obtain new lambda symmetries.Moreover some invariant solutions can be obtained by using this approach.