Equivalence Groups and Differential Invariants for (2+1) dimensional Nonlinear Diffusion Equation
aa r X i v : . [ m a t h - ph ] J a n Equivalence Groups and Differential Invariantsfor (2+1) dimensional Nonlinear DiffusionEquation
Saadet ¨Ozer
Istanbul Technical University, Faculty of Science and Letters,Department of Mathematics Engineering, 34469 Maslak Istanbul Turkey;email: [email protected]
Abstract (2+1) dimensional diffusion equation is considered within the frame-work of equivalence transformations. Generators for the group are ob-tained and admissible transformations between linear and nonlinear equa-tions are examined. It is shown that transformations between linear andnonlinear equations are possible provided that the generators of indepen-dent variables depend on the dependent variable. Exact solutions forsome nonlinear equations are obtained. Differential invariants related tothe transformation groups are investigated and the results are comparedwith the direct integration method. keywords:
Lie Group Application, Equivalence Groups , Exact Solution, Non-linear Diffusion Equation, Differential Invariants
Differential equations containing some arbitrary functions or parameters rep-resent actually family of equations of the same structure. Almost all fieldequations of classical continuum physics possess this property related to thebehaviour of different materials. In dealing with such family of differentialequations, Lie symmetry analysis provides some powerful algorithmic methodsfor determination of invariant solutions, conserved quantities and constructionof maps between differential equations of the same family that turns out tobe equivalent [1–3]. To examine such problems, it is convenient consideringequivalence transformation groups that preserve the structure of the family ofdifferential equations but may change the form of the constitutive functions,parameters when appropriate transformations are available.The first systematic treatment that the usual Lie’s infinitesimal invarianceapproach could be employed in order to construct equivalence groups was for-mulated by Ovsiannikov [4]. Then several well-developed methods have beenused to construct equivalence groups. The general theory of determining trans-formation groups and algorithms can be found in the references [4–7].1n the present text we shall examine the (2+1) dimensional diffusion equa-tion. Nonlinear members of the family of diffusion equations have significantimportance in many areas in applied sciences. A great number of members havebeen used to model many physical phenomena in Mathematical Physics, Math-ematical Biology etc. and they have been widely studied not only by means ofnumerical, asymptotic analysis but also as application to Lie Group Analysissince Lie [8]. The simple nonlinear heat equation; u t = [ A ( u ) u x ] x was firstexamined by Ovsiannikov [9] within the frame-work of Lie’s symmetry classifi-cation. The complete classification and form-preserving point transformationsfor the inhomogenous one dimensional nonlinear diffusion are obtained in [10].The conditions for reduction the more general diffusion type equations to theone dimensional heat equation are also examined in [11]. Equivalence transfor-mations of linear diffusion equations into nonlinear equations for some classeshave been considered in Lisle’s PhD thesis [12] widely. Constructing the exactanalytical solutions for some specific problems related to the nonlinear diffusionequation have recently examined by [13–15]. Torrisi et al. in their paper [14]have also studied the developments of bacterial colonies as application to equiv-alence groups. Bruz´on et al. have derived maps between nonlinear dispersiveequations in their detailed work [16].The aim of the present work is to study equivalence transformations fora general family of (2+1) dimensional diffusion equation. We investigate thestructure of the transformation group generators which lead to map linear andnonlinear members.For the convenience of the reader to follow, in section 2, we have obtainedthe generators of the group of equivalence transformations and determined thestructure for admissible transformations. Theorems in the section state theconditions of the appropriate equivalence transformations between linear andnonlinear members . In the next section, we have considered some subgroups ofthe general equivalence groups and by choosing some specific forms we are ableto obtain some classes of nonlinear equations which are equivalent to linear ones.We have also investigated the classes of nonlinear diffusion equations that aremapped onto the classical heat equation. Exact solutions for those nonlinearequations are also obtained. In section 4 we have also examined differentialinvariants for the subgroups and discussed the results with the direct integrationmethod. In the present paper we shall investigate the equivalence group a general familyof (2+1) dimensional diffusion equation u t = f ( x, y, t, u, u x , u y ) x + g ( x, y, t, u, u x , u y ) y (1)which represents a great variety of linear and nonlinear equations. Here u isthe dependent variable of the independent variables x, y and t . Here f and g are smooth nonconstant functions of their variables and subscripts denote thepartial derivatives with respect to the corresponding variables. Definition 1.
With n independent variables x i , N dependent variables u α and smooth functions φ k of independent, dependent variables and their derivatives F ( x i , u α ( p ) , φ k ( q ) ( x i , u α ( p ) )) = 0 is called a family of differential equations. Here i = 1 , ...n, α = 1 , , ...N, k =1 , , ...m and u α ( p ) include both the tuple of dependent variables u = ( u , u , ..., u N ) as well as all the derivatives of u with respect to x i ’s up to order N . By φ k ( q ) we denote the smooth functions φ k and the partial derivatives with respect toboth x i ’s and u α ( p ) ’s. Definition 2.
For a given differential equation of the family F ( x i , u α ( p ) , φ k ( q ) ( x i , u α ( p ) )) = 0 the equivalence group E is the group of smooth transformations of independent,dependent variables, their derivatives and smooth functions preserving the struc-ture of the differential equation but transforms it into F (¯ x i , ¯ u α (¯ p ) , ¯ φ k (¯ q ) (¯ x i , ¯ u α (¯ p ) )) = 0 . More precisely, equivalence transformations associated with the (2+1) di-mensional most general diffusion equation (1) transform the equation into u t − f ( x, y, t, u, u x , u y ) x − g ( x, y, t, u, u x , u y ) y = 0 −→ ¯ u ¯ t − ¯ f (¯ x, ¯ y, ¯ t, ¯ u, ¯ u ¯ x , ¯ u ¯ y ) ¯ x − ¯ g (¯ x, ¯ y, ¯ t, ¯ u, ¯ u ¯ x , ¯ u ¯ y ) ¯ y = 0 . where ¯( . ) represents the transformed variables and functions.Let M = N × R be a (2+1) dimensional manifold with a local coordinatesystem x = ( x i ) = ( x, y, t ) which we shall call as the space of independentvariables. Consider a trivial bundle structure ( K , π, M ) with fibers are the realline R . Here M is the base manifold and K , called the graph space is globallyin form of a product manifold M × R . We equip the four dimensional graphspace K with the local coordinates ( x , u ) = ( x, y, t, u ).A vector field on the graph space K is a section of its tangent bundle andlocally in form V = ξ ∂∂x + ξ ∂∂y + ξ ∂∂t + η ∂∂u (2)where ξ i ( i = 1 , ,
3) and η are coordinate functions on K .In order to construct the equivalence groups E for the equation (1), firstwe extend the graph space K by adding the auxiliary variables and the onesrepresenting the functional dependencies of the smooth functions f and g ˜ K = { x, y, t, u, f, g, u x , u y , u t , f x , f y , f t , f u , g x , g y , g t , g u , f u x , f u y , g u x , g u y } . (3)The prolongation vector ˜ V over the extended manifold covered by ˜ K can bewritten as˜ V = V + µ ∂∂f + µ ∂∂g + ζ ∂∂u x + ζ ∂∂u y + ζ ∂∂u t + X j =1 µ j ∂∂f j + X j =1 µ j ∂∂g j + X j =1 ν j ∂∂f u j + X j =1 ν j ∂∂g u j (4)3here in the last four summations j = 1 , .. x, y, t and u and allcoefficient functions are smooth functions of the coordinates of the extendedmanifold. Theorem 1 ( [7]) . Let a vector field on an n dimensional differentiable manifold M be given by V ( p ) = v i ( x ) ∂∂x i , p = ϕ ( − ( x ) , i = 1 , ..., n where ( U, ϕ ) is the chart to which p ∈ M belongs. A curve γ is an integralcurve of the vector field V iff the coordinate functions x i ( t ) are solutions of thefollowing system of local ordinary differential equations in R n dx i dt = v i ( x ( t ))Precisely, the equivalence transformations can be determined by solving thefollowing system of autonomous ordinary differential equations on the extendedmanifold (3) d ¯ xdǫ = ξ (¯ x, ¯ y, ¯ t, ¯ u ) , d ¯ ydǫ = ξ (¯ x, ¯ y, ¯ t, ¯ u ) , d ¯ tdǫ = ξ (¯ x, ¯ y, ¯ t, ¯ u ) ,d ¯ udǫ = η (¯ x, ¯ y, ¯ t, ¯ u ) , d ¯ fdǫ = µ , d ¯ gdǫ = µ (5)under the initial conditions¯ x (0) = x, ¯ y (0) = y, ¯ t (0) = t, ¯ u (0) = u, ¯ f (0) = f, ¯ g (0) = g (6)where µ and µ depend on (¯ x, ¯ y, ¯ t, ¯ u, ¯ u ¯ x , ¯ u ¯ y , ¯ u ¯ t , ¯ f , ¯ g ).Coefficients of the prolonged vector field (4), namely the infinitesimal gen-erators for the equivalence group can be evaluated by the very well-known pro-longation formula (for detailed information, theorems and detailed applicationssee [5,17]). For more details we refer the reader the papers [18–20] and referencestherein which are concerned with equivalence transformations. ζ = D x ( η ) − u x D x ( ξ ) − u y D x ( ξ ) − u t D x ( ξ ) ,ζ = D y ( η ) − u x D y ( ξ ) − u y D y ( ξ ) − u t D y ( ξ ) ,ζ = D t ( η ) − u x D t ( ξ ) − u y D t ( ξ ) − u t D t ( ξ ) (7)where D x , D y , D t denote the total derivatives with respect to their parameters: D i = ∂∂x i + u x i ∂∂u . And µ j = ˜ D j ( µ ) − X i =1 f i ˜ D j ( ξ i ) − f u ˜ D j ( η ) − X i =1 f u i ˜ D j ( ζ i ) ,µ j = ˜ D j ( µ ) − X i =1 g i ˜ D j ( ξ i ) − g u ˜ D j ( η ) − X i =1 g u i ˜ D j ( ζ i ) ,ν j = ˜ D u j ( µ ) − X i =1 f u i ˜ D u j ( ζ i ) ,ν j = ˜ D u j ( µ ) − X i =1 g u i ˜ D u j ( ζ i ) (8)4here ˜ D j = ∂∂x j and ˜ D u j = ∂∂u j .These expressions do not impose any restriction on functional dependenciesof the smooth functions f and g in the main equation (1). If some variablesdo not appear in the coordinate cover of the extended manifold (3), due to aparticular structure of the given differential equation, that might entail somerestrictions on the extended vector field components because the correspondingcomponents must then be set to zero. Note that in equation (1) the free pa-rameters f and g do not depend on u t . Thus their corresponding componentsin (4) must vanish: µ = µ = 0 . (9) Theorem 2.
A nonlinear (2+1) dimensional diffusion equation can be mappedonto a linear equation by a point equivalence transformation, if and only if it isin the following form: ¯ x = φ ( x, y, t, u ) and/or ¯ y = ψ ( x, y, t, u ) where φ, ψ ∈ C and ∂φ∂u = 0 , ∂ψ∂u = 0 .Proof. To generate the transformations of the equivalence group for the diffusionequation given by (1), we shall apply the restrictions (9) to the given formulas(7) and (8). Then we have ξ = ξ ( x, y, t, u ) , ξ = ξ ( x, y, t, u ) , ξ = ξ ( t ) , η = η ( x, y, t, u ) ,ζ = η x + ( η u + ξ x ) u x + ξ u ( u x ) + ξ x u y + ξ u u x u y ,ζ = η y + ( η u + ξ y ) u y + ξ y u x + ξ u u x u y + ξ u ( u y ) ,ζ = η t + ( η u + ˙ ξ ) u t + ξ t u x + ξ u u x u t + ξ t u y + ξ u u y u t ,µ = ( η u + ˙ ξ − ξ x + ξ u u y ) f − ( ξ y + ξ u u y ) g + γu y + κ ,µ = ( η u + ˙ ξ − ξ y + ξ u u x ) g − ( ξ x + ξ u u x ) f − γu x + κ (10)and η u = ξ x + ξ y − ˙ ξ + s ( t ) (11)where s ( t ) is an arbitrary continuous function, γ, κ and κ depend on theindependent and dependent variables and satisfy the following relations ξ t = − γ y + κ u , ξ t = γ x + κ u , η t = κ x + κ y . Equation (11) points out that η can not depend on u nonlinearly, unless ξ and/or ξ depend on u . Thus to have any transformation between linear andnonlinear equations ξ and/or ξ must involve the dependent variable u whichmeant to be the transformed variables ¯ x and /or ¯ y involve u : ξ = ξ ( x, y, t, u ) , and/or ξ = ξ ( x, y, t, u ) , ∂ξ i ∂u = 0 , i = 1 , . Thus transformed variables are obtained via the solution of the set of ordinarydifferential equations (5) as¯ x = φ ( x, y, t, u ) and/or ¯ y = ψ ( x, y, t, u ) . Theorem 3. (2+1) dimensional diffusion equation does not admit the followingtype of transformation: ¯ x = ψ ( x, y, t ) , ¯ y = ψ ( x, y, t ) , ¯ t = ψ ( t ) , ¯ u = ψ ( x, y, t, u ) where ψ ( x, y, t, u ) is nonlinear in u .Proof. One can easily see from the equation (11), unless ξ or ξ depend on u , η can not involve u . Thus ¯ u can not be nonlinear in u when ¯ x or ¯ y does notinvolve u . Corollary 1.
Any integrable transformation related to the generators; ξ = ξ ( x, y, t ) , ξ = ξ ( x, y, t ) , ξ = ξ ( t ) , η = h ( x, y, t ) u map a linear equationonto another linear equation with various different coefficient functions. Readers may see that equations (10) generate all set of admissible equivalencetransformations, related to the general (2+1)dimensional diffusion equation (1).The equivalence transformations for some particular members of the family of(1) can be evaluated from these by integrating the system of equations (5).
Theorem 2 also refers that the admissible equivalence transformations relatedto the diffusion equation (1) may involve some arbitrary functions. As a conse-quence of this result, in addition to the maps between single linear and nonlinearequations, one can say that transformations between a single linear equation anda class of nonlinear equations can also be constructed. That is to say we canset such transformations which map a single linear equation into a particularfamily of nonlinear equations or vice versa. f x + g y − u t = 0 ←→ ¯ f k ¯ x + ¯ g k ¯ y − ¯ u k ¯ t = 0where the subscript k denotes the class of equations.In this section we shall investigate some applications to that circumstanceand study such maps by considering the transformations involving some ar-bitrary differentiable functions. Hereby solutions of some particular family ofnonlinear equations can be obtained from an appropriate linear equation. ξ = m ( u ) , ξ = h ( u ) , ξ = η = 0 Here we shall examine a subgroup of the admissible equivalence groups generatedby the infinitesimal generators: ξ = m ( u ) , ξ = h ( u ) , ξ = η = 0 where m ( u )6nd h ( u ) are arbitrary differentiable continuous functions. Then the prolongedvector field (4) can be written via (10) as follows˜ V = m ( u ) ∂∂x + h ( u ) ∂∂y + ( h ′ ( u ) f − m ′ ( u ) g ) u y ∂∂f + ( m ′ ( u ) g − h ′ ( u ) f ) u x ∂∂g + · · · (12)By integrating the system of equations (5) under the initial conditions (6) wehave the following class of transformations for the subgroup¯ x = x − ǫm ( u ) , ¯ y = y − ǫh ( u ) , ¯ t = t, ¯ u = u, ¯ u x = u x − ǫ ( u x m ′ ( u ) + u y h ′ ( u )) , ¯ u y = u y − ǫ ( u x m ′ ( u ) + u y h ′ ( u )) , ¯ u t = u t − ǫ ( u x m ′ ( u ) + u y h ′ ( u )) , ¯ f = (1 − ǫu x m ′ ( u )) f − ǫu y m ′ ( u ) g − ǫ ( u x m ′ ( u ) + u y h ′ ( u )) , ¯ g = (1 − ǫu y h ′ ( u )) g − ǫu x h ′ ( u ) f − ǫ ( u x m ′ ( u ) + u y h ′ ( u )) . (13)Note that here ǫ is the group parameter. By substituting u x and u y in terms ofthe transformed variables, ¯ f and ¯ g can now be written as¯ f = (1 + ǫ ¯ u ¯ y h ′ (¯ u )) ˜ f − ǫ ¯ u ¯ y m ′ (¯ u )˜ g, ¯ g = (1 + ǫ ¯ u ¯ x m ′ (¯ u )) ˜ g − ǫ ¯ u ¯ x h ′ (¯ u ) ˜ f (14)where ˜ f (¯ x, ¯ y, ¯ t, ¯ u, ¯ u ¯ x , ¯ u ¯ y ) and ˜ g (¯ x, ¯ y, ¯ t, ¯ u, ¯ u ¯ x , ¯ u ¯ y ) represent f and g in terms ofthe transformed variables.It is clear that for every different choice of m ( u ) , h ( u ) and linear functions f and g in the dependent variable u and its derivatives, the transformations(14) map a single linear differential equation f x + g y − u t = 0 onto a class ofnonlinear equations of the form ¯ f k ¯ x + ¯ g k ¯ y − ¯ u k ¯ t = 0, and a solution of the linearequation φ ( x, y, t, u ) = 0generates solutions to the corresponding nonlinear equations as φ k (¯ x + ǫm (¯ u ) , ¯ y + ǫh (¯ u ) , ¯ t, ¯ u ) = 0 . Example 1:
As a particular example, we search the nonlinear diffusionequations of the class (1) that can be mapped onto the very well known heatequation. The (2+1) dimensional heat equation can be represented as a memberof (1) by choosing f = u x , g = u y ; u xx + u yy = u t . (15)The transformed form of f = u x and g = u y can be obtained from (14) as¯ f = (1 + ǫh ′ (¯ u )¯ u ¯ y )¯ u ¯ x − ǫm ′ (¯ u )¯ u y ǫh ′ (¯ u )¯ u ¯ y + ǫm ′ (¯ u )¯ u ¯ x , ¯ g = (1 + ǫm ′ (¯ u )¯ u ¯ x )¯ u ¯ y − ǫh ′ (¯ u )¯ u x ǫh ′ (¯ u )¯ u ¯ y + ǫm ′ (¯ u )¯ u ¯ x which generate the following class of nonlinear equations A ¯ u ¯ x ¯ x + B ¯ u ¯ x ¯ y + C ¯ u ¯ y ¯ y + D = E ¯ u ¯ t (16)7here the coefficients are A = 1 + 2 ǫh ′ (¯ u )¯ u ¯ y + ǫ (cid:0) h ′ (¯ u ) + m ′ (¯ u ) (cid:1) ¯ u y B = − ǫ (cid:2) m ′ (¯ u )¯ u ¯ y + h ′ (¯ u )¯ u ¯ x + ǫ (cid:0) h ′ (¯ u ) + m ′ (¯ u ) (cid:1) ¯ u ¯ y ¯ u ¯ x (cid:3) C = 1 + 2 ǫm ′ (¯ u )¯ u ¯ x + ǫ (cid:0) h ′ (¯ u ) + m ′ (¯ u ) (cid:1) ¯ u x D = − ǫ ( m ′′ (¯ u )¯ u ¯ x + h ′′ (¯ u )¯ u ¯ y ) (cid:0) ¯ u x + ¯ u y (cid:1) E = (1 + ǫ ( m ′ (¯ u )¯ u ¯ x + h ′ (¯ u )¯ u ¯ y )) (17)It is obvious that via the transformations of (13), we can construct various mapsbetween the single heat equation (15) and the nonlinear equations of the classof (16). And any solution of the heat equation (15) would generate solutions ofnonlinear equations.For the reader to see more clearly how a solution to nonlinear problemscan be constructed via equivalence transformations we consider the followingproblem. Example 2:
A subclass of nonlinear diffusion equations (16) can be consideredby taking m ( u ) = u n , h ( u ) = u r in the previous example, where n, r ∈ R . h ǫru r − u y + + ǫ ( r u r − + n u n − ) u y i u xx − ǫ (cid:2) nu n − u y + ru r − u x + ǫ ( r u r − + n u n − ) u x u y i u xy + (cid:2) ǫnu n − u x + ǫ ( r u r − + n u n − ) u x i u yy − ǫ ( u x + u y )( r ( r − u r − u y + n ( n − u n − u x ) = (1 + ǫ ( nu n − u x + ru r − u y )) . (18)Here ¯( ∗ )’s are omitted for the simplicity. And an implicit solution for (18) canbe written as u − ¯ F sin ( µ ( x + ǫu n )) sin ( η ( y + ǫu r ) e − ( µ + η ) t = 0from a solution like u = F sin ( µx ) sin ( ηy ) e − ( µ + η ) t of the heat equation (15) byapplying the equivalence transformations; x = ¯ x + ǫ ¯ u n , y = ¯ y + ǫ ¯ u r , t = ¯ t, u = ¯ u .Here F and ¯ F are some constants. Example 3 : For the reader to follow the procedure better and check the cal-culations, let us consider another particular member of the nonlinear equation(16) by taking m ( u ) = u and h ( u ) = 0, where we could simply write(1 + ¯ u ¯ u y )¯ u ¯ x ¯ x − u ¯ u ¯ x )¯ u ¯ u ¯ y ¯ u ¯ x ¯ y + (1 + ¯ u ¯ u ¯ x ) ¯ u ¯ y ¯ y − (¯ u y + ¯ u x )¯ u ¯ x = (1 + ¯ u ¯ u ¯ x ) ¯ u ¯ t . (19)The reader can easily see, a much more easier function u = x + y + 2 t whichsatisfies the linear heat equation would generate¯ u = 1 ∓ p − x + ¯ y + 2¯ t ))as a solution to the nonlinear equation (19) under the transformations x =¯ x + ¯ u , y = ¯ y, t = ¯ t, u = ¯ u . Example 4:
As another example by choosing m ( u ) = sin u, h ( u ) = cos u the8onlinear equation would become(1 + ¯ u y − u ¯ y sin ¯ u )¯ u ¯ x ¯ x + (1 + ¯ u x + 2¯ u ¯ x cos ¯ u )¯ u ¯ y ¯ y − u ¯ y cos ¯ u − ¯ u ¯ x sin ¯ u + ¯ u ¯ x ¯ u ¯ y )¯ u ¯ x ¯ y + (¯ u ¯ y cos ¯ u + ¯ u ¯ x sin ¯ u )(¯ u y + ¯ u x ) − (1 − ¯ u ¯ y sin ¯ u + ¯ u ¯ x cos ¯ u ) ¯ u ¯ t = 0 . (20)can be transformed into the heat equation (15) by the equivalence transforma-tions: ¯ x = x − sinu, ¯ y = y − cosu, ¯ u = u . Any solution of the linear heatequation would generate a solution of the equation. ξ = m ( u, y ) , ξ = ξ = η = 0 Similarly to the previous subsection, transformations related to a subgroup gen-erated by the infinitesimal generators ξ = m ( u, y ) , ξ = ξ = η = 0 are con-sidered. The corresponding equivalence transformations of the subgroup areexplicitly determined as:¯ x = x − ǫm ( u, y ) , ¯ y = y, ¯ t = t, ¯ u = u, ¯ u x = u x − ǫm u ( u, y ) u x , ¯ u y = u y + ǫm y ( u, y ) u x − ǫm u ( u, y ) u x , ¯ u t = u t − ǫm u ( u, y ) u x , ¯ f = f − ǫ ( m y ( u, y ) + m u ( u, y ) u y ) g − ǫm u ( u, y ) u x , ¯ g = g − ǫm u ( u, y ) u x where subscripts indicate partial differentiation with respect to the correspond-ing variable. Example 5:
For the simplicity, let us again seek the nonlinear diffusion equa-tions can be mapped onto the linear heat equation (15). For f = u x and g = u y ,¯ f and ¯ g turn to be:¯ f = ¯ u ¯ x − ǫ ( m ¯ y + m ¯ u ¯ u ¯ y ) (¯ u ¯ y − ǫm ¯ y ¯ u ¯ x )1 + ǫm ¯ u ¯ u ¯ x , ¯ g = ¯ u ¯ y − ǫm ¯ y ¯ u ¯ x where m = m (¯ u, ¯ y ). Thus the following class are the class of nonlinear equationsthat mapped onto the heat equation: A ¯ u ¯ x ¯ x + B ¯ u ¯ x ¯ y + C ¯ u ¯ y ¯ y + D = E ¯ u ¯ t −→ u xx + u yy = u t where the coefficients are A = 1 + ǫ m y + 3 ǫ m ¯ u m ¯ y ¯ u ¯ y + 2 ǫ m u ¯ u y B = ǫ (cid:2) ǫ m u ¯ u x ( m ¯ u ¯ u ¯ y + m ¯ y ) − ǫm u ¯ u ¯ x ¯ u ¯ y − ǫm ¯ u m ¯ y ¯ u ¯ x − m ¯ u ¯ u ¯ y − m ¯ y (cid:3) C = 1 + ǫm ¯ u ¯ u ¯ x − ǫ m u ¯ u x − ǫ m u ¯ u x D = ǫ (cid:2) ǫ m ¯ u m ¯ y m ¯ u ¯ y − ǫ m u m ¯ y ¯ y − m ¯ u ¯ u − ǫ m y m ¯ u ¯ u (cid:3) ¯ u x + ǫ [ − m ¯ y ¯ y m ¯ u +2 m ¯ u ¯ y m ¯ y − m ¯ u ¯ y m ¯ u ¯ u ¯ y + 3 m ¯ u ¯ u m ¯ y ¯ u ¯ y ] ¯ u x − ǫm ¯ u ¯ u ¯ u ¯ x ¯ u y − ǫm ¯ u ¯ y ¯ u ¯ x ¯ u ¯ y − ǫm ¯ y ¯ y ¯ u ¯ x E = (1 + ǫm ¯ u ¯ u ¯ x ) (21)Any solution φ ( x, y, t, u ) = 0 of the heat equation (15) generates hereby thesolution φ (¯ x + ǫm (¯ u, ¯ y ) , ¯ y, ¯ t, ¯ u ) = 0 for the nonlinear equation (16) with thecoefficients (21). 9 .3 On the subgroup: ξ = m ( u, x ) , ξ = ξ = η = 0 Completely similar to the previous subsections, the infinitesimal generators onthe vector field of (4) yield the transformations¯ x = x, ¯ y = y − ǫm ( u, x ) , ¯ t = t, ¯ u = u, ¯ u x = u x + ǫm x ( u, x ) u y − ǫm u ( u, x ) u y , ¯ u y = u y − ǫm u ( u, x ) u y , ¯ u t = u t − ǫm u ( u, x ) u y , ¯ f = f − ǫm u ( u, x ) u y , ¯ g = g − ǫ ( m x ( u, x ) + ǫm u ( u, x ) u x ) g − ǫm u ( u, x ) u y . (22)As the procedure is already given in detail in the previous subsections withmany examples, here we will only consider the following example which is alittle different than the previous ones. Example 6:
Even though the first order PDE, u x + u y − u t = 0is not a diffusion equation, we may still examine it under the equivalence trans-formations generated for the diffusion equation (1) by taking f = g = u . Thefollowing class of nonlinear equations are mapped onto the given constant coef-ficient PDE under the transformation group (22):¯ u ¯ x + (1 − ǫm ¯ x (¯ u, ¯ x ))¯ u ¯ y − ¯ u ¯ t = 0 . (23)And a solution to the equation (23) can be written as¯ u − ψ (¯ t + ¯ x, ¯ y + ǫm (¯ u, ¯ x ) − ¯ x ) = 0from the general solution of the linear equation u = ψ ( t + x, y − x ). Here weshould warn the reader that even though the solution of the linear equation isits general solution, the transformed solution is not the general solution to thenonlinear equation. ξ = ξ = ξ = 0 , η = m ( x, y, t ) As the last case, here we shall consider the equivalence transformations of linearor nonlinear equations which do not map between each other, but map one intoanother with different coefficient functions. We can construct many of suchtransformations, but as an example here will investigate one subgroup by takingthe infinitesimal generators drive the following equivalence transformations¯ x = x, ¯ y = y, ¯ t = t, ¯ u = u + ǫm ( x, y, t ) , ¯ u x = u x + ǫm x ( x, y, t ) , ¯ u y = u y + ǫm y ( x, y, t ) , ¯ u t = u t + ǫm t ( x, y, t ) , ¯ f = f + ǫ Z m t ( x, y, t ) dx, ¯ g = g. Under such transformations f x + g y − u t = 0 is mapped onto (cid:20) f + ǫ Z m t ( x, y, t ) dx (cid:21) x + g y − u t = ǫm t ( x, y, t ) . Differential Invariants of Lie groups of continuous transformations play impor-tant role in mathematical modelling, differential geometry and nonlinear fieldequations. In recent years, differential invariants admitting equivalence trans-formations have been mostly studied for constructing maps between linear andnonlinear differential equations. The reader may look at [24] and [21] for theapplication of differential invariants to the linearization problem for nonlinearwave equation. The linearization problem via differential invariants for one di-mensional diffusion equation was investigated in [25–28]. Recently the problemis studied for third order evolution equation by Tsaousi et al. in [29]. Like theequivalence transformation for the general class of diffusion equation (1) hasnot been studied yet its differential invariants have not been considered in anyresearch as well.One can understand that determining the differential invariants for the com-plete group of equivalence transformations for the equation (1) is almost impos-sible. Thus we will here investigate the differential invariants admitting thespecial subgroups of equivalence transformations represented in section 3.1 anddiscuss about the results. ξ = m ( u ) , ξ = h ( u ) The subgroup which are examined in Section 3.1 for the class of (2+1) dimen-sional diffusion equation (1) is infinite dimensional and spanned by the vectorfields V m = m ( u ) ∂∂x + m ′ ( u ) (cid:20) u x ∂∂u x + u x u y ∂∂u y + u x u t ∂∂u t − u y g ∂∂f + u x g ∂∂g (cid:21) ,V h = h ( u ) ∂∂y + h ′ ( u ) (cid:20) u x u y ∂∂u x + u y ∂∂u y + u y u t ∂∂u t + u y f ∂∂f − u x f ∂∂g (cid:21) (24)where [ V m , V h ] = V m ( V h ) − V h ( V m ) = 0 . Definition 3.
A function J = J ( x, y, t, u, u x , u y , u t , f, g ) is called the invariant of order zero of the (2+1) dimensional diffusion equation (1) , if it is invariant under the equivalence groups V m and V h given by (24) . V m ( J ) = 0 and V h ( J ) = 0 yield the invariant functionof order zero to be J = J ( t, u, ξ , ξ , ξ )where ξ = u y u x , ξ = u t u x , ξ = f ξ + g. (25)One can easily see that these differential invariants are consistent with the results(13) obtained by the direct integration method u y u x = ¯ u ¯ y ¯ u ¯ x , u t u x = ¯ u ¯ t ¯ u ¯ x , f u x u y + g = ¯ f ¯ u x ¯ u y + ¯ g. Definition 4.
A function J ( x, y, t, u, u x , u y , u t , f, g, f x , f y , f t , f u , f u x , f u y , g x , g y , g t , g u , g u x , g u y ) is called the invariant of order one for the diffusion equation (1) , if it is invariantunder the equivalence groups related to the prolonged vector fields ˜ V m and ˜ V h . Computation of the differential invariants of order one for this subgrouptoo complicated as they involve two free functions. For the simplicity, here wewill examine the procedure finding the first order invariants by considering thesubgroup in which h ( u ) = 0.In addition to V m given by (24), substituting the additional components ofthe vector field (4) which determined by (8) yields the prolongation vector ˜ V m to be e V m = − (cid:2) m ′ ( u ) ( f x + u y g u ) + m ′′ ( u ) (cid:0) u x f u x + u x u y f u y (cid:1)(cid:3) ∂∂f u − (cid:2) m ′ ( u ) ( g x − u x g u ) + m ′′ ( u ) (cid:0) u x g u x + u x u y g u y − u x g (cid:1)(cid:3) ∂∂g u − m ′ ( u ) u y g x ∂∂f x − m ′ ( u ) u y g y ∂∂f y − m ′ ( u ) u y g t ∂∂f t + m ′ ( u ) u x g x ∂∂g x + m ′ ( u ) u x g y ∂∂g y + m ′ ( u ) u x g t ∂∂g t − m ′ ( u ) (cid:0) u x f u x + u y f u y + u y g u x (cid:1) ∂∂f u x − m ′ ( u ) (cid:0) u x f u y + u y g u y + g (cid:1) ∂∂f u y − m ′ ( u ) (cid:0) u x g u x + u y g u y − g (cid:1) ∂∂g u x . (26)Since m ( u ) is an arbitrary function, we apply the invariant test V m ( J ) = 0 , e V m ( J ) = 0and obtain the invariant function of order one as a function depending on 15invariants J ( y, t, u, ξ i ) , i = 1 , , ... ξ = u y u x , ξ = u t u x , ξ = gu x , ξ = f + ξ g,ξ = g x + 1 ξ f x , ξ = g y g x ξ , ξ = f y − g y g x f x ,ξ = g t g x ξ , ξ = f t − g t g x f x , ξ = g u y ,ξ = − ( ξ ξ + ξ ) f x + f u y g x ξ , ξ = ( ξ − ξ g u y ) f x + ξ g x g u x ,ξ = (cid:2)(cid:0) g u y f x − g u x g x (cid:1) f x + (cid:0) f u x g x − f u y f x (cid:1) g x (cid:3) ξ . (27)A symbolic software is used to compute the differential invariants. Differentialinvariants related to the prolonged vector field ˜ V h can also be obtained butbecause the procedure is the same we neglect that part. In this work we considered the equivalence transformations for a general (2+1)dimensional diffusion equation with no restriction on the functional dependen-cies of free functions. The goal was to construct the most general infinitesimalgenerators for the transformations associated equivalence group and investigatethe structure of admissible transformations between linear and nonlinear equa-tions. We showed that such transformations were only possible when the trans-formed independent variables involve the dependent variable. Similar analysiscan either be applied to some smaller classes directly or the results obtainedhere can be applied by appropriate restrictions.Second, we considered some subgroups and chose some particular trans-formations to generate maps between linear and nonlinear equations and wedetermined the class of nonlinear diffusion equations that can be mapped ontothe linear heat equation. We have not interested in algebraic structure and theclassification problem. Classification for some members of the diffusion equationcan be a subject of another study.Third, in the last section we investigated the differential invariants of orderzero and of order one for a subgroup which was examined in the previous sec-tion. And were able to show that the zeroth order differential invariants werecompatible with the results we obtained by direct integration method. The de-termination of the differential invariants can easily be extended by taking someother particular subgroups by running the similar calculations.
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