Symmetry analysis of the cylindrical Laplace equation

Transcription

1 Symmetry analysis of the cylindrical Laplace equation Mehdi Nadjafikhah Seyed-Reza Hejazi Abstract. The symmetry analysis for Laplace equation on cylinder is considered. Symmetry algebra the structure of the Lie algebra of the symmetries and some related topics such as invariant solutions one-parameter subgroups one dimensional optimal system and differential invariants are given. M.S.C. 2000: 49J30 49K30 55S45. Key words: Laplace equation invariant solution optimal system. 1 Introduction Pierre-Simon marquis de Laplace March March was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. He summarized and extended the work of his predecessors in his five volume Mcanique Cleste Celestial Mechanics This seminal work translated the geometric study of classical mechanics used by Isaac Newton to one based on calculus opening up a broader range of problems. He formulated Laplace s equation and invented the Laplace transform which appears in many branches of mathematical physics a field that he took a leading role in forming. The Laplacian differential operator widely used in applied mathematics is also named after him. In mathematics and physics the Laplace operator or Laplacian denoted by or 2 and named after Pierre-Simon de Laplace is a differential operator specifically an important case of an elliptic operator with many applications. In physics it is used in modeling of wave propagation the wave equation is an important second-order linear partial differential equation that describes the propagation of a variety of waves such as sound waves light waves and water waves. It arises in fields such as acoustics electromagnetics and fluid dynamics. Historically the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d Alembert Leonhard Euler Daniel Bernoulli and Joseph-Louis Lagrange. an important partial differential equation which describes the distribution of heat or variation in temperature in a given region over time. For a function of three spatial variables x y z and one time Balkan Journal of Geometry and Its Applications Vol.14 No pp c Balkan Society of Geometers Geometry Balkan Press 2009.

2 54 Mehdi Nadjafikhah Seyed-Reza Hejazi variable t; heat flow forming the Helmholtz equation. It is central in electrostatics and fluid mechanics anchoring in Laplace s equation and Poisson s equation. In quantum mechanics it represents the kinetic energy term of the Schrödinger equation. The kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity. Having gained this energy during its acceleration the body maintains this kinetic energy unless its speed changes. Negative work of the same magnitude would be required to return the body to a state of rest from that velocity. In mathematics functions with vanishing Laplacian are called harmonic functions; the Laplacian is at the core of Hodge theory and the results of de Rham cohomology [9]. The aim is to analysis the point symmetry structure of the Laplace equation on cylinder i.e. cylindrical Laplace equation which is u = 1 r r r u r + 1 r 2 2 u θ u z 2 = 0 where u is a smooth function of r θ z. The symmetry algebra of 1.1 is determined in the next section and some results yield from the structure of the Lie algebra of symmetries are given. 2 Lie symmetries of the equation The method of determining the classical symmetries of a partial differential equation is standard which is described in [1 4 5] and [8]. To obtain the symmetry algebra of the 1.1 we take an infinitesimal generator of symmetry algebra of the form X = ξ 1 r θ z u r + ξ 2r θ z u θ + ξ 3r θ z u z + ηr θ z u u. Using the invariance condition i.e. applying the second prolongation X 2 to 1.1 the following system of 18 determining equations yields: ξ 2u = 0 ξ 3u = 0 η uu = 0 ξ 3rzz = 0 ξ 3θzz = 0 ξ 3zzz = 0 ξ 3rr + ξ 3zz = 0 ξ 3θ + r 2 ξ 2z = 0 ξ 3θ rξ 3rθ = 0 ξ 3rz + 2η ru = 0 ξ 3θz + 2η θu = 0 ξ 3zz + 2η zu = 0 ξ 1 + rξ 2θ ξ 3z = 0 ξ 2θ + rξ 2rθ ξ 3rz = 0 ξ 2θθ rξ 2r ξ 3θz = 0 ξ 3θθ + rξ 3r + rξ 3zz = 0 ξ 3θz + r2ξ 2r rξ 2rr = 0 η θθ + rη r + rη rr + rη zz = 0. The solution of the above system gives the following infinitesimals X 1 = θ X 2 = z X 3 = u u X 4 = r r + z z X 5 = rz r z2 r 2 z 1 2 zu u X 6 = sin θ r + 1 r cos θ θ X 7 = cos θ r 1 r sin θ θ

3 Symmetry analysis of the cylindrical Laplace equation 55 X 8 = z sin θ r + 1 r z cos θ θ r sin θ z X 9 = z cos θ r 1 r z sin θ θ r cos θ z X 10 = 1 2 z2 r 2 sin θ r + 1 2r z2 + r 2 cos θ θ rz sin θ z + 1 ru sin θ 2 u X 11 = 1 2 z2 r 2 cos θ r 1 2r z2 + r 2 sin θ θ rz cos θ z + 1 ru cos θ 2 u. The commutation relations of the 11 dimensional Lie algebra g spanned by the vector fields X 1 X 11 are shown in following table. X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X X 7 X 6 X 9 X 8 X 11 X 10 X X 2 1 X X X 6 X 7 X 8 X 9 X X 4 0 X X 5 X 7 X X 10 X X 5 0 X3 X4 0 X5 0 X8 X9 X10 X X 6 X X 7 X X 2 0 X 2 3 X 4 X 1 X 7 X X 7 X X 2 X X 4 X 8 X 9 X X 10 X X 1 X 5 0 X 9 X 8 X X 11 0 X 2 X X 5 X 10 X 11 X 8 0 X X3 + X4 X1 X X 11 X 10 X 9 0 X 11 0 X 1 1 X X 4 0 X Differential invariants Suppose that G is a transformation group. It is well known that a smooth real differential function I : J n R where J n is the corresponding n th jet space is a differential invariant for G if and only if for all X g its n th prolongation annihilates I i.e. X n I = 0. Here we are going to find the differential invariants of order 0 1 and 2 and some of their results. For more details and some practical examples see [3] and [8]. The symmetry group of the cylindrical Laplace equation has not any nonconstant ordinary differential invariants i.e. it does not have any differential invariants of order zero thus the orbit of generic points the points which lies in the orbit of maximal dimension has dimension four and its isotropy subgroup is seven dimensional. The first prolonged action of the symmetry group has one differential invariant which is a function of u u r u θ u z so the action has a generic orbit of dimension six and its isotropy subgroup is five dimensional. Finally the second prolonged action has another differential invariant which is a function of u u r u θ u z u rr u rθ u rz u θθ u θz u zz consequently the dimension of the generic orbit is equal to the dimension of the group; thus the second prolonged action is transitive and has a discrete isotropy subgroup. According to the relations between above subjects and [4] we can obtain the number of functionally independent differential invariants up to order n.

4 56 Mehdi Nadjafikhah Seyed-Reza Hejazi Theorem 2.1. The number of functionally independent differential invariants up to order n 3 for the symmetry group of cylindrical Laplace equation is 3+n n 8 and the number of strictly differential invariants which occur in each prolongation is 2+n 3+n n n 1. Proof. According to the above discussion for n 3 the action is transitive so the number of independent differential invariants i n is 3 + n 3 + n i n = = 8. n n If j n is the number of strictly differential invariants then 3 + n 3 + n n j n = = n n 1 n 2.2 Reduction of the equation 2 + n. n 1 The Eq. 1.1 maybe regarded as a submanifold M of the jet space J 2 R 3 R. Doing as section 3.1 of [5] and find in a sense the most general group-invariant solutions to the Eq Theorem 2.2. The one-parameter groups g i t : M M generated by the X i i = 1 11 are given in the following table: g 1 t : r θ z u r θ + t z u g 2 t : r θ z u r θ z + t u g 3 t : r θ z u r θ z ue t g 4 t : r θ z u re t θ ze t u g 5 t : r θ z u rzt + r θ t 2 z2 r zut + u g 6 t : r θ z u t sin θ + r t r cos θ + θ z u g 7 t : r θ z u t cos θ + r t r sin θ + θ z u g 8 t : r θ z u tz sin θ + r t r z cos θ + θ rt sin θ + z u g 9 t : r θ z u tz cos θ + r t r z sin θ + θ rt cos θ + z u t g 10 t : r θ z u 2 z2 r 2 sin θ t z 2 2 r + r cos θ + θ g 11 t : r θ z u rzt sin θ + z 1 2 rut sin θ + u t 2 z2 r 2 cos θ t z 2 2 r + r sin θ + θ rzt cos θ + z 1 2 rut cos θ + u

5 Symmetry analysis of the cylindrical Laplace equation 57 where entries give the transformed point exptx i r θ z u = r θ z ũ. The transformations g 1 and g 2 demonstrate the radius- and space-invariance and radius- and angle-invariance solutions of the equation. The one-parameter g 3 reflects the linearity of the cylindrical laplace equation and the well-known scaling symmetry turns up in g 4. The most general one-parameter group of symmetries is obtained by considering a general linear combination of the given vector fields; the explicit formula for the group transformations is very complicated but alternatively we can use theorem 2.52 in [4] and represent an arbitrary group transformation g as the composition of transformations in theorem 2.2. Theorem 2.3. Let i = 1 11 and t R. If u = Ur θ z is a solution of Eq. 1.1 so are the functions u i r θ z = Ur θ z where u 1 = Ur θ + t z u 2 = Ur θ z + t u 3 = e t Ur θ z u 4 = Ue t r θ e t z u 5 = zt Urzt + r θ t 2 z2 r 2 u 6 = Ut sin θ + r t r cos θ + θ z u7 = U t cos θ + r t r cos θ + θ z u 8 = U tz sin θ + r t r z cos θ + θ rt sin θ + z u 9 = U tz cos θ + r t r z sin θ + θ rt cos θ + z u 10 = 1 rt 2 sin θ t U 2 z2 r 2 t sin θ 2r z2 + r 2 cos θ + θ z rzt sin θ u 11 = 1 rt 2 cos θ U t 2 z2 r 2 cos θ t 2r z2 + r 2 sin θ + θ z rzt cos θ. For example if ur θ z = 1 be a constant solution of Eq. 1.1 we conclude that the trivial functions g i t 1 = 1 for i = g 3 t 1 = e t and 2a g 5 t 1 = 2 + zt and by applying g 5 b g 3 ln b on u = 1 we conclude the linear solution u = az + b. The two nontrivial solutions for Eq. 1.1 are: g 10 t 1 = 2 rt sin θ g 11 t 1 = 2 rt cos θ. 8+2c Now by summing g 1 g 5 d g 3 ln d 1 g 10 a 1 and g 11 b 1 we conclude the following result: Corollary 2.4. The function ur θ z = ar sin θ + br cos θ + dz + c is a solution of Eq. 1.1 where a b c and d are arbitrary constants. 2.3 Some invariant solutions The first advantage of symmetry group method is to construct new solutions from known solutions. Neither the first advantage nor the second will be investigated here

6 58 Mehdi Nadjafikhah Seyed-Reza Hejazi but symmetry group method will be applied to the Eq. 1.1 to be connected directly to some order differential equations. To do this a particular linear combinations of infinitesimals are considered and their corresponding invariants are determined. The cylindrical Laplace equation expressed in the coordinates r θ z so to reduce this equation is to search for its form in specific coordinates. Those coordinates will be constructed by searching for independent invariants y v corresponding to an infinitesimal generator. so using the chain rule the expression of the equation in the new coordinate allows us to the reduced equation. Here we will obtain some invariant solutions with respect to symmetries. 1. First consider X 6. To determine the independent invariants I we need to solve the first order partial differential equation X 6 {Ir θ z u} = 0 that is sin θ I r + cos θ I r θ + 0 I z + 0 I u = 0 which is a homogeneous first order PDE. Thus we solve the associated characteristic ordinary differential equations dr cos θdθ = = dz sin θ r 0 = du 0. Hence we obtain two functionally independent invariants y = z and v = u. r sin θ If we treat v as a function of y we can compute formulae for the derivatives of u with respect to x and t in terms of y v and the derivatives of v with respect to y along with a single parametric variable which we designate to be t so that x will be the corresponding principal variable. Using the chain rule we find that if u = v = vy = vz/r sin θ then u r = v y y r u rr = y rr v y + v yy y r u θθ = y θθ v y + v yy y θ u zz = y zz v y + v yy y z. Substituting u r u rr u θθ and u zz in 1.1 we obtain an ODE with solution u = C 1 arctan + C 2 which is called an invariant solution for Eq z r sin θ 2. Similarly the invariants of X 7 are y = z and v = u and the solution of r cos θ reduced equation invariant solution by substituting and 2.4 is u = C 1 arctan + C 2. z r cos θ 3. The invariants of X 8 are r2 + z 2 and v = u. The invariant solution constructed r sin θ r with and 2.4 is u = C 1 arctanh 2 +z 2 r sin θ + C 2 obtained from ordinary differential equations

7 Symmetry analysis of the cylindrical Laplace equation 59 dr z sin θ = rdθ a cos θ = dz r sin θ = du 0. The above mentioned method in sections 1 and 2 can be followed for X 9 with invariants r2 + z 2 r r sin θ and v = u and invariant solution u = C 1 arctanh 2 +z 2 r cos θ + C 2. We can proceed for other X i s as above. 2.4 The Lie algebra of symmetries In this section we determine the structure of full symmetry Lie algebra g of Eq Theorem 2.5. The full symmetry Lie algebra g of Eq. 1.1 has the following semidirect decomposition: 2.5 g = R g 1 where g 1 is a simple Lie algebra. Proof: The center z of g is Span R {X 3 }. Therefore the quotient algebra g 1 = g/z is Span R {Y 1 Y 10 } where Y i := X i + z for i = The commutator table of this quotient algebra is given in following table: Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 9 Y 10 Y Y 7 Y 5 Y 8 Y 7 Y 10 Y 9 Y Y 2 Y Y 5 Y 6 Y 7 Y 8 Y 3 0 Y 2 0 Y 4 Y 5 Y Y 9 Y 10 Y 4 0 Y 3 Y 4 0 Y 7 Y 8 Y 9 Y Y 5 Y 7 0 Y 5 Y Y 2 0 Y 3 Y 1 Y 6 Y 5 0 Y 6 Y Y 2 Y 1 Y 3 Y 7 Y 8 Y 5 0 Y 9 Y Y 1 Y 4 0 Y 8 Y 7 Y 6 0 Y 10 0 Y 2 Y Y 4 Y 9 Y 10 Y 7 Y 9 0 Y 3 Y 1 Y Y 10 Y 9 Y 8 Y 10 0 Y 1 Y 3 0 Y The Lie algebra g is non-solvable because g 1 = [g g] = Span R { X 1 X X 3 X 4 X 5 X 11 } g 2 = [g 1 g 1 ] = g 1. Similarly g 1 is semi-simple and non-solvable because g 1 1 = [g 1 g 1 ] = Span R {Y 1 Y 10 } = g 1. The Lie algebra g admits a Levi decomposition as the following semi-direct product g = r s where r = Span R {X 3 } is the radical of g the largest solvable ideal contained in g and

8 60 Mehdi Nadjafikhah Seyed-Reza Hejazi s = {X 1 X 2 12 X 3 X 4 X 5 X 11 }. r is a one-dimensional subalgebra of g therefore it is isomorphic to R; Thus the identity g = r s reduces to g = R g Optimal system of sub-algebras As is well known the theoretical Lie group method plays an important role in finding exact solutions and performing symmetry reductions of differential equations. Since any linear combination of infinitesimal generators is also an infinitesimal generator there are always infinitely many different symmetry subgroups for the differential equation. So a mean of determining which subgroups would give essentially different types of solutions is necessary and significant for a complete understanding of the invariant solutions. As any transformation in the full symmetry group maps a solution to another solution it is sufficient to find invariant solutions which are not related by transformations in the full symmetry group this has led to the concept of an optimal system [6]. The problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. For one-dimensional subalgebras this classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation. This problem is attacked by the naive approach of taking a general element in the Lie algebra and subjecting it to various adjoint transformations so as to simplify it as much as possible. The idea of using the adjoint representation for classifying group-invariant solutions is due to [5] and [6]. The adjoint action is given by the Lie series 2.6 AdexptY i Y j = Y j t[y i Y j ] + t2 2 [Y i [Y i Y j ]] where [Y i Y j ] is the commutator for the Lie algebra t is a parameter and i j = We can write the adjoint action for the Lie algebra g 1 and show that Theorem 2.6. A one dimensional optimal system of the g 1 is given by 1 Y 1 2 Y 3 3 Y 4 4 Y 7 5 Y 8 6 Y 9 7 ay 1 + by 3 8 ay 1 + by 4 9 ay 1 + by 7 10 ay 1 + by 8 11 ay 1 + by 9 12 ay 3 + by 7 13 ay 3 + by 8 14 ay 4 + by 9 where a b c d R are arbitrary numbers. 15 ay 7 + by 8 16 ay 1 + by 4 + cy 9 17 ay 1 + by 3 + cy 8 18 ay 1 + by 7 + cy 8 19 ay 3 + by 7 + cy 8 20 ay 1 + by 3 + cy 7 21 ay 1 + by 3 + cy 7 + dy 8 Proof. F t i : g 1 g 1 defined by Y AdexptY i Y is a linear map for i = The matrices M t i of F t i i = 1 10 with respect to basis {Y 1 Y 10 } are

9 Symmetry analysis of the cylindrical Laplace equation 61 M1 t = M2 t = M3 t = M4 t = M5 t = sin t 1 cos t t 1 cos t 1 t sin t cos t sin t sin t cos t cos t sin t sin t cos t t t 2 /2 t t t t 2 /2 0 t t 2 /2 0 t e t e t e t e t e t e t t t 2 / t t 0 t 2 / t 0 t 2 / t t t 2 / t t t 2 / t t 0 t 2 / t t 3 / t 2 /

10 62 Mehdi Nadjafikhah Seyed-Reza Hejazi M6 t = M7 t = M8 t = M9 t = M10 t = t t t 2 / t t t 2 / t 0 0 t 2 / cos t t sin t cos t 0 0 sin t cos t sin t 0 0 sin t 0 0 cos t sin t cos t sin t cos t cos t sin t cos t sin t cos t sin t sin t cos t sin t cos t sin t cos t t t 2 /2 0 0 t t t t 2 /2 0 t t 2 / t t t 2 / t t t t 2 / t t 2 / t

11 Symmetry analysis of the cylindrical Laplace equation 63 Let Y = 10 i=1 a i Y i then F t10 10 F t9 9 F t1 1 : Y cos t 7 cos t 8 + t 5 sin t t2 5 cos t 7 + t 6 sin t 7 t 9 sin t 8 2 t2 5 sin t 7 t 6 cos t 7 t 10 a t2 5 sin t 7 t 6 cos t 7 t 9 t t2 5 cos t 7 + t 6 sin t 7 t 2 10 sin t 8 + cos t 7 cos t 8 + t 5 sin t 8 t t2 5 cos t 7 + t 6 sin t 7 t 2 9 sin t 8 a 10 Y 1 + {[ t2 2t 2 4e s t 1 cos t 1 t 2 t 4 cos t 1 t t2 2t 2 4e t 3 1 sin t 1 t 2 t 4 sin t 1 + e t 3 1 sin t 1 + e t 3 1 cos t 1 t 6 cos t t2 2t 4 e t 3 1 cos t 1 t 2 cos t 1 sin t 7 cos t t2 2t 2 4e t 3 1 cos t 1 t 2 t 4 cos t 1 t 3 5 sin t t2 2t 2 4e t 3 1 cos t 1 t 2 t 4 cos t 1 t 5 t t2 2e t 3 1 sin t t2 2t 2 4e t 3 1 sin t 1 t 2 t 4 sin t 1 + e t 3 1 sin t 1 + e t 3 1 cos t 1 t t2 2t 2 4e t3 1 sin t 1 t 2 t 4 sin t 1 + e t3 1 sin t 1 + e t 3 1 cos t 1 t 2 6 [ 1 cos t 7 t 10 ]a t2 2t 2 4e t 3 1 sin t 1 t 2 t 4 sin t 1 + e t3 1 sin t 1 + e t3 1 cos t 1 t e t3 1 sin t 1 + t t2 2t 2 4e t3 1 sin t 1 t 2 t 4 sin t 1 + e t3 1 sin t 1 + ] e t 3 1 cos t t2 2t 2 4e t 3 1 cos t 1 t 2 t 4 cos t 1 cos t 8 a 10 }Y 10. Then if a i = 0 for i = and a 1 a 7 a 8 are non-zero then we can make the coefficients of Y 8 vanish by F t 1 2 and F t 2 3. Scaling Y if necessary we can assume that a 1 = a 7 = a 8 = 1. And Y is reduced to Case and 18. If a i = 0 for i = and a 1 a 3 a 7 a 8 are non-zero then we can make the coefficients of Y 8 vanish by F t 1 2 and F t 2 3. Scaling Y if necessary we can assume that a 3 = a 9 = 1. And Y is reduced to Case and 15.

12 64 Mehdi Nadjafikhah Seyed-Reza Hejazi If a i = 0 for i = and a 1 a 4 a 8 a 9 are non-zero then we can make the coefficients of Y 4 and Y 8 vanish by F t 1 2 and F t 2 3. Scaling Y if necessary we can assume that a 4 = 1. And Y is reduced to Case and 16. If a i = 0 for i = and a 1 a 3 a 7 a 8 are non-zero then we can make the coefficients of Y 8 vanish by F t 1 2 and F t 2 3. Scaling Y if necessary we can assume that a 1 = 1. And Y is reduced to Case and 21. According to our optimal system of one-dimensional subalgebras of the full symmetry algebra g 1 we need only find group-invariant solutions for 21 one-parameter subgroups generated by Y as Theorem Conclusions In this paper by using the criterion of invariance of the equation under the infinitesimal prolonged infinitesimal generators we find the most Lie point symmetry group of the cylindrical Laplace equation. Looking the adjoint representation of the obtained symmetry group on its Lie algebra we have found the preliminary classification of group-invariants solutions. References [1] N. Bila Symmetry Groups and Conservation Laws of Certain Partial Differential Equations Balkan Journal of Geometry and its Applications [2] M. Nadjafikhah Lie Symmetries of Burger s Equation Adv. Appl. Clifford Alg. DOI /s [3] M. Nadjafikhah and S.R. Hejazi Differential Invariants of SL2 and SL3- Action on R 2 Mathematical Science Vol. 1 No [4] P.J. Olver Equivalence Invariant and Symmetry Cambridge University Press Cambridge [5] P.J. Olver Applications of Lie Groups to Differential equations Second Edition GTM Vol. 107 Springer Verlage New York [6] L.V. Ovsiannikov Group Analysis of Differential Equations Academic Press New York [7] A.D. Polyanin Handbook of Linear Partial Differential Equations for Engineers and scientists Chapman and Hall/CRC [8] C. Udriste and N. Bila Symmetry Lie Group of Monge-Ampere Equation Balkan Journal of Geometry and its Applications [9] Authors addresses: Mehdi Nadjafikhah and Seyed-Reza Hejazi School of Mathematics Iran University of Science and Technology Narmak-16 Tehran I.R.IRAN. URL: webpages.iust.ac.ir\m nadjafikhah m nadjafikhah@iust.ac.ir reza hejazi@iust.ac.ir