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 sym-metries 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 23, 1749 - March 5, 1827) was a French mathematician and astronomer whose work was pivotal to the development of math-ematical astronomy. He summarized and extended the work of his predecessors in his five volume Mcanique Cleste (Celestial Mechanics) (1799-1825). 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, theLaplace operator orLaplacian, denoted by ∆ or ∇2and named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of anelliptic operator, with many applications. In physics, it is used in modelingofwave 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, elec-tromagnetics, 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, andJoseph-Louis Lagrange.), an important partial differen-tial 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.2, 2009, pp. 53-64. c
variablet;heat flow, forming theHelmholtz equation. It is central inelectrostaticsand fluid mechanics, anchoring in Laplace’s equationandPoisson’s equation. Inquantum mechanics, it represents thekinetic energy term of theSchr¨odinger equation. The ki-netic 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 calledharmonic functions; the Laplacian is at the core ofHodge theory and the results ofde 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
∇2u= 1
r ∂ ∂r
³
r∂u ∂r
´ + 1
r2
∂2u
∂θ2 +
∂2u
∂z2 = 0, (1.1)
whereuis 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 =ξ1(r, θ, z, u)
∂
∂r +ξ2(r, θ, z, u) ∂
∂θ +ξ3(r, θ, z, u) ∂
∂z +η(r, θ, z, u) ∂ ∂u.
Using the invariance condition, i.e., applying the second prolongationX(2) to (1.1), the following system of 18 determining equations yields:
ξ2u= 0,
ξ3u= 0,
ηuu= 0,
ξ3rzz= 0,
ξ3θzz= 0,
ξ3zzz= 0,
ξ3rr+ξ3zz= 0,
ξ3θ+r2ξ2z= 0,
ξ3θ−rξ3rθ= 0,
ξ3rz+ 2ηru= 0,
ξ3θz+ 2ηθu= 0,
ξ3zz+ 2ηzu= 0,
ξ1+r(ξ2θ−ξ3z) = 0,
ξ2θ+r(ξ2rθ−ξ3rz) = 0,
ξ2θθ−rξ2r−ξ3θz = 0,
ξ3θθ+r(ξ3r+rξ3zz) = 0,
ξ3θz+r(2ξ2r−rξ2rr) = 0,
ηθθ+r(ηr+rηrr+rηzz) = 0.
The solution of the above system gives the following infinitesimals,
X1 =
∂
∂θ, X2= ∂
∂z, X3=u ∂ ∂u,
X4 = r
∂ ∂r +z
∂
∂z, X5=rz ∂ ∂r +
1 2(z
2−r2)∂
∂z −
1 2zu
∂ ∂u,
X6 = sinθ ∂
∂r +
1
rcosθ ∂
∂θ, X7= cosθ ∂ ∂r −
1
X8 = zsinθ
∂ ∂r +
1
rzcosθ ∂
∂θ −rsinθ ∂ ∂z,
X9 = zcosθ ∂
∂r −
1
rzsinθ ∂
∂θ −rcosθ ∂ ∂z,
X10 = 1 2(z
2−r2) sinθ∂
∂r +
1 2r(z
2+r2) cosθ ∂
∂θ −rzsinθ ∂ ∂z+
1 2rusinθ
∂ ∂u,
X11 = 1 2(z
2−r2) cosθ ∂
∂r −
1 2r(z
2+r2) sinθ ∂
∂θ −rzcosθ ∂ ∂z+
1 2rucosθ
∂ ∂u.
The commutation relations of the 11−dimensional Lie algebra g spanned by the vector fieldsX1,· · ·, X11, are shown in following table.
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
X1 0 0 0 0 0 X7 −X6 X9 −X8 X11 −X10
X2 0 0 0 X2 − 1
2X3+X4 0 0 X6 X7 X8 X9
X3 0 0 0 0 0 0 0 0 0 0 0
X4 0 −X2 0 0 X5 −X7 −X7 0 0 X10 X11
X5 0 1
2X3−X4 0 −X5 0 −X8 −X9 −X10 −X11 0 0
X6 −X7 0 0 X7 X8 0 0 −X2 0 1
2X3−X4 −X1
X7 X6 0 0 X7 X9 0 0 0 −X2 X1 1
2X3−X4
X8 −X9 −X6 0 0 X10 X2 0 0 X1 −X5 0
X9 X8 −X7 0 0 X11 0 X2 −X1 0 0 −X5
X10 −X11 −X8 0 −X10 0 −
1
2X3+X4 −X1 X5 0 0 0
X11 X10 −X9 0 −X11 0 X1 − 1
2X3+X4 0 X5 0 0
2.1
Differential invariants
Suppose thatG is a transformation group. It is well known that a smooth real dif-ferential function I:Jn→R, where Jn is the corresponding n−th jet space, is a differential invariantforGif and only if for all X∈g, itsn−th prolongation annihi-latesI, 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, ur, uθ, uz), 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, ur, uθ, uz, urr, urθ, urz, uθθ, uθz, uzz), 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.
Theorem 2.1. The number of functionally independent differential invariants up to ordern ≥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 ¡3+n
n ¢
−¡2+n n −1 ¢
.
Proof.According to the above discussion forn≥3 the action is transitive, so the number of independent differential invariantsin, is
in = 3 + µ3 +n
n
¶ −11 =
µ3 +n
n
¶ −8.
Ifjn is the number of strictly differential invariants then,
jn = µ3 +n
n
¶ −
µ3 +n−1
n−1 ¶
= µ3 +n
n
¶ −
µ2 +n
n−1 ¶
. ✷
2.2
Reduction of the equation
The Eq. (1.1) maybe regarded as a submanifoldM of the jet space J2(R3,R). Doing as section 3.1 of [5] and find (in a sense) the most general group-invariant solutions to the Eq. (1.1).
Theorem 2.2. The one-parameter groups gi(t) :M →M generated by the Xi, i = 1,· · ·,11, are given in the following table:
g1(t) : (r, θ, z, u)7−→(r, θ+t, z, u),
g2(t) : (r, θ, z, u)7−→(r, θ, z+t, u),
g3(t) : (r, θ, z, u)7−→(r, θ, z, uet),
g4(t) : (r, θ, z, u)7−→(ret, θ, zet, u),
g5(t) : (r, θ, z, u)7−→ ³
rzt+r, θ,t
2(z
2−r2),−1
2zut+u ´
,
g6(t) : (r, θ, z, u)7−→ ³
tsinθ+r,t
rcosθ+θ, z, u
´
,
g7(t) : (r, θ, z, u)7−→ ³
tcosθ+r, t
rsinθ+θ, z, u
´
,
g8(t) : (r, θ, z, u)7−→ ³
tzsinθ+r, t
rzcosθ+θ,−rtsinθ+z, u
´
,
g9(t) : (r, θ, z, u)7−→ ³
tzcosθ+r,−t
rzsinθ+θ,−rtcosθ+z, u
´
,
g10(t) : (r, θ, z, u)7−→ Ã
t
2(z
2−r2) sinθ,t 2
³z2
r +r
´
cosθ+θ,
−rztsinθ+z,1
2rutsinθ+u !
,
g11(t) : (r, θ, z, u)7−→ Ã
t
2(z
2−r2) cosθ,−t 2
³z2
r +r
´
sinθ+θ,
−rztcosθ+z,1
2rutcosθ+u !
where entries give the transformed pointexp(tXi)(r, θ, z, u) = (˜r,θ,˜ z,˜ u˜).
The transformationsg1andg2demonstrate the radius- and space-invariance and radius- and angle-invariance solutions of the equation. The one-parameterg3 reflects the linearity of the cylindrical laplace equation, and the well-known scaling symmetry turns up in g4. 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=U(r, θ, z) is a solution of Eq. (1.1) so are the functionsui(r, θ, z) =U(r, θ, z), where
u1 =U(r, θ+t, z), u2=U(r, θ, z+t),
u3 =e−tU(r, θ, z), u4=U(etr, θ, etz),
u5 =³1 +1 2zt
´
U(rzt+r, θ,t
2(z
2−r2)),
u6 =U(tsinθ+r,t
rcosθ+θ, z), u
7=U³tcosθ+r,t
rcosθ+θ, z
´
,
u8 =U³tzsinθ+r, t
rzcosθ+θ,−rtsinθ+z
´
,
u9 =U³tzcosθ+r,−t
rzsinθ+θ,−rtcosθ+z
´
,
u10 =³1−rt 2 sinθ
´
U
Ã
t
2(z
2−r2) sinθ, t 2r(z
2+r2) cosθ+θ, z−rztsinθ !
,
u11 =³1−rt 2 cosθ
´
U
Ã
t
2(z
2−r2) cosθ, t 2r(z
2+r2) sinθ+θ, z−rztcosθ !
.
For example, if u(r, θ, z) = 1 be a constant solution of Eq. (1.1), we conclude that the trivial functions gi(t)·1 = 1 for i = 1,2,4,6,7,8,9, g3(t)·1 = e−t and
g5(t)·1 = 2 +zt, and by applying g5 ³
2a b
´
·g3(lnb) onu= 1, we conclude the linear solutionu=az+b. The two nontrivial solutions for Eq. (1.1) are:
g10(t)·1 = 2−rtsinθ, g11(t)·1 = 2−rtcosθ.
Now by summingg1,g5 ³
−8+2c d
´
·g3lnd·1,g10(−a)·1 andg11(−b)·1, we conclude the following result:
Corollary 2.4. The functionu(r, θ, z) =arsinθ+brcosθ+dz+c, is a solution of Eq. (1.1), wherea, b, canddare arbitrary constants.
2.3
Some invariant solutions
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 X6. To determine the independent invariantsI, we need to solve the first order partial differential equationX6{I(r, θ, z, u)}= 0, that is
sinθ∂I ∂r +
cosθ r
∂I ∂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
sinθ =
cosθdθ
r =
dz
0 =
du
0 .
Hence we obtain two functionally independent invariantsy= z
rsinθ andv=u.
If we treat v as a function of y, we can compute formulae for the derivatives of uwith respect to x and t in terms of y, v and the derivatives ofv with respect to y, along with a single parametric variable, which we designate to be t, so that
xwill be the corresponding principal variable. Using the chain rule we find that if
u=v=v(y) =v(z/rsinθ), then
ur = vyyr, (2.1)
urr = yrrvy+vyyyr, (2.2)
uθθ = yθθvy+vyyyθ, (2.3)
uzz = yzzvy+vyyyz. (2.4)
Substituting ur, urr, uθθ and uzz in (1.1) we obtain an ODE with solution u =
C1arctan ³
z rsinθ
´
+C2,which is called an invariant solution for Eq. (1.1).
2. Similarly, the invariants of X7 are y=
z
rcosθ and v = u, and the solution of
reduced equation (invariant solution) by substituting (2.1), (2.2), (2.3) and (2.4) is
u=C1arctan ³
z rcosθ
´ +C2.
3. The invariants ofX8 are
r2+z2
rsinθ andv =u. The invariant solution constructed
with (2.1), (2.2), (2.3) and (2.4) is u = C1 arctanh ³√
r2+z2
rsinθ ´
dr zsinθ =
rdθ acosθ =
dz
−rsinθ = du
0 .
The above mentioned method in sections1and2can be followed forX9with invari-ants r
2+z2
rsinθ and v =u, and invariant solutionu=C1 arctanh
³√ r2+z2
rcosθ ´
+C2. We
can proceed for otherXis as above.
2.4
The Lie algebra of symmetries
In this section, we determine the structure of full symmetry Lie algebra g, of Eq. (1.1).
Theorem 2.5. The full symmetry Lie algebrag, of Eq. (1.1) has the following semidi-rect decomposition:
g =R ⋉g1, (2.5)
whereg1 is a simple Lie algebra.
Proof: The center z of g is SpanR{X3}. Therefore the quotient algebra g1 = g/z is SpanR{Y1,· · ·, Y10}, whereYi :=Xi+ z for i= 1,· · ·,10. The commutator table of this quotient algebra is given in following table:
Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10
Y1 0 0 0 0 Y7 −Y5 Y8 −Y7 Y10 −Y9
Y2 0 0 Y2 Y3 0 0 Y5 Y6 Y7 Y8
Y3 0 −Y2 0 Y4 −Y5 −Y6 0 0 Y9 Y10
Y4 0 −Y3 −Y4 0 −Y7 −Y8 −Y9 −Y10 0 0
Y5 −Y7 0 Y5 Y7 0 0 −Y2 0 −Y3 −Y1
Y6 Y5 0 Y6 Y8 0 0 0 −Y2 Y1 −Y3
Y7 −Y8 −Y5 0 Y9 Y2 0 0 Y1 −Y4 0
Y8 Y7 −Y6 0 Y10 0 Y2 −Y1 0 0 −Y4
Y9 −Y10 −Y7 −Y9 0 Y3 −Y1 Y4 0 0 0
Y10 Y9 −Y8 −Y10 0 Y1 Y3 0 −Y4 0 0
The Lie algebra g is non-solvable, because
g(1) = [g,g] = SpanR ½
X1, X2, 1
2X3−X4, X5,· · · , X11 ¾
,
g(2) = [g(1),g(1)] = g(1).
Similarly, g1 is semi-simple and non-solvable, because
g(1)1 = [g1,g1] = SpanR{Y1,· · · , Y10}= g1.
s = ½
X1, X2, 1
2X3−X4, X5,· · ·X11 ¾
.
r is a one-dimensional subalgebra of g, therefore it is isomorphic to R; Thus the
identity g = r⋉s reduces to g =R ⋉g1. ✷
2.5
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
Ad(exp(tYi)Yj) =Yj−t[Yi, Yj] +
t2
2[Yi,[Yi, Yj]]− · · ·, (2.6)
where [Yi, Yj] is the commutator for the Lie algebra, t is a parameter, and i, j = 1,· · ·,10. We can write the adjoint action for the Lie algebra g1, and show that
Theorem 2.6. A one dimensional optimal system of theg1 is given by 1) Y1,
2) Y3, 3) Y4, 4) Y7, 5) Y8, 6) Y9, 7) aY1+bY3,
8) aY1+bY4, 9) aY1+bY7, 10) aY1+bY8 11) aY1+bY9, 12) aY3+bY7, 13) aY3+bY8, 14) aY4+bY9,
15) aY7+bY8, 16) aY1+bY4+cY9, 17) aY1+bY3+cY8, 18) aY1+bY7+cY8, 19) aY3+bY7+cY8, 20) aY1+bY3+cY7, 21) aY1+bY3+cY7+dY8,
wherea, b, c, d∈R are arbitrary numbers.
Proof. Ft
i : g1→g1 defined by Y 7→ Ad(exp(tYi)Y) is a linear map, for i = 1,· · ·,10. The matrices Mt
i of F t
LetY =
3 . Scaling Y if necessary, we can assume thata1=a7=a8= 1. And Y is reduced to Case 1, 4, 5 and 18.
Ifai= 0, for i= 2,4,5,6,9,10, anda1, a3, a7, a8, are non-zero then we can make the coefficients ofY8, vanish, byF2t1 andF
t2
Ifai= 0, for i= 2,3,5,6,7,10, anda1, a4, a8, a9, are non-zero then we can make the coefficients ofY4 andY8 vanish, byF2t1 andF
t2
3 . Scaling Y if necessary, we can assume thata4= 1. AndY is reduced to Case 3, 6, 8, 11, 14 and 16.
Ifai= 0, for i= 2,4,5,6,9,10, anda1, a3, a7, a8, are non-zero then we can make the coefficients ofY8, vanish, byF2t1 andF
t2
3 . ScalingY if necessary, we can assume thata1= 1. AndY is reduced to Case 10, 17, 19, 20, and 21. ✷
According to our optimal system of one-dimensional subalgebras of the full sym-metry algebra g1, we need only find group-invariant solutions for 21 one-parameter subgroups generated byY as Theorem (2.6).
3
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
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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,
