Equivalence Groups and Differential Invariants for (2+1) dimensional Nonlinear Diffusion Equation

0.18 MB·

by  Saadet Özer

Equivalence Groups and Differential Invariants
for (2+1) dimensional Nonlinear Diffusion
Equation
7 Saadet O¨zer
1
0
2

n Istanbul Technical University, Faculty of Science and Letters,
a Department of Mathematics Engineering, 34469 Maslak Istanbul Turkey;
J email: [email protected]

1

] Abstract
h
(2+1) dimensional diffusion equation is considered within the frame-
p
- work of equivalence transformations. Generators for the group are ob-
h tainedandadmissibletransformationsbetweenlinearandnonlinearequa-
at tions are examined. It is shown that transformations between linear and
m nonlinearequationsarepossible providedthat thegeneratorsof indepen-
dent variables depend on the dependent variable. Exact solutions for
[
some nonlinear equations are obtained. Differential invariants related to

1 the transformation groups are investigated and the results are compared
v with the direct integration method.
4
6
2 keywords: LieGroupApplication,EquivalenceGroups,ExactSolution,Non-
0 linear Diffusion Equation, Differential Invariants
0
.
1
1 Introduction
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7
1 Differential equations containing some arbitrary functions or parameters rep-
v: resent actually family of equations of the same structure. Almost all field
i equations of classical continuum physics possess this property related to the
X
behaviour of different materials. In dealing with such family of differential
r
equations, Lie symmetry analysis provides some powerful algorithmic methods
a
for determination of invariant solutions, conserved quantities and construction
of maps between differential equations of the same family that turns out to
be equivalent [1–3]. To examine such problems, it is convenient considering
equivalence transformation groups that preserve the structure of the family of
differential 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 invariance
approach could be employed in order to construct equivalence groups was for-
mulated by Ovsiannikov [4]. Then several well-developed methods have been
used to construct equivalence groups. The generaltheory of determining trans-
formation groups and algorithms can be found in the references [4–7].
1
In the present text we shall examine the (2+1) dimensional diffusion equa-
tion. Nonlinear members of the family of diffusion equations have significant
importanceinmanyareasinappliedsciences. Agreatnumberofmembershave
been usedto modelmany physicalphenomena in MathematicalPhysics,Math-
ematical Biology etc. and they have been widely studied not only by means of
numerical, asymptotic analysis but also as application to Lie Group Analysis
since Lie [8]. The simple nonlinear heat equation; u = [A(u)u ] was first
t x x
examined by Ovsiannikov [9] within the frame-work of Lie’s symmetry classifi-
cation. The complete classification and form-preserving point transformations
for the inhomogenous one dimensional nonlinear diffusion are obtained in [10].
The conditions for reduction the more general diffusion type equations to the
one dimensional heat equation are also examined in [11]. Equivalence transfor-
mations of linear diffusion equations into nonlinear equations for some classes
have been considered in Lisle’s PhD thesis [12] widely. Constructing the exact
analyticalsolutionsforsomespecificproblemsrelatedtothenonlineardiffusion
equation have recently examined by [13–15]. Torrisi et al. in their paper [14]
havealsostudiedthedevelopmentsofbacterialcoloniesasapplicationtoequiv-
alence groups. Bruz´on et al. have derived maps between nonlinear dispersive
equations in their detailed work [16].
The aim of the present work is to study equivalence transformations for
a general family of (2+1) dimensional diffusion equation. We investigate the
structure of the transformation group generators which lead to map linear and
nonlinear members.
For the convenience of the reader to follow, in section 2, we have obtained
the generators of the group of equivalence transformations and determined the
structure for admissible transformations. Theorems in the section state the
conditions of the appropriate equivalence transformations between linear and
nonlinearmembers. Inthenextsection,wehaveconsideredsomesubgroupsof
the generalequivalence groupsand by choosingsome specific forms we are able
toobtainsomeclassesofnonlinearequationswhichareequivalenttolinearones.
We have also investigated the classes of nonlinear diffusion equations that are
mapped onto the classical heat equation. Exact solutions for those nonlinear
equations are also obtained. In section 4 we have also examined differential
invariantsforthesubgroupsanddiscussedtheresultswiththedirectintegration
method.
2 Equivalence Transformations
Inthe presentpaperwe shallinvestigatethe equivalence groupa generalfamily
of (2+1) dimensional diffusion equation
u =f(x,y,t,u,u ,u ) +g(x,y,t,u,u ,u ) (1)
t x y x x y y
which represents a great variety of linear and nonlinear equations. Here u is
the dependent variable of the independent variables x,y and t. Here f and g
are smooth nonconstant functions of their variables and subscripts denote the
partial derivatives with respect to the corresponding variables.
Definition 1. With n independent variables x , N dependent variables u and
i α
2
msmoothfunctionsφ ofindependent, dependentvariables andtheirderivatives
k
F(x ,u ,φ (x ,u ))=0
i α(p) k(q) i α(p)
is called a family of differential equations. Here i=1,2...n, α=1,2,...N, k =
1,2,...mandu includeboththetupleofdependentvariablesu=(u ,u ,...,u )
α(p) 1 2 N
as well as all the derivatives of u with respect to x ’s up to order N. By φ
i k(q)
we denote the smooth functions φ and the partial derivatives with respect to
k
both x ’s and u ’s.
i α(p)
Definition 2. For a given differential equation of the family
F(x ,u ,φ (x ,u ))=0
i α(p) k(q) i α(p)
the equivalence group E is the group of smooth transformations of independent,
dependentvariables, theirderivatives andsmoothfunctionspreservingthestruc-
ture of the differential equation but transforms it into
F(x¯ ,u¯ ,φ¯ (x¯ ,u¯ ))=0.
i α(p¯) k(q¯) i α(p¯)
More precisely, equivalence transformations associated with the (2+1) di-
mensional most general diffusion equation (1) transform the equation into
u −f(x,y,t,u,u ,u ) −g(x,y,t,u,u ,u ) =0 −→
t x y x x y y
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 coordinate
system x = (x ) = (x,y,t) which we shall call as the space of independent
i
variables. Consider a trivialbundle structure (K,π,M) with fibers are the real
line R. Here M is the base manifold and K, called the graph space is globally
in form of a product manifold M×R. We equip the four dimensional graph
space 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 and
locally in form
∂ ∂ ∂ ∂
V =ξ1 +ξ2 +ξ3 +η (2)
∂x ∂y ∂t ∂u
where ξi (i=1,2,3) and η are coordinate functions on K.
In order to construct the equivalence groups E for the equation (1), first
we extend the graph space K by adding the auxiliary variables and the ones
representing the functional dependencies of the smooth functions f and g
K˜ ={x,y,t,u,f,g,u ,u ,u ,f ,f ,f ,f ,g ,g ,g ,g ,f ,f ,g ,g }. (3)
x y t x y t u x y t u ux uy ux uy
The prolongation vector V˜ over the extended manifold covered by K˜ can be
written as
4
∂ ∂ ∂ ∂ ∂ ∂
V˜ =V +µ1 +µ2 +ζ1 +ζ2 +ζ3 + µ1
∂f ∂g ∂u ∂u ∂u j∂f
x y t Xj=1 j
(4)
4 3 3
∂ ∂ ∂
+ µ2 + ν1 + ν2
j∂g j∂f j ∂g
Xj=1 j Xj=1 uj Xj=1 uj
3
where in the last four summations j = 1,..4 represent x,y,t and u and all
coefficient functions are smooth functions of the coordinates of the extended
manifold.
Theorem1([7]). Letavectorfieldonanndimensionaldifferentiablemanifold
M be given by
∂
V(p)=vi(x) , p=ϕ(−1)(x), i=1,...,n
∂xi
where (U,ϕ) is the chart to which p ∈ M belongs. A curve γ is an integral
curve of the vector field V iff the coordinate functions xi(t) are solutions of the
following system of local ordinary differential equations in Rn
dxi
=vi(x(t))
dt
Precisely,the equivalence transformationscanbe determined by solving the
followingsystemofautonomousordinarydifferentialequationsonthe extended
manifold (3)
dx¯ dy¯ dt¯
=ξ1(x¯,y¯,t¯,u¯), =ξ2(x¯,y¯,t¯,u¯), =ξ3(x¯,y¯,t¯,u¯),
dǫ dǫ dǫ
(5)
du¯ df¯ dg¯
=η(x¯,y¯,t¯,u¯), =µ1, =µ2
dǫ dǫ dǫ
under the initial conditions
x¯(0)=x, y¯(0)=y, t¯(0)=t, u¯(0)=u, f¯(0)=f, g¯(0)=g (6)
where µ1 and µ2 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-knownpro-
longationformula (for detailed information,theorems and detailedapplications
see[5,17]). Formoredetailswereferthereaderthepapers[18–20]andreferences
therein which are concerned with equivalence transformations.
ζ1 =D (η)−u D (ξ1)−u D (ξ2)−u D (ξ3),
x x x y x t x
ζ2 =D (η)−u D (ξ1)−u D (ξ2)−u D (ξ3), (7)
y x y y y t y
ζ3 =D (η)−u D (ξ1)−u D (ξ2)−u D (ξ3)
t x t y t t t
where D ,D ,D denote the total derivatives with respect to their parameters:
x y t
D = ∂ +u ∂ . And
i ∂xi xi∂u
3 3
µ1 =D˜ (µ1)− f D˜ (ξi)−f D˜ (η)− f D˜ (ζi),
j j i j u j ui j
Xi=1 Xi=1
3 3
µ2 =D˜ (µ2)− g D˜ (ξi)−g D˜ (η)− g D˜ (ζi),
j j i j u j ui j
Xi=1 Xi=1
(8)
3
ν1 =D˜ (µ1)− f D˜ (ζi),
j uj ui uj
Xi=1
3
ν2 =D˜ (µ2)− g D˜ (ζi)
j uj ui uj
Xi=1
4
where D˜ = ∂ and D˜ = ∂ .
j ∂xj uj ∂uj
These expressions do not impose any restriction on functional dependencies
of the smooth functions f and g in the main equation (1). If some variables
do not appear in the coordinate cover of the extended manifold (3), due to a
particular structure of the given differential equation, that might entail some
restrictionsonthe extended vectorfield componentsbecause the corresponding
components must then be set to zero. Note that in equation (1) the free pa-
rameters f and g do not depend on u . Thus their corresponding components
t
in (4) must vanish:
µ1 =µ2 =0. (9)
3 3
Theorem 2. A nonlinear (2+1) dimensional diffusion equation can be mapped
onto a linear equation by a point equivalence transformation, if and only if it is
in the following form:
x¯=φ(x,y,t,u) and/or y¯=ψ(x,y,t,u)
where φ,ψ ∈C2 and ∂φ 6=0, ∂ψ 6=0.
∂u ∂u
Proof. Togeneratethetransformationsoftheequivalencegroupforthediffusion
equation given by (1), we shall apply the restrictions (9) to the given formulas
(7) and (8). Then we have
ξ1 =ξ1(x,y,t,u), ξ2 =ξ2(x,y,t,u), ξ3 =ξ3(t), η =η(x,y,t,u),
ζ1 =η +(η +ξ1)u +ξ1(u )2+ξ2u +ξ2u u ,
x u x x u x x y u x y
ζ2 =η +(η +ξ2)u +ξ1u +ξ1u u +ξ2(u )2,
y u y y y x u x y u y
(10)
ζ3 =η +(η +ξ˙3)u +ξ1u +ξ1u u +ξ2u +ξ2u u ,
t u t t x u x t t y u y t
µ1 =(η +ξ˙3−ξ1+ξ2u )f −(ξ1+ξ2u )g+γu +κ1,
u x u y y u y y
µ2 =(η +ξ˙3−ξ2+ξ1u )g−(ξ2+ξ2u )f −γu +κ2
u y u x x u x x
and
η =ξ1+ξ2−ξ˙3+s(t) (11)
u x y
where s(t) is an arbitrary continuous function, γ, κ1 and κ2 depend on the
independent and dependent variables and satisfy the following relations
ξ1 =−γ +κ1, ξ2 =γ +κ2, η =κ1 +κ2.
t y u t x u t x y
Equation (11) points out that η can not depend on u nonlinearly, unless ξ1
and/or ξ2 depend on u. Thus to have any transformation between linear and
nonlinear equations ξ1 and/or ξ2 must involve the dependent variable u which
meant to be the transformed variables x¯ and /or y¯involve u:
∂ξi
ξ1 =ξ1(x,y,t,u), and/or ξ2 =ξ2(x,y,t,u), 6=0, i=1,2.
∂u
Thus transformed variables are obtained via the solution of the set of ordinary
differential equations (5) as
x¯=φ(x,y,t,u) and/or y¯=ψ(x,y,t,u).
5
Mapsbetweennonlinearandlinearequationsmightsometimesbegenerated
by some nonlinear dependencies on only the dependent variable; like the very
well-knownwaveequation(see [21–23]). The followingtheorempoints out such
transformations are not admissible for diffusion equation.
Theorem3. (2+1)dimensional diffusionequationdoesnotadmitthefollowing
type of transformation:
x¯=ψ (x,y,t), y¯=ψ (x,y,t), t¯=ψ (t), u¯=ψ (x,y,t,u)
1 2 3 4
where ψ (x,y,t,u) is nonlinear in u.
4
Proof. One can easily see from the equation (11), unless ξ1 or ξ2 depend on u,
η can not involve u. Thus u¯ can not be nonlinear in u when x¯ or y¯ does not
involve u.
Corollary 1. Any integrable transformation related to the generators; ξ1 =
ξ1(x,y,t), ξ2 = ξ2(x,y,t), ξ3 = ξ3(t), η = h(x,y,t)u map a linear equation
onto another linear equation with various different coefficient functions.
Readersmayseethatequations(10)generateallsetofadmissibleequivalence
transformations,relatedtothegeneral(2+1)dimensionaldiffusionequation(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).
3 Applications to exact solutions of nonlinear
diffusion equations
Theorem 2 also refers that the admissible equivalence transformations related
to the diffusion equation (1) may involve some arbitrary functions. As a conse-
quenceofthisresult,inadditiontothemapsbetweensinglelinearandnonlinear
equations,onecansaythattransformationsbetweenasinglelinearequationand
a class of nonlinear equations can also be constructed. That is to say we can
set such transformations which map a single linear equation into a particular
family of nonlinear equations or vice versa.
f +g −u =0←→f¯ +g¯ −u¯ =0
x y t kx¯ ky¯ kt¯
where the subscript k denotes the class of equations.
In this section we shall investigate some applications to that circumstance
and study such maps by considering the transformations involving some ar-
bitrary differentiable functions. Hereby solutions of some particular family of
nonlinear equations can be obtained from an appropriate linear equation.
3.1 On the subgroup: ξ1 = m(u), ξ2 = h(u), ξ3 = η = 0
Hereweshallexamineasubgroupoftheadmissibleequivalencegroupsgenerated
by the infinitesimal generators: ξ1 =m(u), ξ2 =h(u), ξ3 =η =0 where m(u)
6
and h(u) are arbitrary differentiable continuous functions. Then the prolonged
vector field (4) can be written via (10) as follows
∂ ∂ ∂ ∂
V˜ =m(u) +h(u) +(h′(u)f−m′(u)g)u +(m′(u)g−h′(u)f)u +···
y x
∂x ∂y ∂f ∂g
(12)
By integrating the system of equations (5) under the initial conditions (6) we
have the following class of transformations for the subgroup
x¯=x−ǫm(u), y¯=y−ǫh(u), t¯=t, u¯=u,
u u
u¯ = x , u¯ = y ,
x 1−ǫ(u m′(u)+u h′(u)) y 1−ǫ(u m′(u)+u h′(u))
x y x y
u¯ = ut , (13)
t 1−ǫ(u m′(u)+u h′(u))
x y
(1−ǫu m′(u))f −ǫu m′(u)g (1−ǫu h′(u))g−ǫu h′(u)f
f¯= x y , g¯= y x .
1−ǫ(u m′(u)+u h′(u)) 1−ǫ(u m′(u)+u h′(u))
x y x y
Note thathere ǫ is the groupparameter. By substituting u andu in terms of
x y
the transformed variables, f¯and g¯ can now be written as
f¯=(1+ǫu¯ h′(u¯))f˜−ǫu¯ m′(u¯)g˜, g¯=(1+ǫu¯ m′(u¯))g˜−ǫu¯ h′(u¯)f˜ (14)
y¯ y¯ x¯ x¯
where f˜(x¯,y¯,t¯,u¯,u¯ ,u¯ ) and g˜(x¯,y¯,t¯,u¯,u¯ ,u¯ ) represent f and g in terms of
x¯ y¯ x¯ y¯
the 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 +g −u = 0 onto a class of
x y t
nonlinearequations ofthe formf¯ +g¯ −u¯ =0, andasolutionofthe linear
kx¯ ky¯ kt¯
equation
φ(x,y,t,u)=0
generates solutions to the corresponding nonlinear equations as
φ (x¯+ǫm(u¯),y¯+ǫh(u¯),t¯,u¯)=0.
k
Example 1: As a particular example, we search the nonlinear diffusion
equations of the class (1) that can be mapped onto the very well known heat
equation. The(2+1)dimensionalheatequationcanberepresentedasamember
of (1) by choosing f =u , g =u ;
x y
u +u =u . (15)
xx yy t
The transformed form of f =u and g =u can be obtained from (14) as
x y
f¯= (1+ǫh′(u¯)u¯y¯)u¯x¯−ǫm′(u¯)u¯2y¯, g¯= (1+ǫm′(u¯)u¯x¯)u¯y¯−ǫh′(u¯)u¯2x¯
1+ǫh′(u¯)u¯ +ǫm′(u¯)u¯ 1+ǫh′(u¯)u¯ +ǫm′(u¯)u¯
y¯ x¯ y¯ 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)
7
where the coefficients are
A=1+2ǫh′(u¯)u¯ +ǫ2 h′(u¯)2+m′(u¯)2 u¯2
y¯ y¯
B =−2ǫ m′(u¯)u¯ +h′((cid:0)u¯)u¯ +ǫ h′(u¯)2(cid:1)+m′(u¯)2 u¯ u¯
y¯ x¯ y¯ x¯
C =1+2(cid:2)ǫm′(u¯)u¯x¯+ǫ2 h′(u¯)2+(cid:0)m′(u¯)2 u¯x2¯ (cid:1) (cid:3) (17)
D =−ǫ(m′′(u¯)u¯ +h′′(u¯(cid:0))u¯ ) u¯2 +u¯2 (cid:1)
x¯ y¯ x¯ y¯
E =(1+ǫ(m′(u¯)u¯ +h′(u¯)u¯ (cid:0)))2 (cid:1)
x¯ y¯
Itisobviousthatviathetransformationsof (13),wecanconstructvariousmaps
between the single heat equation (15) and the nonlinear equations of the class
of (16). And any solutionof the heat equation(15) would generatesolutions of
nonlinear equations.
For the reader to see more clearly how a solution to nonlinear problems
can be constructed via equivalence transformations we consider the following
problem.
Example 2: Asubclassofnonlineardiffusionequations(16)canbeconsidered
by taking m(u)=un, h(u)=ur in the previous example, where n, r ∈R.
1+2ǫrur−1u ++ǫ2(r2u2(r−1)+n2u2(n−1))u2 u −2ǫ nun−1u
y y xx y
h i (cid:2)
+rur−1u +ǫ(r2u2(r−1)+n2u2(n−1))u u u + 1+2ǫnun−1u
x x y xy x
i (cid:2) (18)
+ǫ2(r2u2(r−1)+n2u2(n−1))u2 u −ǫ(u2 +u2)(r(r−1)ur−2u
x yy x y y
i
+n(n−1)un−2u )=(1+ǫ(nun−1u +rur−1u ))2.
x x y
¯
Here (∗)’s are omitted for the simplicity. And an implicit solution for (18) can
be written as
u−F¯sin(µ(x+ǫun))sin(η(y+ǫur)e−(µ2+η2)t =0
fromasolutionlikeu=F sin(µx)sin(ηy)e−(µ2+η2)toftheheatequation(15)by
applyingtheequivalencetransformations;x=x¯+ǫu¯n, y =y¯+ǫu¯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)=u2 and h(u)=0, where we could simply write
(1+u¯2u¯2)u¯ −2(1+u¯u¯ )u¯u¯ u¯ +(1+u¯u¯ )2u¯ −(u¯2+u¯2)u¯
y¯ x¯x¯ x¯ y¯ x¯y¯ x¯ y¯y¯ y¯ x¯ x¯
(19)
=(1+u¯u¯x¯)2u¯t¯.
The reader can easily see, a much more easier function u = x+y2+2t which
satisfies the linear heat equation would generate
u¯=1∓ 1−2(x¯+y¯2+2t¯))
p
as a solution to the nonlinear equation (19) under the transformations x =
x¯+ 1u¯2, y =y¯, t=t¯, u=u¯.
2
Example 4: As another example by choosing m(u) = sinu, h(u) = cosu the
8
nonlinear equation would become
(1+u¯2−2u¯ sinu¯)u¯ +(1+u¯2 +2u¯ cosu¯)u¯
y¯ y¯ x¯x¯ x¯ x¯ y¯y¯
−2(u¯ cosu¯−u¯ sinu¯+u¯ u¯ )u¯ +(u¯ cosu¯+u¯ sinu¯)(u¯2+u¯2) (20)
y¯ x¯ x¯ y¯ x¯y¯ y¯ x¯ y¯ x¯
−(1−u¯y¯sinu¯+u¯x¯cosu¯)2u¯t¯=0.
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 heat
equation would generate a solution of the equation.
3.2 On the subgroup: ξ1 = m(u,y), ξ2 = ξ3 = η = 0
Similarlytotheprevioussubsection,transformationsrelatedtoasubgroupgen-
erated by the infinitesimal generators ξ1 = m(u,y), ξ2 = ξ3 = η = 0 are con-
sidered. The corresponding equivalence transformations of the subgroup are
explicitly determined as:
x¯=x−ǫm(u,y), y¯=y, t¯=t, u¯=u,
u u +ǫm (u,y)u u
u¯ = x , u¯ = y y x, u¯ = t ,
x y t
1−ǫm (u,y)u 1−ǫm (u,y)u 1−ǫm (u,y)u
u x u x u x
ǫ(m (u,y)+m (u,y)u )g g
f¯=f − y u y , g¯=
1−ǫm (u,y)u 1−ǫm (u,y)u
u x u x
where subscriptsindicate partialdifferentiationwith respectto the correspond-
ing variable.
Example 5: For the simplicity, let us again seek the nonlinear diffusion equa-
tionscanbemappedontothelinearheatequation(15). Forf =u andg =u ,
x y
f¯and g¯ turn to be:
u¯ −ǫ(m +m u¯ )(u¯ −ǫm u¯ )
f¯= x¯ y¯ u¯ y¯ y¯ y¯ x¯ , g¯=u¯ −ǫm u¯
1+ǫm u¯ y¯ y¯ x¯
u¯ x¯
wherem=m(u¯,y¯). Thusthefollowingclassaretheclassofnonlinearequations
that mapped onto the heat equation:
A u¯x¯x¯+B u¯x¯y¯+C u¯y¯y¯+D =E u¯t¯ −→ uxx+uyy =ut
where the coefficients are
A=1+ǫ2m2+3ǫ2m m u¯ +2ǫ2m2u¯2
y¯ u¯ y¯ y¯ u¯ y¯
B =ǫ ǫ2m2u¯2(m u¯ +m )−2ǫm2u¯ u¯ −ǫm m u¯ −3m u¯ −2m
u¯ x¯ u¯ y¯ y¯ u¯ x¯ y¯ u¯ y¯ x¯ u¯ y¯ y¯
C =1(cid:2)+ǫm u¯ −ǫ2m2u¯2 −ǫ3m3u¯3 (cid:3)
u¯ x¯ u¯ x¯ u¯ x¯
D =ǫ 2ǫ2m m m −ǫ2m2m −m −ǫ2m2m u¯3 +ǫ2[−2m m (21)
u¯ y¯ u¯y¯ u¯ y¯y¯ u¯u¯ y¯ u¯u¯ x y¯y¯ u¯
+2m(cid:2) m −3m m u¯ +3m m u¯ ]u¯2 −2ǫm (cid:3)u¯ u¯2
u¯y¯ y¯ u¯y¯ u¯ y¯ u¯u¯ y¯ y¯ x u¯u¯ x¯ y¯
−3ǫm u¯ u¯ −ǫm u¯
u¯y¯ x¯ y¯ y¯y¯ x¯
E =(1+ǫm u¯ )2
u¯ x¯
Any solution φ(x,y,t,u) = 0 of the heat equation (15) generates hereby the
solution φ(x¯ +ǫm(u¯,y¯),y¯,t¯,u¯) = 0 for the nonlinear equation (16) with the
coefficients (21).
9
3.3 On the subgroup: ξ2 = m(u,x), ξ1 = ξ3 = η = 0
Completely similar to the previous subsections, the infinitesimal generators on
the vector field of (4) yield the transformations
x¯=x, y¯=y−ǫm(u,x), t¯=t, u¯=u,
u +ǫm (u,x)u u u
x x y y t
u¯ = , u¯ = , u¯ = ,
x 1−ǫm (u,x)u y 1−ǫm (u,x)u t 1−ǫm (u,x)u (22)
u y u y u y
f ǫ(m (u,x)+ǫm (u,x)u )g
f¯= , g¯=g− x u x .
1−ǫm (u,x)u 1−ǫm (u,x)u
u y u y
As the procedure is already given in detail in the previous subsections with
many examples, here we will only consider the following example which is a
little different than the previous ones.
Example 6: Even though the first order PDE,
u +u −u =0
x y t
is notadiffusionequation,we maystillexamine itunder the equivalencetrans-
formations generated for the diffusion equation (1) by taking f = g = u. The
following class of nonlinear equations are mapped onto the givenconstant coef-
ficient PDE under the transformation group (22):
u¯x¯+(1−ǫmx¯(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¯)=0
from the general solution of the linear equation u = ψ(t+x,y−x). Here we
should warn the reader that even though the solution of the linear equation is
its general solution, the transformed solution is not the general solution to the
nonlinear equation.
3.4 On the subgroup: ξ1 = ξ2 = ξ3 = 0, η = m(x,y,t)
Asthelastcase,hereweshallconsidertheequivalencetransformationsoflinear
ornonlinearequationswhichdo notmapbetweeneachother,butmapone into
another with different coefficient functions. We can construct many of such
transformations,butasanexampleherewillinvestigateonesubgroupbytaking
the infinitesimal generators drive the following equivalence transformations
x¯=x, y¯=y, t¯=t, u¯=u+ǫm(x,y,t),
u¯ =u +ǫm (x,y,t), u¯ =u +ǫm (x,y,t), u¯ =u +ǫm (x,y,t),
x x x y y y t t t
f¯=f +ǫ m (x,y,t)dx, g¯=g.
t
Z
Under such transformations f +g −u =0 is mapped onto
x y t
f +ǫ m (x,y,t)dx +g −u =ǫm (x,y,t).
t y t t
(cid:20) Z (cid:21)
x
10