DI‘FFEREN'HAL EQUATlONS INVARIANT
UNDER ONE-PARAMETER
TRANSFORMATION GROUPS

Thesis for the flegree of Ph. D.
MICHIGAN STATE UNIVERSITY
SUCHAT CHANTIP
1973

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This is to certify that the
thesis entitled
DIFFERENTIAL EQUATIONS INVARIANT

UNDER ONE-PARAMETER
TRANSFORMATION GROUPS

presented by

Suchat Chantip
has been accepted towards fulfillment

of the requirements for

PhoDo degree in Mathematics

[ Major professor

Dam 11-2-73

0-7639

 

 

 

    
 

   

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" ammua av
m .. m
we.
UBRARY BINDERS '
gullogopy. mam“;

ABSTRACT

DIFFERENTIAL EQUATIONS INVARIANT UNDER

ONE-PARAMETER TRANSFORMATION GROUPS

BY
Suchat Chantip

The present thesis is concerned with differential
equations which are invariant under one-parameter trans-
formation groups. After an introduction and some back-
ground material this idea is introduced in section 3, in
which a definition of invariance of general differential
equations (O.D.E.'s, or, P.D.E.'s) is given. This defini-
tion is a generalization of Lie's definition of invariance
of the first order ordinary differential equations. The
author derives a criterion for invariance of differential
equations under one-parameter transformation groups. It
is shown in section 4 that this definition can be reduced
to Lie's definition of invariance of linear homogeneous
partial differential equations of the first order. The
author also gives in section 5 a definition of invariance
of systems of differential equations and obtains a cri-
terion. Section 6 is a method of determining the one-
parameter transformation groups leaving the given differen-

tial equations invariant, which utilizes the obtained criteria.

Suchat Chantip

In section 7, the author gives a new proof of
Lie‘s theorem of reduction of order of ordinary differen-
tial equations. Section 8 is the discussion of Morgan's
theorem of reduction of the number of independent vari-
ables in partial differential equations. This theorem
is generalized in this paper. In section 9, the author
uses the groups found in section 6 together with the
modified Morgan theorem to reduce independent variables
in the system of equations of nonsteady rotational plane
flow of incompressible fluid. The author also obtains
some classes of solutions of this system. In the last
section there is obtained a simplification of the form
of the system of differential equations of plane flow
of polytropic gas. The author starts by reducing the
system to a canonical form and then finds the one-
parameter transformation group leaving the canonical
system invariant and finally uses the obtained group to
reduce the canonical system to a system of ordinary

differential equations.

DIFFERENTIAL EQUATIONS INVARIANT UNDER

ONE-PARAMETER TRANSFORMATION GROUPS

BY

Suchat Chantip

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1973

AC KNOWLEDGMENTS

I wish to express my gratitude to Dr. Robert H.
Wasserman for suggesting this investigation, for his
kind consideration, and for the use of his library during
the research. I also wish to express my gratitude to Dr.
Carl C. Ganser for his suggestions and comments.

I am grateful to the Thai Government for the
financial support throughout my master and doctoral

programs of study.

ii

TABLE OF CONTENTS

Section Page
1 . INTRODUCTION . . C C . . . C . . . . C C O C C 1
2. ONE—PARAMETER TRANSFORMATION GROUPS .'. . . . 6

3. INVARIANCE OF DIFFERENTIAL EQUATIONS UNDER .
ONE-PARAMETER TRANSFORMATION GROUPS . . . . .19

4. THE LINEAR HOMOGENEOUS PARTIAL DIFFERENTIAL
EQUATIONS OF THE FIRST ORDER . . . . . . . 28

5. INVARIANCE OF SYSTEMS OF DIFFERENTIAL
EQUATIONS UNDER ONE-PARAMETER TRANS-
FORMAT ION GROUP S C O O O O O O I O O O O I 3 3

6. DETERMINATION OF ONE-PARAMETER
TRANSFORMATION GROUPS LEAVING
GIVEN DIFFERENTIAL EQUATIONS
INVARIANT .'. . . . . . . . . . . . . . . . 45

7. REDUCTION OF ORDER OF ORDINARY DIFFERENTIAL
EQUATIONS O O O I O O O O O O O O C O O O O 7 4

8. REDUCTION OF NUMBER OF INDEPENDENT
VARIABLES IN PARTIAL DIFFERENTIAL
. EQUATIONS O O O O C O O C O O O O O O I Q I 8 4

9. SOME SOLUTIONS OF THE SYSTEM OF DIFFERENTIAL
EQUATIONS OF NONSTEADY ROTATIONAL FLOW
OF INCOMPRESSIBLE FLUID . . . .,. . . . . . 94
10. REDUCTION OF INDEPENDENT VARIABLES OF THE
EQUATIONS OF STEADY PLANE FLOW OF
POLYTROPIC GAS . . . . . . . . . . . . . . 106

BIBLIOGRAPHY . O O O O O O O O O I O O O 0 O 0 O I O 120

iii

l . INTRODUCTION

The idea of integrating differential equations with
the aid of continuous transformation groups was first con-
sidered by Sophus Lie (1842-1899), the founder of the
theory of continuous transformation groups. Lie discovered
magnificent methods of integrating ordinary differential
equations of the first and the second order and linear
partial differential equations of the first order. More-
over, he discovered an important theorem, the theorem of
reduction of order of ordinary differential equations. These
methods and the theorem are based on one-parameter trans-
formation groups. All of such work of Lie is given by G.
Scheffers in the book entitled Differentialgleichungen
[1]. Later, the Lie theory of differential equations was
published in English by many authors namely A. Cohen [6],

E. L. Ince [7], L. E. Dickson [8], and K. O. Friedrichs [9].

In 1950 G. Birkhoff [5] suggested that the con-
tinuous one-parameter transformation groups could be used
to reduce the number of independent variables of some
partial differential equations. At about the same time
A. J. A. Morgan [3] and A. D. Michal [4] published results

on reducing the number of independent variables in systems

of partial differential equations, which constitute a
generalization of Birkhoff's suggestion.

The present paper deals with differential equations
which are invariant under one-parameter transformation
groups. Section 1 is introductory, Section 2 contains a
general idea of the theory of one-parameter transformation
groups.

In Section 3, Lie's idea of invariance of differ-
ential equations is generalized by giving a definition of
invariance of general differential equations (general
O.D.E., or general P.D.E.) under one-parameter transforma-
tion groups. This definition is a generalization of Lie's
definition for invariance of the first order ordinary dif—
ferential equations. A theorem which gives another
property of differential equations is proved.

The new property is called a criterion of invariance of
differential equations under one-parameter transformation
groups. Our criterion is the same as Lie's criterion for
ordinary differential equations of the first and the second
order, even the ways of obtaining the criteria are different.

In Section 4, it is shown that the definition,
which is set in Section 3, can be reduced to Lie's defini-
tion for invariance of linear homogeneous partial differen-
tial equations of the first order.

In Section 5 a definition of invariance of systems

of differential equations is defined. As in Section 3, we

obtain a criterion. It is shown by example that the invari-
ance in our sense agrees with the invariance of differential
equations in the sense of H. A. Lorentz; that is, agrees
with the invariance of Maxwell's equations under Lorentz's
transformations.

Section 6 contains a method of determining the
groups for given differential equations. As one-parameter
transformation groups furnish us a new tool for integrating
differential equations and for simplifying the work of
integrating differential equations, the methods of finding
groups for given differential equations are important.

Many authors developed different methods of finding such
groups. Lie [1] actually started with given groups and
found the class of all ordinary differential equations
which are invariant under each of those groups. Thus,
having a large table of classes of differential equations
and their groups, one could try to find in it the groups
corresponding to a given ordinary differential equation.
L. V. Ovsjannikov [l4] discovered an algebraic method for
determining groups. M. Z. V. Krzyworblocki and H. Roth
[15], who paid attention to Morgan's method of reduction
of independent variables in partial differential equations,
developed a method of determining groups by finite trans-
formations. G. W. Bluman and J. D. Cole [13] found a
method of finding infinitesimal transformations in their

work of finding similarity solution of heat equation. The

method of finding groups in this paper, enable us to find
all groups leaving given differential equations (O.D.E., or
P.D.E.) invariant. Our method is the utilization of the
criteria of invariance of differential equations and of
systems of differential equations (Theorem 3.2 and Theorem
5.1), which make use of the extended groups of one-
parameter transformations.

In Section 7 a new proof of Lie's theorem of
reduction of order of ordinary differential equations is
given. This proof is clearer than the original proof of
Lie ([1], pp. 386-387). However, the interesting matter
in this section is a lemma, which is a key for proving
Lie's theorem. This lemma shows an important property
of one-parameter transformation groups.

Section 8 contains a discussion of Morgan's
theorem of reduction of the number of independent vari-
ables in systems of partial differential equations. This
theorem is slightly modified so that the more general
groups (one-parameter transformation groups) can be used
in place of the groups which appear in Morgan's theorem.
And so the more general classes of solutions of partial
differential equations can be obtained from Morgan's
method.

In 1961 E. A. Mfiller and K. Matschat [12] applied
transformation groupstx>the equations of one-dimensional

nonsteady flow of isentropic gas, and obtained some exact

solutions. In Section 9 of this paper, one-parameter
transformation groups are applied to the system of equa-
tions of nonsteady rotational plane flow of incompressible
fluid to reduce the number of independent variables and
eventually obtained some exact solutions of the system.

The last section, Section 10, we apply one-
parameter transformation group to the system of equations
of steady plane flow of polytropic gas to reduce it into
a system of ordinary differential equations. A reduction
of independent variable in this system has been made
before by P. Kucharczyk [16], who uses Lie derivatives
to find coordinate system in which this system can be
transformed into a system of ordinary differential equa-
tions. Our reduced system is simpler than that obtained
by Kucharczyk.

Following the spirit of Lie, work in this area
has focused relatively more on algebraic and geometric
properties of differential equations than on analytic
properties In particular, domain of definition, continuity
and differentiability properties, etc. of functions which
are introduced are implicitly assumed to be adequate for
each stage of each argument. Usually, no explicit assump-
tion about these properties are inserted in the course of

the discussion. We shall follow this spirit.

2. ONE-PARAMETER TRANSFORMATION GROUPS

Consider parametric transformation in k-

dimensional space

k

21 = ¢l(zl, ... , z , a) (i = 1....,k)

(2.1)
where a is the parameter. The above set of transformations
is called a one-parameter transformation group if the fol-
lowing properties hold:

1) The T's are continuous functions of their
arguments, and the Jacobian of the T's with respect to
the 2's is not zero, i.e.,

det (33-59:) + o,

323

which implies that we can solve (2.1) for the 2's in terms

of the E‘s in the form

21 = *®l(E-l' no. I Ek] a) (i = 1' ... ' k)

1

2) For values a1, a of a such that 5' = ¢l(z, a1),

2

:1 = 91(5, a2), (i = l, ... , k), there exists a function
f(al, a2) such that

=i
2

ll

...;
‘

O

O
‘

7V
V
O

: ¢i(-z-r 3.2) (1)1(2’ f(all 32)) (i

3) There exists a value of a, say a0, corre-

sponding to the identity transtrmation, i.e.,

z1 = ¢l(zl, ... , zk; a (i = l,...,k)

O)
‘ 4) For any value a of the parameter which yields
a transformation from a point z to a point 5, there exists

a* corresponding to the inverse transformation from z to z,

i.e., we have
4’

i -1 -k i -l -k

i
(2,...,z;a)=<l>(z,...,z;a*).

z = *®
L. P. Eisenhart has shown,1 in the case of the
set of transformations (2.1) form a group, that the deriva-

tive of the T's with respect to the parameter a can be

written in the form

 

-i i . _
(2.2) 9525- = 3" gg' 3‘) = €l(z)-A(a) (i=l,...,k)

for some functions gl(2) and A(a). After defining a new .

parameter t by

the relations (2.2) become

-i i .
(2.2!) gag? = 34’ (tha(t).) ___ 51(2).

 

 

1[2]I p. 320

Observe that the value t = 0 gives the identity transforma-

tion.

From now on, we shall assume that the set of
transformations (2.1) has the mentioned group properties.
For some advantage, we substitute the a in (2.1) in terms

of t and denote the results by
(2.1') G: 2 = ¢ (2 , ... , z ; t) (i = l,...,k),

where the symbol G indicates that the transformations
form a group.

Let us expand the O's in (2.1') as Taylor series
at t = 0, i.e., at the value of t which yields the identity

transformation:

 

 

 

 

. i

1 _ i 8(1) (Zpt)
4) (Z, t) — ¢ (2: O) + t[ at ]t=0

+ Ei.[32¢l(z' t)] + ,
' = O. .0
from which we obtain
-i i 3¢i(z t)
... I
' = 0.... O

For a small change of t from 0, say 6t, so that the powers

greater than one can be neglected, (2.3) becomes

. i
_ 3(1) (Zr t)
z + 6t [ 3t ]
With consideration of (2.2') the last result can be

written as
(2.4) 2 = z + €l(z)-5t (i = l,...,k)

This is called the infinitesimal transformations of (2.1').
The relations (2.4) tell us that the vector ERZ) is tangent
to the path of transformation. We call such path the
trajectory of the group. Thus the trajectories of the

group are characterized by

 

(2.5) :21 = —%33— = . . . = —%EE— .
E (2) E (Z) 5 (z)

. . . -l
Con81der a continuous function f(z , ... , z )

which is composed with the group, i.e.,

f(El. ... . 2k) = f(¢1(z, t). ... . ¢k(z.t)).

Expanding this function as a power series of t, we have

2 2 -
(2.6) f(2) = f(z) + t[df(z)]F +%T [9—£§3L]t=0 + ...
° dt
n n -
t d f(z)
...'*"I"1-!--|:--—-—---'-n ]t=0+..... .

dt

Since

10

 

 

[arm] =l: 1; af(§) dgi] = [1; 328;)- 51(2)]
dt. t=0 . "l dt t;0 - '1 t=0

i=1 32 1:1 32
= 1; g1(2) 3f(2)
i=1 321
and denoting
(2.7) X = Z Ei (Z) '—_—i-
i=l 32
or
k i a
(2.7') x = 2 a (z) ——; .
i=1 32
we have
[dg‘2)1t_ = [Yf(z)1t=0 = Xf(z).
2 - _ __ __ _
[933§311t=0 = [§% x f(z)]t=0 = [x (x f(z))1t=0
t
2
= X(Xf(z)) == X f(z),
n —
[ddfgz>1t=o - [§% in 1f(E)1t=0 = [i“f(z)1t=0 = xnf(z).
t

Then (2.6) becomes

ll

2

(2.8) f(2) = f(z) + tXf(2) + gT-x2f(z) + ...
n
... + %T an(z) + ..... .

Setting f(z) = 21 in (2.8), we get

2

(2.9) 21 = zi + tEi(z) + §T.xgi(z) + ...
n .
... + £7 xn'lgl(z) + ... (i = l,...,k)

This is the other form of (2.1').

Absolute Invariants

Definition: A function u(zl,...,2k) which is

 

unaltered by all transformations of the group (2.1), that
is, such that u(21,...,2k) = u(zl,...,zk), is called an
.absolute invariant of the group.

There is a theorem helping us to find absolute

invariants for a given group of transformation, that is:

Theorem 2.1 ([2], p. 62): A necessary and suffi-

 

cient condition for a function u(zl,...,zk) to be an abso-

lute invariant of a group generated by X = 51(2)-iI-+ ...
32

... +€k(2)-2E- is that
32

3“(z) + ... + gk(z) EElEl-= o.

32 32

(2.10) Xu(z) = €l(z)

12

dz1 d2k
Since any integral of = ... = is a solution of

l k
E (2) E (2)
(2.10), it follows that the absolute invariants of the

 

 

group X can be found from the system of equations

. 1 k
(2.11) dz = . . . . . = dz .

51(2) «5km

 

 

Note that the system (2.11) is the system of differential
equations for the trajectories of the group X (cf. (2.5)).
We also note that, since there are only k-l independent
solutions of the equation (2.10), there are only k—lindepen-

dent absolute invariants of the given group X (see [2] , p. 62) .

Extended Groups

Given a one-parameter group of transformations

i l m l n
¢ (X ,...,x : y ,.--.y ; t)

(2.12) G: (

l l n
= wr(x I"'Ixm' Y I---IY 7 t)

X
ll

i=l'000’m)

r=1,...,n

Suppose the y's are considered as functions of the X's.
Then these transformations induce definite transformations
of the derivatives of y's with respect to the x's of the

form:

V; = wr'uxj. ys, y? . t)

J
-r ' ..' ' s s s
YI . I = wr'll 19 (X3: y p yjI-O-IYj ..j 3 t)
1 ° 9 1 8

 

 

i, 11, .. , 16’ j, jl' .. , 36 = l,...,m;
r, s = l, ... , n
where
-r s e-r
-£E§L.,YSE§'ZTI§£ *2 tayr

1 3x1 3 3x3 l1"19 sill..ax19

and
Y5 : aeys
J1"36 3x31..ax36

The method of determining the functions wr'i,...,wr'il"ie
is described by L. P. Eisenhart ([2], pp. 102-104). And it
is shown by him that the induced transformations (2.13)
have the same group properties as the given transforma—
tions (2.12). So, the set of transformations (2.12)
together with (2.13) form a one-parameter transformation

group and will be called the extended group (of order 6)

 

of G, and so will be denoted by G(6)' i.e.,
r" . . , S
x1 = ¢1(x3, y ; t)
§r = wr(x3. ys; t)
(2.14) G(ef( . .
§§ = wr’l(xj. ys. yg; t)
J
-r ' ..i ' s s S
Y: i = wrrll 6(XJ' y ' yj,...,yj j 3 t).
L. 1" 6 1" e

 

l4

wr,il..i

 

 

 

 

The formula for determining the functions a+l
from wr,11..1a is that
r,il..ia r'il"ia . r’il"ia
3
(2.15) W? 1 + 31. s y: + a), s Ygi
3x 3y 3yj 3
r,i ..i
l
.3)” .y% .
1" a
m k k
-r 3 3 s
-ZYIIIEJT+_(P's-°yi=o
k=1 1" a 3x 3y
(j, jl’°°°'ja = i,...,m ; s = l,...,n)

where summation convention is used in (2.15). This equa-

tion can be written shortly as

 

r,i ..i
l a m k
dx k=1 1" d dx

When 1 runs from 1 to m, the system (2.15) gives m equa—

tions which we can solve for y; i I,...,§§ to
O. a

I a
l I.

l a

r'ilooial

get the functions w , ... , wr'll"lam.

Let the operator of the given group (2.12) be

m l n 3
(X ,...,X ,y 'ooopy )'—-'-

1 3x1

n
l 3
+ Z nr(X1'ooopxm' y IOO°Iyn) _—

l 3yr

15

where

3 i 1 m l n
g = [3? ¢ (X I°°°IX I Y I°°°Iy 7 t)]t=0

nr = [1 (Ir

1 1
at (X ,...,xm,y loo-ryn7 t)]

t=0

are known, t = 0 signifies the identity transformation of
the group (2.12). Denoting the operator of the group G(6)

in (2.14) by

(2.12) X(e) = x + znr'l '23—]: + + znr'll°'1e r3 ,
3Y1 3Y1 . i
l ' e

the coefficients nr’l,..., nr'll" 9 can be found directly

from the group G(3) if the transformation laws of G(6) are

known, i.e.,

r,i

_ 3 r,i
n - [3? W (

j S S.
X I y I Yj’ t)]t:0

r i ..i 3 r i ..i ' s s s
n ' l 8 = [5? w I 1 e (X)! Y I yjI-OOij '7t)]
We also can find nr'l,...,nr'll’°19 from the known operator

X by recurrent formula of Eisenhart ([2], p. 106):

 

 

 

 

r,il..i r,il..i
(2.18) nr’11°'lal = 3n i + 3n 8 . y:
3x 3y
r,i ..i r,i ..i
l l
+ an S . yS.i + .00 + an S ‘ ys 1
3y. 3 3y. . 31"3
J JINJOL
m r 3€k 3€k s
' 2 Y1 i k (“I + "3" Y1)
k=1 1" a 3x 3y
(j, jl"°"ja = l,...,m ; s = l,...,n)

where the summation convention is used in (2.18). We can

write (2.18) shortly as

 

(2 18') nr'i1'°iai = d” _ z r d:

' 1 _ Yi ..i k "‘I '
dx k—l l d dx

Commutators

 

Given two operators

n . .
3
X = Z €1(X1'ooo'xn) if , Y = Z nj(Xl,...,Xn)—3’
i=1 3x j=l 3x

we define the commutator of X and Y by

(2.19) [X, Y] = XY - YX.

As a consequence of (2.19), we have

(2.20) [Y, X] = -[X, Y].

17

The commutator can be written precisely in the form

(2.21) (x, Y] = (xn1 - ygl)—§—-+...+(xnn - ygn)—3— ,

3xl 3xn
Since by direct calculation:
n i 3 n 1 3 n i n 32
i=1 3x i=1 3x i=1 3:1 3x 3x3
n i a n i a n 1 n a
yx = Y( 2 g “I) = 2 (ya "“? + 2 g z n3 __31_1_,
i=1 3x i=1 3x i=1 3:1 3x 3x]
we have
n i a n i a
[X, Y] = XY - YX = Z (XT) ) ___‘f " 2 (Y5 ) ___]:-
1—1 3x i=1 3x

which can be written as (2.21).

Differential Invariants

 

A function f(x, y, y') which actually involves
the derivative y' is called a differential invariant of

the first order of the group of transformations:

G: i = ¢(x, y; t), § = ¢(X, Y; t);

if it is an absolute invariant of the group G the

(l)'
first extended group of the group G. In the same way,

a function F(x, y, y',...,y(k)) which actually involves

18

y(k) is called a differential invariant of order k of

the group G, if it is an absolute invariant of the
extended group G(k)' We then have a theorem (cf. theorem
2.1).

Theorem 2.2: A necessary and sufficient condition
for a function F(x, y, y',...,y(k)), which actually

(k)

involves y , to be a differential invariant of order
k of the group generated by X = €(x, 37);}? 4- (fix, 10.3.3.1;

is that

X(k)F(xr Yr Y',...,Y(k)) = 0

where X(k) is the kth extended operator of X.

3. INVARIANCE OF DIFFERENTIAL EQUATIONS
UNDER ONE-PARAMETER

TRANSFORMATION GROUPS

The definitions for differential equations to be
invariant under transformations are given by S. Lie for
the first and the second order ordinary differential
equations and for linear homogeneous partial differential
equations of the first order. His definition for the
first order ordinary differential equation is:

Definition 3.1 (Lie's definition, [1]. p. 101):

 

It is said that a differential equation
(3.1) M(x, y)dx - N(x, y)dy = 0

is invariant under the transformations
(3.2) i = ¢(x. y) .17 = ((x. y)

if its form is unaltered, save for a factor, by the trans-
formations, i.e., it may be written, in terms of the new

variables, in the form

96:. 5?) (Mai. §)d§ - mi. 17m?) = o.

Lie proved a theorem which gives another property-
equivalent to the property in the above definition.

19

20

Theorem 3.1 (Lie's theorem, [13, p. 101): The

 

form of the equation (3.1) is unaltered, save for a factor,
by the transformations (3.2) if and only if each integral
curve of (3.1) is transformed by (3.2) to some integral
curve of the same equation.

For invariance of the second order ordinary dif-
ferential equations, Lie made an analogous definition.

We shall now make a generalization of the Lie
idea of invariance of differential equations. Let us
consider a general differential equation (general O.D.E.,
or, general P.D.E.) of order r with independent variables

1

x ,...,xn (for P.D.E. n > 1, for O.D.E. n = 1) and dependent

variable y:

 

i
(303) F(X I YI yi,...,yi ..i ) = 0
1 r
where
_ a _ 3r
Y1 2 'AT ' Yi i z i 2* i ’
3x 1" r 3x 1"3x r

together with a one-parameter transformation group

' l
¢l(x ,...,xn, y; t)

(3.4) G: (i=l,...,n)

- 1
Y W(x ...-.xn. y; t).

-i
x

Let the rth extended group of the group G be

21

n

¢i(x1.....x . y; t) = ¢i(xj. y: t)

w(xll'°°lxnl Y: t) = W(ij Y; t)

Kl
H

J

. - i j .
G(r)°fi Yi 1D (XIYI Y-I t)

- i1..i O
Y: ? = W (X): Y! Y°'°°°'y‘

j rt)
1" r 3 31" r

 

 

where
.- rc—
_- = 3 §_ _ _ 'ay
1 -1 ' i1..1r _11 1r °
3x ..3x

Definition 3.2: It will be said that the differ-

 

ential equation (3.3) is invariant under the transformation
group G, (3.4), if and only if under the transformations of

G(r) the following relations hold:

(§J'§'coc'§7 '7': ,t))

i-j- -3'—
(3-5) F(X (X IYIt)IY(X IYIt)I°--IY- '
ll"lr 31'°Jr

= V(;ll §l t) F(§ll §I YII"°I§I ? ) '

or,

.i (XJIY000'IY' - ,t))

(3.6) F(§1(x3.y.t).§<x3:th)v--~v§1 J
l' r 1'Ojr

= “(XII YI t)°F(xlI YI Y°

1'00'Iyi ..i )I
r

1

where v and u are not zero.

22

Note that (3.5) and (3.6) are equivalent rela-
tions. We shall now derive a property equivalent to that
given in definition 3.1. First, we assume that for the
given differential equation (3.3) and the group (3.4),
the relations (3.5) or (3.6) hold. Since (cf. Section 2)
(3.7) [§%.F(§i, §, §I'°"'§I

? )1 _
1.. r t—O

_. _. _. d§ e
3F(xl,...) dx3 + + a§(x1,...) i1"3r
332 (E Byil. 3r at t=0

 

 

aF(xi{000)[3¢j 3F(xipooo) [awj1..jr]
t

+...+
3x3 3t]t:0 3yj at =0

lo.jr

i
X(r)F(x . Y: yi. ... .yi ..i )

l r

where t=0 signifies the identity transformation of the

group (3.4), X( ) denotes the operator of the group G(r),

r

the summation convention is used in (3.7); we have from

(3.6) and (3.7)

(3'8) X(r)F(Xl'Y'Y1"°"Yil..ir’

3 i i
[rt 1J(X IYIt)]t=o F(X IYIin'OOIYi i )
1" r

i .
A(x ,y)'F(x1,y,yi,...,yi i ), say.
1" r

23

Conversely, if the differential form

F(xl,y,yi,...,yi i ) in (3.3) satisfies a relation of
1" r
the form (3.8), i.e.,

i
(3.9) x F(X ’y’y-pooo’y. o )
(r) 1 11.. r

i i
= 2(x ,y)F(x ,y,yi,...,yi
1" r

we claim that the differential form F(x1,y,yi,...,yi )
1

also satisfies the relation (3.6). Since (cf. Section 2)

..i
r

- t t2 2
(3.10) F(V) = F(V) + FX(r)F(v) + z—l-X(r)F(V) + co.
k
k
°°'+ETX(I‘)F(V) +... ,
where F(5) E F(§i,§, §e,...,§e e ),
1 ll..1r
F(V) E F(X1IYIYiI---Iyi i )I
1" r

and since

F
2 _ _
X(r)F(v) — X(r)[£(x,y)F(v)] — [X(r)£(x,y)
+ £(X.y)°£(X.y)]F(v) = 21(X.y)F(v)
(3.11)$ .u.............................
k _ _
+ £k_2(X.y)-£(X.Y)IF(V) = £k_l(X.y)F(V)

 

24

we obtain by substitution from (3.9) and (3.11) into
(3.10)
- t2 tk
F(v) = [1 + t£(x,y) + -2—1- 21(x,y) +...+ ET £k__l(X.y)

+ ...]F(v) = m(x,y,t)F(v)

where m(x,y,t) denotes the sum in the bracket which is

not zero. Thus the F(v) satisfies the relation (3.6),

and the claim is proved. We now have conclusion:
Theorem 3.2: Given the differential equation

F(xl,y,yi,...,yi ) = 0 and the group of transforma-

..i
1 r
tions

G: ;1 r ¢1(XJI YI t)! § = W(XJIYI t);

the relations

(;JI§I°°°I§?

i-j- -j- -
(3.5) F(x (x .y.t). y(x 'y't)""'y11"ir 31..jr.t))

= v(;ll§lt).F(;1I§I§i'.°'l§i T )

or

(3-6) F(;1(XJIYIt)I §<XJIYIt)I°°°I§i 7 (XJIYI°°°IYj ' It))

loolr loojr

i i
= U(X IYIt)'F(X IYIYiI'°°Iyil..ir)

where v and u are not zero, hold if and only if a relation

25

(3.8) X(r)F(xl.y.....y. ..i ) = A(x1.y)-F(xl.y.....y. )

1l r l1"lr
holds fin:some function A. In other words, the differential
equation F(xl,y,yi,...,yi . ) = 0 is invariant under the

..i
1 r
group G if and Only if the relation of the form

i i '
X(r)F(x IYIyi10‘01yi - ) = A(X (Y)F(X1IYIin°°°Iyi )

l

1.. r l..lr

holds.

We shall call the relation (3.8) the criterion
for differential equations to be invariant under one-
parameter groups of transformations. Lie obtained the
same criterion as (3.8) for the invariance of ordinary
differential equations of the first order, although the
ways of obtaining the criteria are different. Actually,
Lie obtained the criterion by making use of his theorem
3.1.

Example 3.1 (from [1], p. 278): Lie found that

 

the differential equation for a one—parameter family of

circles (x - a)2 + y2 = r2 where r is fixed, is

(3.12) F(v) 5 Y2(l + y'2) - r2 = 0.

Since this family of circles is invariant1 under the group

of translations parallel to the x-axis, i.e., the group

 

lBy invariance of a family of curves under a trans—
formation we mean that each curve of the family is trans-
formed into some curve of the same family.

26

3
(3.13) X = 5;" I
Lie concluded (by using theorem 3.1) that the differential

equation (3.12) is invariant under the group (3.13). From

(3.13) we find the following:

_ 3 _
(3.14) X(l) — 5; — x,
(3.15) G: §=x+t,§=y,

-_ -_ d§=§1
(3.16) G(1)° x — x + t, y — y, d; dx .

Then, by substitution, we have

F(V(\7))

172(1 + (gig-)2) - r 1-F(\7).

F(x7(v)) y2(1 + (gy) - r 1oF(v).

We also have from (3.14) and (3.12)

X(1)F(V) E 0 = O-F(V).

Example 3.2: Consider a differential equation
(3.17) F(v) :~.'.- yu - u = O,
and a group of transformations

_ a a
(3.18) X - a—X- + qu-l-l- c

27

We obtain from (3.18) the following:

_ 3 3
(3019) X(z) _X+yuXW+ coo +yuXXW—'+ coo ,
x xx
_ _. _ ty
(3.20) G: x = x + t, y = y, u = e u ,
- - - t - t
(3.21) G(2): x = x+t, y=y, u=e yu, u§=e yux,.. ,
E-- = etyu
xx xx
We then have
F(v(\7)) = §-e‘tya- - e'tYa-—=e'tY-F(x7) .
x xx
’ = . t-Y .. W = tY.
F(v(v)) y e uX e uXX e F(v) ,
X(2)F(v) = y-yuX - yuXX = y-(yuX - uxx) = y-F(v).

Thus the equation (3.17) is invariant under the group (3.18).

4. THE LINEAR HOMOGENEOUS PARTIAL
DIFFERENTIAL EQUATIONS OF THE

FIRST ORDER

S. Lie gave a definition for the linear homogeneous
partial differential equations of the first order to be
invariant under transformations as follows:

Lie's Definition ([1], p. 311): It is said that

 

the differential equation

(4.1) A s al(x1,...,xn) 2l’I+...+ an(xl,...,xn) 3L: o
y 3X an

is invariant under the transformations

(4.2) 21 = ¢i(xl,...,xn) (i=l,...,n),

if these transformations preserve its solutions.

Observe that the transformations in the above

definition are the transformations of independent variables

only, while the set of transformations in our definition 3.2
involve the dependent variables also.

It is our purpose here to verify that our defini-
tion 3.2 can be reduced to the above Lie definition. That
is, for the special case where our differential equation

(3.3) is of the form (4.1) and our group (3.4) is of the form

28

29

n

_- ' 1 .
xl ¢l(x ,...,x , t) (1=1,...,n)

(4.3)

Y = Yr

if differential equation is invariant under the group
according to our definition, then it is invariant under
the group according to Lie's definition. The definition
of Lie that differential equation (4.1) is invariant
under the group (4.3) is the same as the above defini-
tion. That is, the differential equation (4.1) is
invariant under the group (4.3) if the transformations

of the_group preserve its solutions. Lie proved directly

 

from this definition a theorem which gives a criterion
of invariance of the equation (4.1). Before stating this
theorem, let us point out some consequences of the group

(4.3). The operator of the group (4.3) is in the form

(4.4) x = 51(x1,...,xn) ‘31”
3x

n l n 3 3
ooo+€ (X ,ooo’X)—IT+O76—y_
3x
where
' l ' 1
51(x .....xn) = [3¢l(x ,...,xn: t)/3t]t=0.

Then the first extended operator X(l) can be found, with

the help of the formula (2.18), in the form:

30

n 35k 3
(405) x =x- 2 y ' - 000
(l) k=1 k 3xI 5y1
_ 2 y .333 _§_
k=1 k ax“ 8yn
where y. E 33; .
l X1

Lie's theorem ([1], p. 316): The differential

 

equation (4.1) is invariant under the group generated
by the operator (4.4) (i.e., the group (4.2)) if and

only if a relation of the form
1 n
(4.6) [X, A]y = 0(X I"°IX )AY

holds identically.

If we can show that our definition 3.2 implies
the property (4.6) in the Lie theorem, then we can con-
clude that our definition 3.2 can be reduced to the Lie
definition. We shall now show that implication. From

(4.5) we have

_ l n
(4.7) X(1)Ay — X(l) {a yl + ... + a yn}

1 n

n k n k
+ a1 - Z yk ééf + ... + an - Z yk fiéfi- .
k=1 3x -. k=1 3x
n . l
X(l)d1 - Z a1 3;: yl
i=1 3x
n n
n 1 35
+0.. + X G. 2 O. ——'l" y I
(1) i=1 3xl n

. i . 1 n
Since a are functions of x ,...,x only, we have from

(4.5) that x )a1 = Xal. Thus (4.7) becomes

(1

(4.8) X Ay = (X04l - A51) ifi; +...+ (X04n - Ag“) in; .
(1) 3x1 3xn

And thus, by (2.21), (4.8) can be written as

(4.9) X(1)Ay = [X, A]y.

Since we assume that the equation (4.1) is invari-
ant under the group (4.3), by theorem 3.2 we have a rela-

tion of the form
1 n
(4.10) X(1)Ay = A(X 'ooo'X , Y)Ay

for some function A. Since from (4.8) and (4.10), we

must have the relations of the form:

32

(4.11) A(x. y) ai(x) = Xai(x) - Aai(x)

for i = l,...,n. The right hand member of (4.11) is a

function of xl,...,xn. Thus, the A is not a function of

y, i.e.,
l n
(4.12) A = A(x ,...,x ).
Now from (4.9), (4.10) and (4.12), we have
1 n
[X, A]y = .A(x ,...,x )Ay,

which is a relation of the form (4.6). Therefore, our

definition 3.2 can be reduced to Lie's definition.

Remark: We see from (4.9) that the relation

(A, X]y = p(xl,...,xn)Ay implies the relation

1 n
X(1)Ay = p(x ,...,x )Ay.

5. INVARIANCE OF SYSTEMS OF DIFFERENTIAL
EQUATIONS UNDER ONE-PARAMETER

TRANSFORMATION GROUPS

Consider a system of differential equations
(P.D.E.'s, or, O.D.E.'s) of order 8 with independent
variables x1,...,xm (for P.D.E.'s m > 1, for O.D.E.'s

m = l) and dependent variables yl,...,yn:

i r r
F1(X I Y I in 0-0 I Y: ..i ) = 0
1 8
(5.1) .....OIOOOIOOOOOOOOOOOOOOOOO.
i i r r r
[F2(x ' Y ' Yi' °'° ' Y1 ..i ) ' 0
1, l 8
. i i
where y: E 3yr/3xl, y: i E 3eyr/3x l..3x 8. We asso-
1" 8
ciate the system (5.1) with a one-parameter transformation
group
r‘
-i _ i l m 1 n _ i j s
X _ ¢ (X ,...,X [y ,...,y ; t) : ¢ (x [Y ; t)
(5.2)02
- r 1 m l n _ ' 5
(fr = 0 (X ,...,x ,y ,...,y ; t) : wr(x3,y ; t)

 

(i, 3.1-l,...,m; r,s=l'ooo’n)

and so with the extended group G of G:

(8)

33

34
= ¢1(XJI yS; t)

wr(xj, yS; t)

. -r _ r,i j s s.
G(9).< yi 1!) (X I Y I yjr t)

 

-r i ..i ' s S
Y? 'c- = wr' l 8(X3 ’y 'yo’ooo,yso ° ; t)
11"19 3 31"38
L.
(ilill'°'lieljljll°'°lje = lIoo-Im; rIS = ll°°°ln)°

Definition 5.1: It will be said that the system
of differential equations (5.1) is invariant under the
group G if and only if under the transformationsof the

group G(8)' we have for a = l,...,l

(5.3) Fa(x1(§3.§s.t).yr(§3. §S.t)....

..., y: .._i (i ,§S.....§§ ,_.3 ))
1 e 1 9
£ -i -r -i -r -r
= 3:1 va8(x ,y ,t)‘fi§x ,y ,...,Yil..ie)
with
(5.4) det(va8) + 0,

or, equivalently

35

(5.5) Fa(xl(xj,ys,t), §r(x3,ys,t), ...

- j s s
0.0, Y? 7 (X ,y ,...,y. o , t))
11.018 31.036
£ i r i r r
= 2 u (x .y .t)'F(x .y .....y- . )
8:1 as lloole
with
(5.6) det (“38) + 0.

Remark: The conditions (5.4) and (5.6) are suffi-
cient to imply the equivalence between the relations (5.3)
and (5.5). Moreover, the condition (5.4) assures us that
the transform of the system (5.1) will be a system of 2

independent equations of the form

 

 

(-
2 -i -r -i -r -r
z \) (x y 't) .F (X 'y ,...,y-o- '0- ) = 0
B=l lB ' B 11"18
3 g -i -r -i -r -r
E V (X Iy It)°F (X Iy IO-OIYT 7 ) = 0
L6:1 28 B 11"18

1 )appears on the left hand
1" 8
side of the above system. A similar system of independent

where every Fa(§1,§r,...,§§

equations is obtained, under the condition (5.6), when

the inverse transformation applied to the system

36

-i -r -r -r
F1(X .y [Yil "l I Y: ..i) = 0
1 6
-i -r -r —r
F£(x 'y 'y-i-I 0.. ' YIl..:e) = 00

We shall now obtain another property of invariance
of system of differential equations under one-parameter
transformation group. This new property will be called a
criterion of invariance of system of differential equations
under one-parameter transformation group.

Theorem 5.1: The system of differential equations
(5.1) is invariant under the group G if and only if the

relations

i r r r
(5.7) X F (x 'y ’y-' 0.0 I y. o )
(9) a 1 11"18
£ i r i r r r
= Z >\ (X .y W (x .y ,y..-...y. -)

(OL = 1,..., 8)

hold for some functions AaB’ where X(8) is the operator of
the group G(8)'
Proof: We first assume that the system (5.1) is

invariant under the group G, then the relations (5.3) or

(5.5) hold. As in section 3 (cf.(3.7)), we have

37

V i r r
(5.8) X(6)Fa(x ,y ,...,yi )

1.016

_ 5L —i j s -r _ 3 s s
‘ [dt Fa(x (X Iy It)I---Iyil..i6(x Iy ,...,yjlooje't))]t=0

Then from (5.5) and (5.8) we have

i r r
(5.9) X(e)Fa(X ,y ,...,yi )

1. .16

)

I
"Mb

3 i r i r r
l [El-108““ Iy It)]t=0 FB(X ,y ,ooo'yil..ie

Writing
i r _ 3 i r
ACIB (X ,Y ) _ [E 110:8 (X Iy It)]t=0

in (5.9), we obtain the relation of the form (5.7).
Conversely, if the relations (5.7) hold for the

given system of differential equations (5.1) and the

given group G, then we claim that the relations (5.3)

or (5.5) hold. Since we have from (5.7)

 

 

38

here faB E fa8(xl,yr), it follows that for positive inte-

ger k:
(5 10) xk F ( i r r )
' (e) ax'y""'yi..i
1 8
2 i r i r r
= 8:1 awe” .y WW (Y ~~~Yil..ie>
where deB E de' fldB E 1&8. From the fact that (cf.
(2.8))
- t t2 2
(5.11) Fa(V) = Fa(v) + IT X(8)Fa(V) + 2T'X(8)Fd(V) + .....
where
- : -i -r -r _ : i r r .
Fa(v) _ Fa(x ,y ,...,yil"ie)' Fa(v) - Fa(x ,y ,...,yil..ie),

we have after substituting (5.10) into (5.11):

39

F1(;) {1 + gll}Fl(V) + g12F2(v) +...+ 912F2(v)

 

F2(v) = ngFl(V) + {1 + g22}F2(v) +...+ g2£F£(v)
(5.12) 3

‘...“-"~' ”0-

F£(v) = g£1F1(V) + g£2F2(v) +...+ {l + g££}F£(v)

where

2 3

__L _ _
‘ 1: flaB + 1 fZaB + 31 f338 + ""' '

(1'

9&8

N

The system (5.12) is of the type (5.5) in definition 5.1.
Assuming that the functions Aa8(xi,yr) in (5.7) are con-
tinuous, it follows that fkaB are continuous functions
of x's and y's. Then the 906 are continuous functions

of x's, y's and t. Since gaB vanish at t = 0, by continuity

1 + g do not vanish in some neighborhood of t = 0, say
a8

 

 

Nt=0' Thus
l+g11 g12 "° 911
det 921 1+922 "° 922 + o
921 922 1+922
in Nt=0' This is the condition (5.6). Therefore, by

definition 5.1, the system of differential equations (5.1)

is invariant, in Nt=0’ under the group G.

40

Example 5.1:

 

We shall show that the system of

Maxwell's equations of an electromagnetic wave is invari-

ant (in our sense) under the group of Lorentz's transfor-

mations. So, our definition of invariance of a system of

differential equations (definition 5.1) agrees with the

invariance in the sense of Lorentz.

tions of electromagnetic wave are:

 

r-
: 3_ 2_1 1__
F1(v) _ Hy Hz c t-0, F2(v)
: 2_ 1_1 3__
“F3(V) _ HX Hy EEt- 0, F4(V)
(5.13) (,
z 1 3 1 2__
F5(v) — E EX+EHt-0, F6(v)
: l 2 3 _
F7(v) — Ex+Ey+Ez - 0, F8(v)
where H = (H1, H2, H3) and E = (El,

and electric

field intensity vectors.

The Maxwell equa-

: Hz Hx-EEt O

2 3_ 2 1 1=

_ Ey EZ+EHt 0

E E2-E3+-]-'-HB=0
x y c t

2 H1+H2+H3 = o
X Y

E , E3) are magnetic

The Lorentz trans-

formations, which form a one-parameter transformation

group, are

3E=b(x+cat), y = y, 2 =

(5.14) G: E = E , E

Cl:
ll
m
:1:
II

where a =

2, E =
b(E2 + aH3), E
b(H2 - aE3), H

u/c is the parameter such that a =

b(t + §ax),

b(E3 - aHz),

b(H3 + aE2),

0 gives iden-

tity transformation, u is the velocity of the observer

41

(see figure), c is the velocity of light, b = 1/ l-a .

The observer moves with his frame of
reference along x-direction

with velocity u.

Using (2.15), we find from (5.14) the extended group 6(1):

 

”G, is; = Maggi), is; = 13;, 1;: 3;,

5%: = b(Ei-caEi), ii = b2(E:-%aEi+ aH:-%a2H:) ,

E5 = b(E;+aH3), E2 = b(E: + aHi),

12-: = b2(E12_—--caE:+ aHi—cazHi) ,

E; = b2(E:-i Ei- aHi+éa2H:), E; = b(ES-aHi) ,

E3 = b(E3-aH2), fig = b2(E3~caE3-aH2+ca2H2),
(5°15) GHM -: i 12 1 t-l 1t -1 x 1 t X

H; = b(Hx-E-aHt), H37 = Hy, H; = HZ,

fitl: = b(Hi- caHi) , 171% = b2(H:-%:-aH:- aEi+éa2E13=) ,

E; = b(Hi - aES), E; = b(H: - aEz),

fig = b2(H:-caH: - aE:+-ca2E:)p

13% = b2(H:-%a}l:+ aE:--C1?a2E12:) , 1'1; = hm; + aEi):

L33 = b(H: + aEi), fig = b2(H:~caH:4-aEi-ca2E:).

Substituting the unbar variables in Fl(v),...,F8(v) in

terms of the bar variables from (5.15), we get

42

Fl(v<§)) = bFl(§) - ab F7<5>, F2(v(v)) F2<§).

F3(v(\-7)) = -F3(\7) , F4(v(<}>) = b-F4(\7) + ab-th'r).
F5(v<5)) = F5<G) . F6<v(5)) = F6<G),
F7(v(§)) = bF7(v) - abFl(v) , F8(v(v)) = bF8(v) + abF4(v),

which are relations of the type (5.3) in definition 5.1.
It is easy to check that the determinant of the coeffi-
cients of Fi(v) in the above system is not zero. That is,

the condition (5.4) is satisfied. From (5.15) we find the

operator X(l) to be
8 1 8 3 3 2 3 3 3 2 3
(5.16)X =ct +—x +H -H -E—+E
(l) ‘5}? C Ft- 8E2 3E3 3H2 3H3
1 1 8 l 3 l 2 3 3 3 8
-—E —-CE —— -(--E - H) + H
c t 3El x 8E1 c t x 332 y 3E:
x t x y
3 3 2 3 8 1 3 2 3
+ H -(cE - H ) ———-(—E + H )———
z 3E2 x t 8E2 c t x 3E3
z t x
_ H2 3 2 3 -(cE3 + H2) 8 __1H1 3 -cHl 3
y 3E3 2 3E3 t 8E3 c t BHI XBHI
Y z t x t
1 1 3 3 _ 3 8 3 3 2 3 3
'iaHt+Ex"‘7 E "'3' E2 ‘—7 < Hx+Et)"7
3H 8H 8H 3H
x z t
1 3 2 3 2 3 2 3 3 2 3
-(—-H-E)—+E +E -(cH-E) .
C t X 3H: Y 3H3 2 an: x t an:

43

From (5.13) and (5.16) we have
X(1)Fl(v) = F7(v), X(1)F2(v) E 0, X(1)F3(v) E 0,
X(1)F4(v) = -F8(v), X(1)F5(v) E 0, X(1)F6(V) E 0,
X(1)F7(v) = Fl(v), X(1)F8(v) = -F4(v)

which are relations of type (5.7) in the theorem 5.1.

Example 5.2: Consider the system of differential

equations

(

w + w + (n = O

F1(V) xx yy

(5.17) <

F2(V) wt + w w - w 0.

y x xwy =

 

This system defines nonsteady rotational flow of incompres-
sible fluid in (x, y)-p1ane, where w and w are respectively‘fiw
stream function and vorticity of the flow. The system

(5.17) is invariant under the group

_ a a a _ a
(5.18) X — 2t-a—t- '1' X5; + YE; 2mm

Derivation of this group is in the next section. Here,
we only show the relationship between this group and the

system (5.17). From (5.18) we find:

= x - 2w -3— - 3 4w '3"' 3w '3‘

3
(5.19) X ___ - W ___ -
tawt xawx yawy tam xawx

(2) t

3 3

-3w_+ooo-2w ‘21P __+000 0
3 5 3
y my xx wxx yy wyy

(5.20) G: <

 

f - _ -2a -_ _ -a -_ = -a
GI WE - e Wt: WX ‘ e wxr WY 9 WY,
L- -4 - -3 - -
(5.21) G(2f) wE = e a t' w = e awx, y — e 3awy,
- _ -2a -__ = -2a
K... r Wig — e WXXI wyy e wyyr 00- r

 

where a = 0 gives identity transformation. We now have

X(2)F (v) = -2w - 2w - 2w = -2(tpxx + w + w) = -2Fl(v)

1 xx yy yy

X(2)F2(v) = -4wt - wywx + wy(-3wx) + wxwy - wx(-3wy)

-4F2(v)

Substituting from (5.21) in Fl(v) and F2(v), we get

-2a -2a -2a _ -2a
e wxx + e wyy + e w - e F1(V)

F1('\-I(V) )

F2 (G M)

I
m
+
m
€-
m
8
I
m
€-
m
E
H
w
W

N
<1

which are the relations of the type (5.5) in definition

5.1. We also have that

det

+0,

e-4a

 

i.e., the condition (5.6) is satisfied.

6. DETERMINATION OF ONE-PARAMETER

TRANSFORMATION GROUPS LEAVING GIVEN

DIFFERENTIAL EQUATIONS INVARIANT

One-parameter transformation groups furnish an
important tool for integrating differential equations.
They are used in Lie's theory for integrating first and
second order ordinary differential equations, and first
order linear partial differential equations. Transforma-
tion groups also may simplify the work of integrating
differential equations, for example, in the reduction of
the order of ordinary differential equations and in the
reduction of the number of independent variables in
partial differential equations. In this section we
give a method of finding the groups of transformations

which are required for these applications.

The method here is to find one-parameter trans-
formation groups leaving the given differential equations
invariant. That is, given a differential equation (P.D.E.,

or, O.D.E.) of order r with independent variables

45

6. DETERMINATION OF ONE-PARAMETER
TRANSFORMATION GROUPS LEAVING GIVEN

DIFFERENTIAL EQUATIONS INVARIANT

The one-parameter transformation groups furnish
us with an important tool for integrating differential
equations and for simplifying the work of integrating
differential equations; the first case, according to the
Lie-group-methods of integrating ordinary differential
equations of the first and the second order, and linear
partial differential equations of the first order; the
later case, according to the theorem of reduction of the
order of ordinary differential equations and the theorem
of reduction of number of independent variables in partial
differential equations. To achieve the group-methods of
integrating differential equations and the methods of
simplifying the work of integrating differential equa-
tions, we propose to find the required groups of trans-
formations. '

The method here is to find one-parameter trans-
formation groups leaving the given differential equations
invariant. That is, given a differential equation (P.D.E.,

or, O.D.E.) of order r with independent variables

45

46

xl,..,xn (n > 1 for P.D.E., n = l for O.D.E.) and

dependent variable y:

i
(6.1) F(x , y, yi, ... , yi "ir) = 0

l

i
..Bx r; we look for a

i
E Bry/Bx 1

where yi By/Bxl, yi i
1" r

group of the form

1 l 3 l 3
(602) X = g (X I°"IXnIY)—I +000+€n(x I°'°Ixnly)—E
3x 3x
1
+ n(x ,...,xn, y)Ji-

3y

1

l 1 1
Where g (X ,...,Xn,Y)po..,gn(X ,...,XnIY) In(x [0"IxnIY)

are to be determined such that (cf. theorem 3.2)

(6.3) X(r)F(xi, y, y., ... ,y. )

1 1 ..i
l r

= Mxl. y)F(xl. y. y-

l’...'yi ..i)

l

where A(xl, y) is also to be determined. The extended

group X(r) is derived from X, i.e., if we denote X(r) by

i1..i 8
X =X+Zn 'g'y—+oooo+ Z .T) ray .
i 11,..,1r i1..1r

il..i
are calculated, with

then the coefficients nl,...,n
the help of (2.18), in terms of the derivatives of 5'3 and

n with respect to x's and y, and the derivatives of y with

47

)

respect to x's. Since the form of F(xl, y, yi,...,yi i
1.. r

is known, the left hand member of (6.3) is known in terms
of x's, y, 5's, n, the derivatives of 5's and n with respect
to x's and y, and the derivatives of y with respect to x's.
Since A is a function of x's and y, the equation (6.3)
enables us to equate the coefficients of the derivatives
of y with respect to x's. The result of equating these
coefficients is a system of partial differential equations
with n + 2 unknowns 5i, n, A(i = l,...,n); from which we
solve for 5i, n, and A.

In a similar manner, from a given system of differ-
ential equations

f.

 

i r r r
F1(X I y I yil°'°IYi .i ) = 0
. 1 9
(6.4) fl _
l r r r
L F24”: I y I yi'...'yi1"ie) = 0I

we can find groupsof transformations leaving the system

invariant from the relations (cf. theorem 5.1)

 

f-
l r r
X(6)F1(X , Y , coo [yil..ie)
_ z 1 r i r r
_ Z AlS(X ' y )FS(X I y ' on. ' yi i )
=1 10. e

(6.5) { s

21 O I
l r 1. r r
= 2 A£S(X , y )FS(X . y , ... . yil"ie)

 

g

The following are the examples of determination of
groups for some differential equations which appeared in
the previous sections, and some differential equations
which will appear later.
Example 6.1: Consider the differential equation

in example 3.1:

(6.6) F E y2(1 + y'2) - r2 = 0, r = constant,

which is the differential equation of a family of circles
(x - a)2 + y2 = r2. This family of circles is invariant

under the group of translations parallel to the x-axis:

(6.7) x — §%.,

Lie concluded, by using his theorem 3.1, that the differ-
ential equation (6.6) is invariant under the group (6.7).

Ignoring the knowledge of the origin of the equa-
tion (6.6), we propose to find the group leaving the

equation invariant by our method. Let the group be
X = €(x y)-a- + M}: Y)-§-
I UK I 3y I

and write

49

l 3
I

X = X + n By

By the formula (2.18) we find

1. l_ l l
(6.8) n — nX + nyy y (EX + Eyy >
= n + (n - a >y' - a (y')2.
x y X y
We need
X(1)F = A(X, y)F
or

(6.9) 2yn(l+y'2) + 2y2y'nl = A(x,y> [y2(1+y'2) - r21.

Substituting from (6.8) into (6.9), and equating the coef-

ficients of 1*, y', (y')2, and (y')3.

we get, respectiVely:

A
y Y

= n(y). The equation

(6.10) (coeff. of 1): 2yn = A(yz - r2)
. 2 _

(6.11) (coeff. of y ): 2y nx - 0
,2 2

(6.12) (coeff. of y ): 2yn - 2y (EX - n )
.3 2 _

(6.13) (coeff. of y ): 2y 5y — O.

The equation (6.11) implies n
(6.13) implies E = €(x).

 

Then, the equation (6.10) gives

*By the coefficient of l, we mean the terms not
including any derivative of y with respect to x.

50
A = A(y). And then, from (6.12), we have
Ex = constant = k, say.
From which we obtain
(Al) E = €(X) = kx + c

where c is a constant of integration. Subtracting (6.10)

from (6.12), and rearranging, we get

2
(A2) M = r_>‘2(Yi + k.
dy 2y

Substituting the value of A(y) from (6.10), i.e.,

 

_ 2 n( )
(A3) A (y) — 2L1? .
y '-r
into (A2) to get
dn( ) r2n(y)
(A4) ___X_. = 2 2 + k .
dY Y(Y - r )

from which we solve for n(y). When THY) is obtained from

(A4), A(y) is obtained from (A3), we finally have the

. '0
requ1red group

(kX+c) 3— + m(y) —

(6.14) x 3X 3y

with the property

X(1)F = A(y)F.

51

If we choose n(y) = 0, the equations (A3) and
(A4) imply, respectively, A(y) = O, k = O; and the group
(6.14) becomes

_ 3
X — C 3X 0

Setting c = l, we have the group (6.7).

Example 6.2: Given the differential equation

 

(6.15) F E y2y' + x2(y')3 - X4(Y")2 = 0:

we shall find a group

 

a 3
X = €(x, y) 5; + m(X. y) 337
such that
. . l 2 . _
U51ng the formula (2.18) we f1nd n , n in X(2) - X +
l 2 a
n :17" n ,7.- to be
r. 1
_ I _ I _ I 2
n — nx + ny y Exy €y(y )
n2 = n + (2n - a )y' + (n - 26 )(y')2
(6.17)‘1 xx xy xx yy xy
- g (yl)3 + (n _ 26 )yn_3€ nyII.
L. YY Y X Y

The other form of (6.16) is

52

(6.16') 2yny' + yzn1 + 2x€(y')3 + 3x2(Y')2nl

3
— 4x 5(y")2 — 2x4y" n2

= A(x,y){y2y' + x2(y')3 - x4(y")2}.

Substituting from (6.17) into (6.16') and equating the

2 3 4

coefficients<xfl, y', (y') , (y') , (y') , y", ....y'(y")2;

we get:
(6.18) (coeff. of 1): yznx = o
(6.19) (coeff. of y'): 2yn + yzny - yzgx = Ayz
(6.20) (coeff. of y'z): -y2£y + 3x2y2nX = 0
(6.21) (coeff. of y'3): 2x6 + 3x2ny - 3x2€x = )x2
(6.22) (coeff. of y'4): -3x2€y = o
(6.23) (coeff. of y"): —2x4nxx = o
(6.24) (coeff. of y'y"): -4x4nXy + 2x4€xx = 0
(6.25) (coeff. of y'zy"): 4x4€XY - 2x4nyy = o
(6.26) (coeff. of y'3y"): 2x46 = o

. yy
(6.27) (coeff. of y"2): -4x3€ - 2x411y + 4x46X = 51x4
(6.28) (coeff. of y'y"2): 6x46 = 0.

Y

53

The equation (6.18) implies that n is not a

function of x, i.e.,

(Bl) n = My) .

Then the equations(6.23) is satisfied automatically. The
equation (6.22) and (6.28) imply that n is not a function

of y, i.e.,

(B2) 5; = €(X) .

And the equations(6.26) and (6.20) are satisfied. Now,
the equations (6.24) and (6.25) imply, respectively, that
E is a linear function of x, and n is a linear funCtion

of y. Thus,
(B3) 5 = ax + b,

where a, b, c, d are constants.

Substituting from (B3) and (B4) into (6.19), we get

2 2

3cy + 2dy - ay = Ayz

which gives

(B

>2
II

5) 3c - a

54
Substituting from (B3) and (B4) into (6.21), we get
-ax2 + 2bx + 3cx2 = 1x2
which gives

) A

(B' 3c - a

5

(B ) b = 0.

7

Substituting from (B3), (B4), (BS)'(BG) and (B7) into

(6.27) we get

-4ax4 - 2cx4 + 4ax4 = -(3c - a)x4

which implies

where a is arbitrary constant. And the required group is

(6.29) X = ax —— + ay ——

Such that

X F = 2aF.

Example 6.3: We shall find the group under which

 

the two-dimensional Laplace equation is invariant. The

given differential equation is

(6.30) u
Let the group be

1
x = E (x.y.u)
where 51, 52, n are t

X<2)= “”617“

l 2 22
where n , n ,...,n

formula (2.18), to be

55

3 2 3 3-
3'); + g (XIYIu) '5'; + r1(XIYIu) 5T;

0 be determined. We write

 

 

2 8 ll 3 12 3 22 3
5uy Buxx Buxy Buyy

are found, with the help of the

n1 = nx + nuuX - ux(E: + Eiux) ’ uy(€: + giux)

n2 = fly + nuuy ' “X(g: + E: ux) - uy(€§ + giuy)

”11 = nxx + nxuux _ uX(gix + giuux) - uYa?“ + giuux)
+ [nxu + nuuux - ux(€:u + giuux)

- u (E +

I
I:
m
C.

2 l l
guuuxnux + [nu - Ex - 2uxgu

2 2 l 1
[Ex + uxguluxy uxx(gx + E;uux)

56
l l 2
= + u - u + u - + u
n n n X(EYY EJyu Y) uY(€YY EYu Y)

1 1
+[nyu + nuuuy ux(€yu + Sunny)

2 2 l l
- uy(Eyu + EuuuyHuy - [Ey + Euuyluxy

l 2 2 l l
+ [nu uXEX 5y Zuyiuluyy uxy(€y + Euuy)

2 2
- + .
uYY(€Y guuy)
Since we require

(2) {u + u } = A(x,y,u) {uxx + uyy},

or

11 22 _
(6.31) n + n - A(x,y,u) {uxx + uyy}.

Substituting the value of n11, n22 into (6.31) and equat-

ing the coefficients of l, ux, uy,...., we get:

(6.32) (coeff. of 1): ”xx + nyy = 0.
(6 33) (coeff of u )- -g1 + 2n - 51 = 0

° ' x ' xx xu yy °
(6.34) (coeff. of u ): -52 + 2n - 52 = 0.

Y XX Yu YY

(6 35) (coeff of u2)' -2gl + n = o

' ' x ' xu uu °

2 1 1 _

(6.36) (coeff. of uxuy). gxu Eyu guu 0.

(6.37)

(6.38)

(6.39)

(6.38')

(6.39')

(6.40)

(6.41)

(6.42)

(6.43)

(6.44)

(6.44')

(6.43')

(6.43")

(6.44")

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

of

of

of

of

of

of

of u

of

of

of

of

of

of

of

57

-253u + nuu =
-€:u = 0'
-53u = 0'
-g$u = o.
- gin = o.
nu -2€: =
-25: - 26:
nu - 25: =
-3gt = o.
-g: = o.
-2gi = o.
-2gi = o.
—:i = 0.
-3g: = o.

58

The equation (6.43) implies that 51 is not a

function of u, i.e.,

(c1) 51 = 61(x. y).

Similarly, the equation (6.44) implies

(oz) $2 = 62(x. y).

From the equations (6.40) and (6.42) we have

1 _ 2
Ex — 6y.
This and (6.41) yield
1 l =
gxx + 8Jyy 0
(C3)
2 2 _
gxx + gyy — 0.

The equations (Cl) and (6.35), or (C2) and (6.37), show

that n is a linear function of u, i.e.,

(C4) n = f(X. y)u + 9(x, y).

Substituting (C4) in (6.33) and (6.34), and using (C3),

we get

ll
0

fx(xI Y)

I
O

fy(xI Y)

which imply

59
(C5) f(x, y) = constant = 2m, say.

From (C4), (C5) and (6.32), we have

(C6) gxx(x, y) + gyy(x, y) = 0.
Thus
(C7) ’ n = 2mu + g(X. y)

where m is a constant, and g(x, y) satisfies (C6).

The equations (6.40), (6.42) and (C7) yield

ll
>a

(C ) 2m - 26:

8

2
2 - 2 = o
(C ) m Ey A

9
Since from (C1) or (C2), the equations (C8) or (C9) imply

A = A(x, y).

This is the only restriction on A.

If we choose A = 4x, we get from (C8) and (C9)

(C10) 61 = mx - x2 + h1(y),

(C11)

01
ll

my - 2xy + h2(X)-

The equations (C (C11) and (6.41) imply

10) I

hi(Y) = 2y - hé(x).
So

hi(y) - 2y = -hé(x) = constant = n, say.

60

Thus

hl(y) = y + ny + c

-nx + d

h2 (X)

where c and d are arbitrary constants. We now have

(C12) 51 = c + mx + ny - x2 + y2

(C13) 5 = d - nx + my - 2xy.

We can check that the values of n, £1, 52 in (C7), (C12)
and (C13) satisfy all of the remaining equations in
(6.32) - (6.42").

Thus the group

2 3

+y2)—

(6.43) X = (c + mx + ny - x 3x

3
+ (d - nx + my - 2xy)§% + (2mu + g(x,y))gfi ,

where g(x,y) satisfies (C6), leaves the Laplace equation
(6.30) invariant so that

u + u = 4x + u .
X(z)[ xx yy] [uxx yy]

If we set A 0, we get from (C8) and (C9):

M
II

mx + ql(y)
E = my +q2(x).

These and (6.41) imply

61
Qi(Y) = -q§(x).
Thus
qi(y) = -qé(x) = constant = k, say.
Consequently,
ql = kY + a, q2 = -kx + b

where a, b are constants of integrations. Then we have

1

) 5 mx + ky + a

(C14

) 5 my - kx + b.

(C15

And we have the other group:

(6'44) x= (mx+ky+a)§;+ (my-kx+b).§§.

+ (2mu + g(x, y)) g%

leaving the equation (6.30) invariant so that

{u + u } 0.

X(2) xx yy

Example 6.4: Let us consider a system of differen-

 

tial equations in fluid dynamics. Nonsteady rotational
plane flows of incompressible fluid is governed by the

system [11]

62

r“
vX - uy = m
(6.45) 1 wt + uwx + va = 0
u = ny V = -¢x
L.

 

where (u,v), w, w are respectively the velocity vector,
the stream function, the vorticity of the flow. Equiva-

lently, the above system can be written in the form

"EI
III
C—

(6.45') ’5

'11
Ill

wt + wywx - wxwy = 0.

 

Note that (6.45') is the system (5.18) in the example 5.2.

We shall find the group

1

8 2
X=€ 66+5

3 1_§_ 2

a a a
"5324E 87+” 34+” 83'

leaving invariant the system (6.45'). Let us restrict

ourselves to find the group of the form

(6-46) gi = gi(tl xI Y)I ”1 = nl(t: XI YI wr w)

Using (2.18), we find the extended group:

63

 

1 1 3 l 2 3 1 3 3 2 l 3
X _x+nl +nl +nl +n’
(2) awt awx 5¢y amt
2,2 3 2,3 3 1,22 3
+7] am +n 3w_'+ooo+n girl—-
x y xx
1 23 3 1 33 3
+n' 5——+n' W+°°“
WXY YY
where
(.1 2 l 1 l 1 2 3
I _ . . _ _ . _ .
n - nx + nw wx + nw wx wtax wx Ex wy Ex
1,3 1 1 1 1 2 3
= + o + '0.) - o - o - 0
n my nw WY nw y wt 5y wx 5y , wy 5y
2,1 _ 2 2. _ 2. _ , 1 _ . 2 _ , 3
n ‘ nt + ”4 wt ”m wt wt gt wx 5I: my at
2,2 _ 2 2 2. _ . 1 _ . 2 _ . 3
)n — nx + nw°wx + nw wx wt gx wx gx my Ex
(6.47))
2,3 2 2 2 1 2 3
= + - + . - ° - - - -
n ”y “w wy "w my wt 5y “X 5y ”Y 5Y
1,22 _ 1 1 . 1 . - _ . 1 _ , 2
n _ nxx + znxw 1”X + 2nxu) u)x wt Exx wx gxx
3 l 2 1 1
- WY EXX + ")4 I1))! + ZnWwOWwa + nww'w
1 l 2 l 3
L‘ + n(powxx + nw°wxx 2Ex’wxx 2gxq‘uxx 25x wx

64

 

f- .
.0
1,33 1 1 1 1 2
= + 2 o + 2 ow _ o _ o
n nyy "yw q’y nyw y wt E"W wx 5yy
3 1 2 1 1 2
6047 - . + I + 2 o + g
( )( 4y 6yy nW my nww wywy nww my
1 1 3 1 2
+ o + ow — 2 o —- 2 o — 2 o .
K. ”I lpyy ”4 yy Ey wyy Ey wty gy wxy

By the theorem 5.2, the group X must satisfy the relations

x(2)F1 = A11F1 + A12F2
(6.48)

X(1)F2 = A21F1 + A22F2
or,

r

1,22 1,33 2 _
n + n + n — AllFl + Ale2
. 2,1 2,2 1,3 _ 2,3 _ 1,2

(6.48 N n + Wyn + wxn wxn wyn

L. = A21F1 + A22F2

 

where Aij; the functions of t, x, y, w, w; are to be
determined. Substituting from (6.45') and (6.47) into
the first equation of (6.48') and equating the coeffi-

cients of 1, wt, wx' wy' wt, ... , w , w , ...; we

xx xy
obtain:
1 l 2
(6.49) (coeff. of l). nxx + nyy + n — Allw.
(6 50) (coeff of w )- -gl - 51 = o
O ' O t 0 xx yy 0

(6.51)

(6.52)

(6.53)

(6.54)

(6.55)

(6.56)

(6.56')

(6.57)

(6.53')

(6.56')

(6.53")

(6.57')

(6.58)

(6.58')

(6.59)

(6.60)

(6.61)

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

of w ):

of w ):

of w ):

of w ):

of

of w ):

of

of w ):

of w ):

of

of w ):

of w ):

Zniw - gix - giy = 0
-€:x + 2";I 53y 8 0’
o = 112.
2niw = 0.
2n$w = 0.
”$4 = o.
”)4 = 0

2n$w = o

0 = 7312'
”$4 - o.

o = 112.

2n$w = 0
”36 = o
”36 = 0

-26: = 0.

-26; = o.

ni - 25: = All.

3 2 _
(6.63) (coeff. of w ): n1 - 253 = A .
YY W y 11
l _
(6.64) (coeff. of wxx). nw — 0.
(6.64') (coeff. of w ): n1 = 0.
YY w

We have immediately from (6.53):

(D1)

The equations (6.59) and (6.60) imply

1

(D a = 61(t).

2)
and this satisfies the equation (6.50). The equations

(6.61) and (6.63) give

— =1 1-
(D3) 5 - E 2 (nw All)
The equation (6.62) gives

(D4) y x

It follows from (D3) and (D4) that 62 and 53 are conjugate
harmonic fuctions of x and y, and so

' 4

2 2

(05) 6xx + ayy = 0.
3 3 _
(D6) Exx + ayy — 0.

Using (D5), (D6) in (6.51), (6.52) respectively, we obtain

67

(D7) nxw = 0
1 _
(D8) nyw 0.

The equations (6.56) and (6.64) imply that n1 is in the

form

1

(D) n = f(tr XI Y) + g(tr X, Y”)-

9

This satisfies the equations (6.54), (6.55), (6.57) and
(6.58). The equations (D7) and (D8) imply that g is a

function of t only. Thus,

(9;) n1 = f(t. x. y) + g(tw.

Now (D3) becomes:

. 2 _ 3 _ 1 _
(133) EX — 5y - 5 (g(t) All).

3

Since £2, E are functions of t, x, y; it follows from

(D5) that A 1 is a function of t, x, and y; i.e.,

l

The equations (D6) and (6.49) imply

2

(D ) fxx(t' x, y) + fyy(t, x, y) + n =_Allw.

11

We now substitute from (6.45') and (6.47) into
the second equation of (6.48), and equate the coefficients

of 1, wt, wx’ ... , to get

(6.65)

(6.66)

(6.67)

(6.68)

(6.69)

(6.70)

(6.71)

(6.60')

(6.59')

(6.72)

(6.60")

(6.73)

(6.74)

(6.59")

(6.75)

(6.76)

(6.77)

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

(coeff.

of l):

of

of w ):

of w ):

of w ):

of w ):

of

of

of

of

of

22°

69

(6.78) (coeff. of wxx): O = 'A21°

I . =
(6.78 ) (coeff. of wyy). 0 (A21.

The equations (6.65), (6.66), (6.67) and (6.68)
imply that n2 is a function of w only. This enables us

to split (D11) into three equations

(Dll) fxx(t, x, y) + fyy(t, x, y) = constant = m, say,

" 2 — _
(D11) n — Allw m,

wii') A

11 A11“"

ll! .
Now, we have from (D10) and (D11 .

All = constant = a, say,

and then (Dll) becomes

2
n = aw—mo

From the fact that n2 is a function of w only and 51 is

a function of t only (cf. (D2)), we get from (6.69)

2 l
d“ - §§_. =
aw dt 122'

From (6.74) (or (6.75)) and using (05) and (D5), we get

2
+ dn

(D12) A11 duo = A

22'

Thus,

70

dgl = -A = -a
at 11 '
and
51 = -at + k, R is a constant.

Differentiate (Dll) with respect to w and compare the
result with (D12). We get
A22 = lel = 2a.

From (D5) we have

52 = §4g<t> - a)x + h2(t. y)

a = %(g(t) — a)y + h3<t, x)

for some functions h2 and h3. Using these in (D4),

 

 

we find
9h2(t.x) = _ ah3<t. x)
3y 3x
which implies
2
h (t. y) = p(t)y + q(t)

h3(t, x)

-P(t)x + r(t)

where p(t), q(t) and r(t) are arbitrary functions. Thus

(013) $2 = %(g(t) - a)x + p(t)y + q(t)

71

...;

(D14) «53 = —(g(t) - a)y - p(t)x + r(t).

M

We now have from (D ), (6.70) and (D5):

13

(015) fy(t. X. y) = %9'(t)x + p'(t)y + q'(t).

Similarly, from (D ), (6.71) and (D5) we have

14

(016) fx(t, x, y) = - %g'(t)y + p'<t)x - r'(t).

Using the property that fyx(t, x, y) = fxy(t' x, y), we

get

g'(t) = 0

Or,

g(t) = constant = b, say.
‘ ' I
U81ng (D15) and (D16) in (D11), we get

p'(t) =

n43

or
p(t) = t + k, k is a constant.
Now, we have from (D15) and (D16):

m(x2 + y2> - r'<t)x + q'<t>y + s(t)

f(t, X, Y) = I

where s(t) is a new arbitrary function. Finally, (D5),

(D13) and (D14) become

72

(135') n1 = bw + IZ‘Wx2 + yz) - r'(t)x + q'(t)y + S(t)
(D' ) 62 = -l-(b - a)x + (at + k)y + q(t)

13 2 2
(D' ) 53 = l(b - a)y - (EE.+ k)x + r(t)

l4 2 2 '

We now have the required group
(6 79) x = (2 - at)iL + {l(b - a)x + (3t + k)y + q(t)}iL
' 3t 2 2 3x

+ {%(b - a)y - (gt + k)x + r(t)}§%

+

{bw + %(x2 + yz) - r'(t)x + q'(t)y + s(t)}§%

+ (aw - m)§% .

where a, b, k, k, m are arbitrary constants; q(t), r(t),

s(t) are arbitrary functions. This group has the proper-

ties:
(6'80) x(2)F1 = aFl' x(1)F2 = 2an;
Since from (6.48) and All ='a, A12 = 0, 121 = 0, 122 = 2a.

Since the group X involves 5 arbitrary constants and 3
(arbitrary functions, it can be decomposed into 8 smaller

groups under which (6.45') is invariant

(set: a = l, the others = 0): X = -t—— - -x§%

73

_ ,_ . -13.. __ 3
(set. b-l, the others — 0). X2 — EXBX + 2 3y + 03w
(set- k=1 the others = 0): X = yiL - xiL

° ' ° 3 3x 3y

_ _. . __3.
(set. 2—1, the others — 0). X4 - 8t

_ _ . -13. 1.3.
(set. m—l, the others — 0). X5 — Etyax 2txay

1 2 8 8
+Z'(X+Y)-a—w--m
(for: q(t) + constant, the others = 0):
x = q(t)—a— + g'(t)y-§-
6 3x SW
(for: r(t) + constant, the others = 0):

(for:

These small groups

F

X1(2) 1

from (6.80) with a

Xi(2)Fl

from (6.80) with a

Note that the group -2X

the example 5.2.

s(t) + 0, the others

— F

:0,

have the properties:

= 2F

1' x1(1)F2 2
= 1;
xisz = o
= o.
is the

1

r(t)-3-3§- — r'(t)x

3
372'

s(t) 5% .

(i = 2,...,8)

group (5.19) in

7. REDUCTION OF ORDER OF ORDINARY

DIFFERENTIAL EQUATIONS

In this section, we deal with Lie's theorem of
reduction of order of ordinary differential equations,
by the utilization of one-parameter transformation
grbups. This theorem is considered as an important one,
since it enables us to simplify the forms of ordinary
differential equations. It is our purpose here to
clarify this theorem by giving a new proof. We note here
that the main result in this section is the proof of a
lemma, which is a key for proving the Lie theorem.
This lemma shows an important property of the extended
group of transformations of two variables where one vari-
able is regarded as a function of the other.

Egmma: Suppose u(x, y) and v(x, y, y') are,
respectively, anl absolute invariant and a differential

invariant of the first order of the group generated by

 

- i. .3- = dv/dx .... 911
X — g(x, y)3X + T](X, Y) 3y. Then V1 m du ,
2 du7dx chi duz ' '°' ' n du7dx du
dnv

__H are, respectively, differential invariants of the
du

second order, ... , the (n + l)th order of the group

generated by X.

74

75

Proof: From the fact that v(x, y, y') actually

 

. ' _ dv/dx =
involves y (cf. Sect. 2) and from V1 — dfi7dx
3v 3v 3v
— + y. + __T ll
3X 35;. 3 3y , it follows that v1 involves the
._u + _Ey'
3x BY

derivative y". The only thing we have to do is to show

(£2435) vanishes identically, where X =

that X du/dx (2)

(2)

3 3 1 3
€(X, y)§§'+ ”(Xv Y)§§'+ n (X: Yr Y')§§T'+

n2(x, y, y', y")§%w is the second extended operator of X.

Since

(7 1) x dV/dx = 1 x 21‘ _ dV/dx x g3
. (2) du7dx du7dx (2) dx (du/dx)2 (2) dx I

and since

dv _ 3V 3V . 3V " _ 3V
(7'2) X(2) (F13?) " X(2)(‘a‘£ + WY + WY) ‘ X(2)('é'§)

3v

+ X(2)(WY') + X<2)(‘537'Y) '

expanding each term on the right side of (7.2), and using

the formula (2.18) we get

8v- av» -aazy. 9219.1 may.
X(2)(§§) ‘x(1)('a‘x')“a'£ ”((1)“ (Ti x+ x y+ ax y')

=_§£_3_z+.3_1.§x+___3n __,.
.Bxax 3x y x3y ’

= - ' 23 3V ED. y. + ___—an]- 3V
y 3y ’5'; y y 3y 51"
331 3n 3n . _ 33; . _ g; .2
+ By (3; + Eyy axy Byy '
8v ,, _ ,, 3v 3v 2 _ ,, 3v 3v 2
X<2) (Fwy) Y X(2)(a'y"") + By” ‘ Y X(l) (ay') _ay'
__. _ u 301 8V + 3V 301 + anl | + 301 n
Y _Tay “Tay "ray _ax 3y _By'y
_ RE .. .35.: ..
3x - Byy y )'
We see that
. §1=_izi§.i§--l¥.i§3§_::
(7'2 ) x(2) (dx 3x 3x + Byy 3y x + yy y

Similarly,

du _ Bu Bu , _ Bu Bu ,
(7'3) X(2)(a'§) ‘ x(2) (53? + 6275’) ‘ X(z) (E) + X(z) (WY

77

Substituting from (7.2') and (7.3) into the right hand

side of (7.1) we get

X dv/dx = 0
(2) du7dx °
dv _ dv/dx . . . . .
Thus dfi’- 5375; is differential invariant of the second

order of the group generated by X.

Moreover, if Vk-l is the differential invariant
of the kth order of the group generated by X = €(x,y)§%~+
3 d‘vk_1 dvk_l/dx
n(x,y)§§- then we claim that ‘56"‘ = ‘EE7EE" is differen-
tial invariant of the (k + 1)th order of X. We have that
(k) (k)

_ 1
vk_1 — vk_1(x.y.y .....y ) depends on Y ‘r and x(k)"k-1
_ _ a a 1 , a
— 0 where X(k) — g(x, was; + m(x. y)—3y + n (X.y.y )g—y.

+ ... + nk(x,y,y',...,y(k))-—%ET is the kth extended

 

3y dvk_1/dx
Operator of X. As an immediate consequence W =
3v 3v 8v
k-l + k-l , + + k-l (k+l)
TX Ty Y . . . T31 k y “(+1)
3 contains y . We
u +%Ey,
gg‘ y dvk_l/dx
shall now show that X(k+l) —EE75;—— = 0. Since

 

3v 6v 8v
k-l _ k-l _, 8 _ 85 k-l
X(k+l)( 3x ) _ x(k)( 3x ) _ 3‘)?(x(k)VK-1) 33? 3x

 

 

 

+ .33. aVk-l + All]; aVk-l +
X By 3x ay' ° ' °
+ ank 3vk_1

"' 3x (k3,

 

 

 

= _ g; aVk-1 + an aYk-1 + a 1 3Vk 1 +
8x 3x 3x 3y 3x ay'
+ ank aVk-i
o o o I
3x 3y (k)

avk-1 aVk-1) + aVk-l 1

X(k+1)( ay Y') = Y'x(k+1)( 3y

 

 

 

 

 

+ aVk-i 1 = _ . §§_3Yk 1 + 31_3Yk-1
8y n y 8y 3x 3y 3y
+ anl aVk-l + + Bnk aVk-l
Thr “ByTi "' wy S;7ET'
131121 an.m._e_a_._i§.y.2
3y ax ayY 3x 8y '

3v (3v 3v (3v
k-l " _ " k-l k-l 2 = . k-l
x(k+1) (_‘ay' Y) ‘ Y x(k+1) ay' ) + ay‘ ” Y x(k) ay' )

 

 

 

 

 

+ 3Vk-1n2 _ _ " anl 3Yk 1 + an2 3vk-i +
3r Y ay' ay5 —-.-ay _Tay "
+ Bnk 3vk_l + BVk-l an1 + anly'
.0. 3y. 3 (I?) 3YT BX y
y
1
an H ig- ll _3_§. I II
+ §§Ty axY 3yy y ) .
x 3Vk-1y(k+1) = y(k+1)X 3vk-1 + aVk-1 k+l
(k+l) 3;??? (k+l) 8y(k) 3y
= _y<k+1) ank 8Vk-i + aVk-i 33:
k k k
311 . 3n .. 3n (k+1)
+ '3-y—-Y + Y Y 4’ ... + y

79

ag (k+l) _ BE . (k+l)
§§Y §§Y Y .I
we have that
dv 3v 3v 3v
k-l _ k 1 k-l . k 1 (k+10
X(k+1)( <b<')' x(k+1) ( 5x "6§“Y + "' + 5y k Y
3v 3v

_ k-l k-l .

x(k+1) ( 3x ) + x(k+1) ( 3y Y )+ "'
3 V |
k-l (k+1)

+ X(km WY

 

3v 3v

aVk—1 (ea 3 ) y(k+1)

 

... l
... W SSE-+3),
.1 _ gngk-l
dx dx °

In a similar fashion, we can show that

du _ Bu Bu , _ Bu Bu , _ _gg Q3
X(k+l) (a?) " X(k+1)('a'32 + WY ) ‘ x(‘a‘§)* x(1)(3_y"Y) " dx dx '

Thus

x dvk_1/dx = l x (de-l) - de-l/dx X QB
(k+l) du7dx du7dx (k+l) dx (du/dx)2 (k+l) dx

80

de-l dvk_l/dx

We now have a concluSion that _7fir_’= ‘EE7E§“ is a dif-
ferential invariant of order k+l of the group X. By induc-

dvn_1/dx
tion, we have that for pOSitive integer n, vn = _du7d§_-
is a differential invariant of order n+1. The proof of

the lemma is completed.

We now are in position to prove Lie's theorem onthe
reduction of order of ordinary differential equations.

Lie's theorem ([1], pp. 386-387): Suppose that

 

the ordinary differential equation of order m:

(7.4) F(x. y. y'. .y‘m’) = o

and the group

(75) x=e;(x y)-9—+n(x mi
' ' 3x ' 8y

are such that X(m)F(x, y, y', ... , y(m)) vanishes whenever
F(x, y, y', ... ,y(m)) vanishes, and that u(x, y) and

v(x, y, y') are, respectively, an absolute invariant

and a differential invariant of the first order of the
group (7.5). Then the differential equation (7.4) can be
written in the form

m-l
d
(7.6) G(u, v, g—Z, ,-—m—_-‘li) = o,
du

a differential equation of order m-l.

81

(m)

Proof: Since F(x, y, y', ... ,y ) involves
y(m) and since we have
X F(x y y' 11““) = 0
(m) I I I '9' I

in the domain of definition of equation (7.4), by theorem 2.2
F(x, y, y', ... ,y(m)) is a differential invariant of order
m of the group X in the domain of definition of equation
(7.4). Since u is an absolute invariant and v is a dif-

ferential invariant of the first order of the group X, by

 

dv d2v dm‘lv .
the above lemma 55' 3:7 , ... , dum‘l are, respectively,
differential invariants of the second order, ... , of the
th dv d2v dm'lv

m order of X. We now have that u, v, 53, 5:5,..-,EEE:T

are m + 1 independent functions satisfyingthe equation

X(m)f = 0. We also have that F(x, y, y', ... ,y(m)) satis-

fies the equation X(m)f = 0 in the domain of definition of
equation (7.4). Thus, in the domain of definition of

equation (7.4), the F(x, y, y', ... ,y(m)) can be written

91. dm-lv. i e
du' ... ,dum-l, . .'

 

as a function of u, v,

m-l
l d d
F(X, Yr Y I°'°Iy(m)) = G(ur VI a‘g'loo-I—Rtli')o
du

Therefore, equation (7.6) follows from (7.4).
Example: We know, from the example 6.2, that the

differential equation

(7.7) F E yzyl + x2(y')2 - x4(y")2 = o

is invariant under the group

82

_ a a
(7.8) x — ax-B; + ayfi, a =|= 0
so that
X(2)F = 2aF.

Thus X F vanishes whenever F vanishes. An absolute

(2)

invariant of the group (7.8) is found from

(7.9) 9’3 = 91.
ax ay

Solving (7.9), we get £1: constant as a solution. The

first extended operator of the group (7.8) is

We see that any function of y' satisfies the equation
X(1)f = 0, so that we can take y' as a differential

invariant of the first order. We now set

u = y/x , v = y'.

 

Then

91 = dV/dx = __L—" = _L-X "

du du7dx y;'_ 3% v - u '

X )c
or,
.. V - my.
y x du '

Substituting y = xu, y' = v, y" = Z—Z—Ewgl into (7.7),

x du

we obtain

83

 

2 2 2 2 4(v - u dv)2 _
xuv+xv-x -— —O,
x du
or,
, 2 2_ _ 2dv2_
(7.7) uv+v (v u) _du —0

which is an equation of the first order.

8. REDUCTION OF NUMBER OF INDEPENDENT
VARIABLES IN PARTIAL DIFFERENTIAL

EQUATIONS

One of the methods of changing the form of partial
differential equations to simplify the finding of solu-
tions, is the reduction of the number of independent
variables. Here, we shall deal with Morgan's method of
reduction of the number of independent variables in par-
tial differential equations, which is the utilization of

one-parameter transformation groups.

Let
1 m 1 n 3k 1 3k n
(8.1) 0&(x ,...,x ,y ,...,y ,...,-—L-(1)k ,...,—__L—am m)k)=
3 x

be a system of partial differential equations of order k.

Definition 8.1 (Morgan's definition, [3]): The

3

solution y = 03(x1,...,xm) is called invariant solution

with respect to the transformations T: (x, y) + (§, §),

if after transformation the solution becomes §3 = 93(§1,..

..,§m); that is the §3 is exactly the same function of the

3 is of the x's.

x's as the y

Definition 8.2 (Morgan's definition, [3]): The

1 m1 n akl

differential form ¢(x ,...,x ,y ,...,y ,..., 1

8k n 3(x )

( m)k) is said to be conformally invariant under the
3 x

'00.,

84

85

one—parameter transformation group G: x1 = m1(xl,...

l — 1 1
Xmo Y .....yn: t): Yr = \(‘r(X I°°°I Xmo Y o-o-an7 t)?

if it satisfies

-1 -m —1 —n ak'l ak‘n
(8.2) 6(x ,...,x ,y ,...,y "°°"':¥7E"""':%‘E'=
3(X ) 3(X )
1 m 1 n akxl akzn
H(x ,...,x ,y ,...,y ,..., 1 k"°" m k' t)
3(x )' 5(X )
k 1 k n
, 1 m l n a y B y
§(x ,...,x ,y ,...,y ,..., 1 k"'°' m k)
3(x )' 5(X )

where H is not zero (see Appendix).

Observe that the invariance of the differential
equation Q = 0 under the group G is a special case of
the conformal invariance of the differential form 6 under
the group G. We can prove, by following the proof of

theorem 3.2, that the equivalent form of (8.2) is

I l m l n Bk 1 Bk n
(8.2 ) X(k)6(x ,...,x ,y ,...,y ,...,———¥——,...,-——§—E) =
' 3(x )' 5(x )
h(xl Xm y1 yn 3k 1 Bk n )
’ 3(xl)k a(x‘“)k
k 1 k n
°§(x1,...,xm,y1,...,yn,..., a —£L1L—A

a(xl)k"wa(x‘m)k

where X(k) is the Operator of G(k)' the kth extended

group of the group G.

85

one-parameter transformation group G: i1 = ¢l(xl,...,

m 1 n -r r l m l n
x , y ,...,y ; t), y = (I) (X ,..., X , y ,...,y ; t);

if it satisfies

-1 -m —1 -n Bkil 8kYn
(8.2) <I>(x .....x .y ,...,y (Ml)k""' (Em) E =
8

1 m 1 n 3k 1 3k n
H(x.....x.y....yv---rv~~'—('yf)1:3—¥n—E
8 x (x )

3k

1 m 1 n
.¢(X ,...,X 'y ,...,y '00., _3—LllkI°'.I—LE-)

3(X

where H is not zero.

Observe that the invariance of the differential
equation ¢ = 0 under the group G is a special case of the
conformal invariance of the differential form 0 under the

group G. We can prove, by following the proof of theorem

3.2, that the equivalent form of (8.2) is

, l m l n Skyl 3k n
(8.2 ) X(k)§(X ,...,X ,y ,...,y '00., l k’...'——%—E) =
3(x ) 3(X )
h( 1 m 1 n 3k 1 akxn
X ,...,X 'y ,...,y ,...’W'...'a(xm)k)

-¢(xl ,...,x M,yl,...,y ,...,--x-Ev-o-:-—x—E9

3(x1) 3(xm)

where X(k) is the operator of G(k)' the kth extended group

of the group G.

86

Morgan associated with system (8.1) the group of

the form

¢i(x1,...,xm; t) (i=l,...,m)

(D
II

(8.3) G':
§r ’ r(yr; t) (r=1,...,n)

U)
U
I
e

here SI and SD denote, respectively, the set of transfor-

mations of independent variables and dependent variables.

Note that SI form a group in m-dimensional space, and SD
form a group in n-dimensional space. Let 01(x1,...,xm),..

1

..,Um_l(x ,...,xm) be a set of absolute invariants

of 51' These are also absolute invariants of G'. Let the

other absolute invariants of G' 1x2 gl(xl,...,xm,yl,...,yn),.
l m l n

..,gn(x ,...,x ,y ,...,y ); so that 01(x),...,0m_1(x),

gl(x, y),...,gn(x, y) form a set of absolute

invariants of G'. In the method of reduction of the num-

ber of independent variables, we need the set of absolute

invariants such that

3(0 ,.;.,0 _ )
(8.4) R i 2 1 = m _ 1
8(x ,...,x )

 

and

3(gl'ooo'gn)

(8.5)
3(ero~-Iyn)

 

where R indicates the rank of the Jacobian. If we make a

change of variables defined by

87

(8.6) Ci = oi(xl,...,xm) (i=1,...,m-l)

then the condition (8.4) enables us to express m-l of the
x's in terms of 01""’Om-l and the remaining x, say xm,

in the form
(8.7) x3 = fj(ol,...,om_l, xm) (j=l,...,m-l).

We now consider the yr and §r to be implicitly

defined as functions of x1 and E1, respectively, by the

equations
1 l l
(8.8) zk(x ,...,xm) = gk(x ,...,xm,y ,...,yn)
- -l - -1 - -1 -
(8.9) zk(x ,...,xm) = gk(x ,...,xm,y ,...,yn).

Morgan has shown that a necessary and sufficient condition
for the yr implicitly defined as functions of xl,...,xm
by the relations (8.8), to be exactly the same functions

of xl,...,xm as the §r, implicitly defined as functions

of E1,...,§m by the relations (8.9), are of the i1,...,;m
is that
(8.10) Zk(X1,...,Xm) = Ek(;{l'ooop;m) = zk(}-{l'ooo';(m)o

The condition (8.10) can be replaced by

1 )

m
zk(x ,...,x ) = Fk(cl,...,om_l

88

where 01""'Om-1 are absolute invariants of the group
(8.3). Thus, when y1,...,yn are considered as invariant
solutions of partial differential equations, we have the

relations of the form
1
(8.11) Fk(ol,...,om_l) = gk(x1,...,xm,y ,...,yn)
(k=l,...,n).

Note that the condition (8.5) enables us to express the
y's in terms of the x's and the F's defined in (8.11),

i.e., we have

r m
y = Hr(01'°"'°m-1' x , Fl,...,Fn)

(r=1,...,n).

When x1,...,xm l are substituted from (8.7), we obtain the

relations of the form

r m
Y = Hr(01'ooo'0m_l' X ' Fl’ooo,Fn)

(r=1,...,n).

Morgan's theorem ([3]): If each differential

form @6 in (8.1) is conformally invariant under the kth

enlargements (the kth extended group) of the group (8.3),

then the invariant solutions of (8.2) can be expressed in
terms of the solutions of a system of the form

akFl 3an

(8.12) A6(Ol'ooopom_l,F1’ooo,Fn'ooo'ro_—1'E,...,—3'77) = 0,
‘ m-l

89

a system of the kth

order partial differential equations
containing one-less independent variable than that in

(8.2). In the above, 01,...,o are those defined by

m-l
(8.6), and F1""’Fn are defined by the relations (8.11).

Definition 8.3: We shall call the system (8.12)
the reduced system of the system (8.1).

Remarks: (1) In practical problems of reduction
of independent variables by Morgan's method, the solutions
of the system of differential equations in question are
unknown. So, the existence of invariant solutions of the
system is also unknown. If the invariant solutions of
the system (8.1) with respect to the group (8.3) exist,
then the reduced system (8.12) is derivable.

(2) We can make a generalization by replacing the
set SD in the Morgan theorem by a new set of the form SD:
§r = wr(xl,...,xm,y1,...,yn; t); that is, the wr in the
new set are functions involving the independent variables.
Every step of the proof of Morgan for his theorem is still
valid for this generalization.

We now can restate the Morgan theorem as follows:

Morgan's theorem (modified): If each differential form

@6 in (8.1) is conformally invariant under the group

¢i(xl,...,xm; t) (i=l,...,m)

(D
II

(8.13) G:

l m l n .
wr(x I°°-Ix IY loo-017313)

(j=l,...,n)

90

and if the invariant solutions of (8.1) with respect to the
group (8.13) exist, then these invariant solutions can be
expressed in terms of the solutions of a system of the
form (8.12).

Example: We have found in the example 6.3 that

the Laplace equation

u + u = O

(8.14) Q
xx yy

is invariant under the group

(8.15) X — (mx + ky + a)3x + (my kx + b)3y
a
+ (2mu + g(x,y))a—u-

where m, k, a, b are arbitrary constants, and g(x, y)
satisfied the relations gxx + gyy = 0. The equation (8.14)
and the group (8.15) are such that

(8.16) X(2)d> E 0 = 0-8.

We see that the equation (8.14) and the group (8.15) satisfy
the conformally invariant condition of Morgan's theorem.
We set m = O, k = 1, a = O, b = 0, g(x, y) = l - y; to get

a group

— 3 - 2. _ i.
(8.17) x1 — y-é—x- xay-l- (1 y)Bu .

Note that Xl still has the property that

91

(8.18) ¢> E O = 0-¢.

X1(2)

We observe that the finite equations of the group (8.17)
are

x cost + y sint

XI
ll

y cost - x sint

‘<1I
II

(8.17') G: <

u = u + x + t - x cost - y sint

 

L_SD:
(t = 0 yields the identity transformation),
that is, it is the group of the form (8.13)inifluamodified
Morgan theorem. We now assume that the invariant solu-
tions of (8.14) with respect to the group (8.17) exist,
and we shall find the corresponding reduced equation.
The complete set of absolute invariants of the

group (8.17) is found from

 

(8.19) dx EX du
“X

(8.20) _ = _ ,

to be x2 + y2. We find that u + x - tan-11;) =const. is a

1

solution of (8.19), that is, u + x - tan- ($) is an abso-

lute invariant of G. Now, we have

x2 + yz, u + x - tan—11%

as independent absolute invariants of the group G. We set

92

 

 

(8.21) 0 = x2 + yz, F(o) = u + x - tan-1(g).
Then
r
u = F(o) - x + tan-1(g)
ux = 2XF11(0’) ‘ 1 + 7—27
x + y
(8.22) 1 u = 4x2F"(o) + 2F'(o) - ___;3517——
xx 2 2
(x + y)
u = 2yF'(g) - ___§___
y x2 + Y2
2 2xy;
u = 4y F"(o) + 2F'(o) + .
2
L” (x2 + y )2

Substituting the values from (8.22) into (8.14) and

simplifying, we obtain

4(x2 + y2)F"(o) + 4F'(o) = 0
or,
(8.23) oF"(o) + F'(o) = 0

which is the reduced equation of (8.14).

The equation (8.23) gives
(8.24) F(o) = clno + k,

c, k are arbitrary constants. Substituting from (8.21)

into (8.24), we get

- x + k

(8.25) 11 = c].n(x2 + y2) + tan—103-

93

as invariant solution of (8.14) with respect to the group

(8.17). We now substitute u, x, y in (8.25) in terms of

u, §, § and t from (8.17'), to get after rearranging:

1 x -
= " +k.
(Y) X

E = cln(>-:2 + §2) + tan-

This shows the invariant property of the solution (8.25)

under the group (8.17').

9. SOME SOLUTIONS OF THE SYSTEM OF
DIFFERENTIAL EQUATIONS OF NONSTEADY
ROTATIONAL FLOW OF INCOMPRESSIBLE

FLUID

Nonsteady rotational flow of incompressible fluid

is governed by

(9.1)

'11
m
8
+
6
I<2
E
I
€-
8
ll
0

where w is the stream function, w is the vorticity. We
have found in the example 6.4 that the system (9.1) is

invariant under the group

_3_
3y

+ {%(b-a)y - (gt + 2):: + r(t)}

+ {bib + §<x2 + yz) - r'(t>x + g'(t))!

.3. a

+ s(t)}aw + (aw - 1105:)-

where a, b, k, 8, m are arbitrary constants; q(t), r(t),
s(t) are arbitrary functions. The system (9.1) and the

group (9.2) are such that

94

95

(9.3) aF = 2aF .

x(2)F1 1' X(1)F2 2
We also have shown in the example 6.4 that the group X
can be decomposed into the following eight smaller groups

under which the system (9.1) is invariant

—-_3_-.3;_§_-l.3 .3.
X1 ‘ tat 2x8x 2Y8y + “88
_ 1 1 8 8
X2 ' 2x8; + 2Y8y + waw
— .. _3_
X3 - YSE’ XBy
_ 8
x4 ‘ 5?
x 4.31. - itxi.z<3_+_xi_§>__i
5 2 Yax 2 8y 4 8) 8w

x6 = mug); + 88.85%,-

x = r(t)§§- - r'(t)x§%

X = s(t)§% .

We shall use these groups together with Morgan's theorem
(cf. Section 8) to find exact solutions of the system

(9.1).

l.° Consider the group

3 8

9 3 8 = YE; ' ”E?

This group can be obtained from X by setting a = 0, b = O,
c = l, k = 0, q(t) = O, r(t) = 0. Thus, we obtain from

(9.3) that

 

96

O, 0.

X9(2)F1 x9(1)F2

We shall find invariant solutions of (9.1) with respect

to X9. Since

- a a - a 1 .3.
X9- 0fi+y-a—x- Xfi+s(t)3w+03w'

the absolute invariants of X can be found from

 

We find that i

t. x2 + y .w + s(t)tan'1(§).w

is a complete set of absolute invariants of X9. This

suggests that the invariant solutions are in the form

" -1
w = s(t)tan (3%) + W(t, o)
(9.4) < u) = W(t, 0)
O = x2 + y2
&.

 

To find W(t, o) and W(t, c), we substitute from (9.4)

into (9.1) to obtain

F 1
YOU + 6711,00 + W = 0
(9.1) 4
1 1
Wt - 38(t)wo — 0

 

97
The last equation of (9.1)1 yields
2
w = 52(0 +28(t))

where Q is arbitrary function, S(t) = fs(t)dt. Now, at

c + 0, the first equation of (9.1)l can be written as

OWOO +WO+ (IQ = 0,

which can be reduced to
090 + [09(02 + 28(t))do = u(t)

where q(t) is an arbitrary function. Integrating again,

we have
11 = -f61-(f09(02 + 25(t))do)do + oc(t)lno + S(t)

where S(t) is also an arbitrary function. Thus, we have

solution of (9.1) defined for o + 0.
w = -s(t)tan-l(§) - f%(fofl(02 + 28(t))do)do
2 2
(9.4)1 + a(t)£n(x + y) + S(t)

w = moz + 25(t))

where o = x2 + y ; S(t) = fs(t)dt; a, B, Q, s are abri-

trary functions. Note that the solution (9.4)l is an

invariant solution with respect to X8.

98

2°. Let us pay attention to the group X6:

8
= _ ' _-
X6 q(t) + q (t)Y3w
which has the properties

0:

x6(2)F1 X6(1)F2

A complete set of absolute invariants of X is

t
t
tr Y! \P "' q(é )XYI 03;

which suggests that the invariant solution of (9.1) with

respect to the group X6 has the form

W = aa%%%xy + W(t, y)

(9.5)

8
ll

W(t: Y).

Substituting (9.5) into (9.1), we get after simplifying:

r.

W + W = 0
YY
(9.1)2 4

-i'fil =
Wt q(t)YWy °

 

k

or,

'(t)
9.1 ‘P 'LTTW = o
( )3 tYY q t YYY

The equation (9.1)3 can be reduced to

99

_ '(t) '(t) _
(9.1)4 ‘i’t amyq’y + 2357-1279 - a(t)y + b(t)

where a(t), b(t) are arbitrary functions. The equation

(9.1)4 suggests that the solution of (9.1)3 is in the form

‘1’ = _121_(>\_) + B(t)). + C(t)

q (t)

where A = q(t)y; A, B, C are arbitrary functions. From

the first equation of (9.1)2 we find

W = —‘y = -A" (A) o
YY

Finally, we get the solution of (9.1):

 

'(t) A(A)
w = a———Txy + + B(tX\ + C(t)
q(t q2(t)

(9.5)l
_Au (A)

E
II

where A = q(t)y; A, B, C, q are arbitrary functions.

3°. We now take the group

.3.

X = r(t)§a§-- r'(t)x3w

7

which has the properties

0.

x7(2)F1 X7(1)F2

We find that the functions

100

I
t, x, w + £;%%%xy, w

form a complete set of absolute invariants of X Thus,

7.

the invariant solutions of (9.1) with respect to X7 will

be in the form

(— - r'(t)

r(t)

 

w xy + W(t, x)

(9.6) )

 

W(t, x).

LU)

Substituting these in (9.1), we get

WXX + W = 0
(9.1)5
_ r' (t) _
Wt mxwx — 0.

By the same procedure as the case 2°, we find a set of

general solutions of (9.1) to be

¢=_£.'_(_t_)_xy+éf_(_1.*_)_ + B*(t)A* + C*(t)
r(t) r2(t)
(9.6)l
w = -A*"(A*)

where A*, B*, C*, r are arbitrary functions, and 1* = r(t)x.

4°. Let us take the group
x = x + x = q(t) 3 + r(t)—8— + (q'(t)y - r'(t)x)-?—
10 6 7 BEE ay 31))

which has the properties

101

0’ 0'

x10(2)F1 x10(1)F2

from (9.3) when a 0. In the case where q(t) # r(t),
q(t) + 0, r(t) + 0, q(t) and r(t) are not constants

simultaneously, we find that

2
_ _ (g'(t))! " r'(t)x)
tr q(t)}, r(t)xr ID 2(q‘(t)r(t) _ r'(t)q(t)) I (1)

is a complete set of absolute invariants of X10. This set

of absolute invariants suggests invariant solutions of

 

 

 

 

 

the form
(- 2
= (qu - r'x)
w 2(qTr _ r'q) + W(t, 0)
(9.7) i w = W(t, o)
L.0 = qy - rx .
Substituting from (9.7) into (9.1), we get
r‘
2 2
q' + r' 2 2 =
q'r _ rjq + (q + r )woo + W 0
(9.1)6 T
L Wt = O.
This yields
W = Q(or)
w l If8( )d d ___—Sl;+ r.2 2
= - o o o - o
q2 + r5 2(q2+r )(q'r-r'q)

+ G(t)o + H(t)

102

where 9(0), G(t), H(t) are arbitrary functions.

Finally, we obtain the following solutions of (9.1):

 

 

/
w = 9(0)
8 = jq'y - r'x)2 - -———l———ff9(0)d0d0
2(q'r - r'q) q2 + r2
(9.7) .2 ,2
1+ " 2 q2+r (qy-rx)2
2(q + r )(q'r - r'q)

+ G(t)(qy - rx) + H(t)

0 = (t) - (t)
\- q Y r X

 

where q = q(t), r = r(t), 9(0), G(t), H(t) are arbitrary
functions; q(t) + r(t); q(t) + o; r(t) + o; q(t) and r(t)

are not constants simultaneously.

5°. Consider the groups X1, X2:
_-_§_..l_3__l.§_ .2.
x1 ‘ tat 2x3x 2Yay + “aw

._ 1 3 1 3 8
X2 — 2X53? + ZYFY- + WW 0
Both are groups of similarity transformations. To get a

more general group of similarity transformations, we form

a new group

_ =-_?_-.1... 1-1.. .3. _3_
X11 — aXl + bx2 atat 2(a b)xax 2(a b)y3y + bwaw
8

+ _ o
3013‘»

103

Note that X11 can be obtained from the group (9.2) by
setting k = 0, q(t) = 0, c = 0, r(t) = O and s(t) = 0.

Thus, by (9.3) we have

X11(2)Fl aFl, x11(l)F2 2aF2.

The system of differential equations determining absolute

invariants of X11 is

 

 

 

 

or,
d_t = dx = dy = 81 _ 9.9.
-t cw - w

-%(1 - a)x -%(1 - a)y

where c = b/a, a + 0. We find that

-§(1-c) é-(l—o)
(9.8) xt , yt , t w , tw

form a complete set of absolute invariants of X This

11'
suggests that the invariant solutions of (9.1) with respect

to Xll are in the form

f 1

-€~
ll

t-C‘P (0'1, 02) I

E
l

(9.9)
< -%(l-c) -%(1-c)
L01=Xt ' c’2=-‘*’t '

 

104

Substituting from (9.9) into (9.1), we find

(9.1)7

 

 

This is a reduced form of (9.1). We see that it is still
difficult to solve the system (9.1)7.

Let us return to the group Xl

—..i.-l._3_-£_ 1.

x1 ‘ tBt 2x3x 2 8y + ”aw
--i—1.1-1.3.. .9. _3_
— tat 2x3x 2y3y + 03¢ + “aw ‘

One can check that
2
(X + Y) /t, w , tw

are absolute invariants of X1. This set suggests invariant

solutions of (9.1) with respect to Xl of the form*

8 = 8(a) . w = t‘1w(o)

(9.10)
i 2
0=(x+y)/t.

 

k.

 

*The motivation of idea of reducing two or more in-
dependent variables at a time is due to the discussion of
W. F. Ames about extending Morgan's method of reduction of
independent variables ([10L.pp. 141-144).

105
Substituting (9.10) into (9.1), we find
809" + 4W' + w = 0
(9.1)8

W + 0W' = 0

where the prime means differentiation with respect to 0.

The last equation of (9.1)8 gives
(9.11) W = A/0

where A is a constant. Substituting (9.11) into the

first equation of (9.1)8, we have

1 A
w" + —w' = -7-
20 80
This differential equation yields
(9.12) ‘i’ = Alno + 13/? + C

4

where B, C are constants of integration. From (9.10),
(9.11) and (9.12) we get solutions of (9.1)

r-

m A/(x + Y)2

(9. 10)1

 

1“, 9.1,,(11Lt1fi +Bx_:_z+c
k 4 t E '

which are invariant solutions with respect to X1.

10. REDUCTION OF INDEPENDENT VARIABLES
OF THE EQUATIONS OF STEADY PLANE

FLOW OF POLYTROPIC GAS

Let us consider compressible fluid having an

equation of state of the form:
(10.1)1 p = f(p)-9(5)

where o, P and 5 denote, respectively, the density, the
pressure, and the entropy of the fluid, f and g are
given functions. The other equations governing the flow

of compressible fluid are

(10.1)2 pv-Vv = -Vp (equation of motion)
(10.1)3 V-pv = 0 (continuity equation)
(10.1)4 v-Vs = O (entropy is constant along
streamline)
- 1 2 3 .
where v = (v , v , v ) denote the velOC1ty vector of the

flow. We shall now change the system (10.1) into a

canonical form. Let P = f(p) and S = 9(5) so that
p = f-1(P) and s = g'1(s) . Then we let :71 = 65. Define

‘1

P d?

106

107

so that F is a known function of P.
Theorem: The variables W, P and S as functions

of x, y, z satisfy the equations

(10.3)1 fi-vw = -F(P)VP
(10.3)2 V'PW = o
(10.3)3 W-vs = o

if and only if 5, p, s and 0 defined by

1

(10.4) G = fi//§ , p = f' (P) , s = g’1(s) ands): f(p)-g(s)

satisfy the system of equations (10.1).
Proof: First, assume that W, P, s satisfy (10.3)

and we shall show that 5, p, s, p satisfy (10.1).

(i) V‘Vs =

loggg—Vs=}—igS—W'Vs=0,
/§ /s

so (10.1)4 is satisfied.

(ii) From (10.2)l we have

. —1
NGMVQE)==-%%%4P,
or,
(5.5)/gov; + (WEN-WE = - -1},~Vp .

The term (§/§)§-v/§ can be put in the form EEW-Vs which

vanishes by (10.2)3. Thus

108

- - l
Sv-Vv = - §Vp ,

or,

pG-vG Vp
which is (10.1)2.

(iii) From (10.2)2 we have

Vop/‘s‘x'r = 0 ,
or,
VOL-f = 0
/'s'
or,
iv-px‘; + pG-vi = 0.
5 f5
- 1. 1 ]. - 1 1. - .
The term pv'V——' = - -—§770v-Vs = - —7pW-Vs vanishes
/§' 2 s f s
by (10.2)3. Thus,
iv-pv = 0,
/'s'
or,
V-pv = 0

which is (10.1)3. The proof of the first part is com-
pleted. By the same procedure, we can prove the converse

of the theorem.

109

As long as the equation of state has the form
(10.1)l which includes many important cases, we can reduce
the usual system (10.1) of 6 equations to the system
(10.3) of 5 equations. The advantage of the system
(10.3) is that we can solve for W and P from (10.3)1 and
(10.3)2, then using the known value of W in (10.3)3 we
obtain a linear differential equation for determining S.
Once W, P and S are found, we find 5, p, s and p from the
relations (10.4) to get the required flow.

Let us restrict ourselves to the case of plane
flow of polytropic gas which is characterized by the

equation of state
(10.5) p = p1/1e(50 ‘ S)/Cp

where Y = cp/cV is the ratio of specific heats, cp is the

specific heat at constant pressure, c is the specific

v
heat at constant volume, 50 is some constant value of
entropy. Note that (10.5) is in the form (10.1)1. There

corresponds a function F defined in (10.2) for (10.5)

Y -
F(P) =%—%’1—,—=ypyz

Then, from (10.3), we have a canonical form of equations

defining polytropic gas flow

fi.vfi -YPY-2VP

(10.6) v-pfi = o

w.vs = o

110

where W = (W1, W2) for plane flow.

The system of equations of plane flow of polytro-
pic gas has been dealt with before by P. Kucharczyk [16),
who uses Lie derivatives to reduce this system to the
system of ordinary differential equations. It is our
purpose here to get a more simple form of the reduced
system than that obtained by Kucharczyk. For this purpose,
we shall make a reduction of independent variablesof
canonical equations of plane flow of polytropic gas
(10.6). By the reason mentioned previously, we shall

only pay attention to the first three equations of (10.6)

and write them in the form:

 

(
z 1 l 2 l y-2 _
Fl _ w wX + w wy + yP PX — 0
z 1 2 2 2 y-Z =
(10.7)1 F2 _ w wX + w wy + yP py 0
LP 5 P(Wl + W2) + WlP + WZP = 0 .
3 x y x y

P

To satisfy Morgan's theorem of reduction of inde-
pendent variables (sect. 8), we shall utilize our method

to find group X such that

l 3 2 3 1 1 2 3
(10'8) X = g (XIY)§'§ + E (XIY)W + r) (XIYIW IW ring—W1:

3
+ n2<x.y.w1.w2.P)—§7 + n3<x.y.w1.w2.1>)15—-P
aw

and

111

_ 1 2
(10 9) x F = h (x y w1 w2 P)F
" 1 (l) 2 2 ' ' ' ' 2

_ 1 2

 

where X(l) is the first extended group of the group X.
Note that gl,§2 are functions of only the independent
variables and equation (10.9) is less general than
equation (5.7). From (10.8) we find, with the help of

(2.18), the extended group X(l):

X(1) = x + n1.1_§T+ “1'2‘3'1‘” “2,1682 + n2,2382
BWX BWY Wx :Y
3 2
3'1 a + T] ’ _—
+ -- 6P
T) BPX y
where
1.1 _ 1 1 1 1 2 1 1 1 1 2
T) _ “X + Thlwx + anWX + 1'1pr - Wxgx - Wygx
1.2 l l 1 1 2 l l l 1 2
= — — w
’0 fly + rhlwy + nwzwy + nppy wxgy ygy
2:1 _ 2 2 1 2 2 2 _ 2 1 _ 2 2
0 - fix + nwlwx + nWZWX + 7‘pr Wx gx wygx
2 2 2 2 1 2 2 2 2 l 2 2
' = + - w - w
1) T‘y + 11,.71Wy + (WW)? T‘PPy xgy ygy
3.1 _ 3 3 1 3 2 3 1 _ 2
n - nx + nwiwx + n'W2Wx + npr - ngx Pygx
3,2=3 3w1+3W2+3P-P1—P2.
n ny+nw1y nwzy T‘Py ng ygy

The left hand members of (10.9) can be written as

 

111

l 2
(10 9) x F h (x y w1 w2 P)F
. (1) 2 2 I I I I 2
X P h (x y W1 W2 P)F
(1) 3 3 I I I I 3

where X(l) is the first extended group of the group X.

From (10.8) we find, with the help of (2.18), the extended

group X(l):

1 1 3 1 2 8 2 l 3 2 2 0
X = X + n ’ + n ' + n ’ + n ' ———
(1) 8wI 8wI 8w2 3W2
x y X Y

where
n1,1 = n; + nilwi + ”32W: + népx - wigi - wiai
nl,2 = n; + ”$1W; + n%2W; + néPy ' Wig; ' Wig:
n2'1 = n: + 0%lwi + 0%2W: + NSPX ' wig: ' W35:
“2,2 = n: + nélw; + n§2W§ + 0123Py - Wii; - Wig;
n3’l = n: + ”31W: + ngzw: + nng - PXE: - PYE:
”3'2 = n; + nfilw; + nézw; + ngPy - PXE; - Py€;-

The left hand members of (10.9) can be written as

112

1 1 l 1 1 2 1

= I

X(1)F1 n WX + W n + n Wy
_ l 2 1 2,1 2 2

X(1)F2 — n Wx + W n + n Wy
+ Y(Y'

_ 3 l 2 1,1

X(1)F3 — n (Wx + Wy) + P(n

+ anx +

+ wznl'2 + v(v-2)PY'3n3px
+ YPy-2n3,1

+ W2n2,2

2)PY-3n3Py + yPY-2n3'2

+ n2,2)

Wln3'1 + n2Py + w2n3'2 .

We now can equate the coefficients of

l

1' Wx’ coo 'Py in (10.9).

(10.9) we have:

From the first equation of

(10.10) (coeff. of 1): w n + w n + yP ”x — o
1 1 1 1 1 2 1
(10.11) (coeff. of WX). n + W n l W Ex W gy
+ yPY-2n3 = h W1
1 1
w
(10.12) (coeff. of w1)- n2 - w 62 + w2n1 - W252 — n w2
y l y l
(10.13) (coeff. of wz) wln1 + pr'2n3 = o
x 2 2
W W
(10.14) (coeff. of wz). wznl2 = 0
Y w

113

(10.15) (coeff. of PX): y(y-2)pY'3n3 + wln; + yPY-zn:
_ y-2 1 _ 7-2
YP 5X - leP
. 2 l _ y-Z 2 =
(10.16) (coeff. of Py). W nP yP 5x 0.
From the second equation of (10.9), we have
(10.17) (coeff. of 1): wln: + W2“: + pr’zn; = 0
(10.18) (coeff. of WI): wln2 = 0
x Wl
(10.19) (coeff. of WI): wznzl + pr'2n3l — 0
Y w w
(10.20) (coeff. of W2): n1 + Wln2 - ngl - W251 = h W1
x W2 x y 2
(10.21) (coeff. of wz): n2 - wlgz + Wznz - W252
y x W2 y
+ yPY-2n32 = h2W2
w
. l 2 _ y-2 1 _
(10.22) (coeff. of PX). W nP yP 5y — 0
(10.23) (coeff. of Py): W2”: + y(y-2)pY'3n3 + yPY-z 3
y-2 2 _ y-2

The third equation of (10.9) gives

1 2 1 3 2 3 =
(10.24) (coeff. of l). P(nx + ny) + w nx + W ny 0

(10.25)
(10.26)
(10.27)
(10.28)

(10.29)

(10.30)

n3
W2

0.

114

(coeff. of W1): n3 + Pnl - P51 + Wln3 = h P
x 1 x l 3
W W
(coeff. of W1): -P52 + Pr)2 + W2n3 = 0
Y x l l
W W
(coeff. of W2): Pn1 - P51 + Wln3 = 0
x 2 y 2
W W
(coeff. of wz): n3 + pnz - p82 + w2n3 = h p
y w2 y W2 3
l l 3 1 l 2 1
(coeff. of PX). Pn + n + W nP W Ex W 5y
_ l
— h3W
2 2 2 3 2 2 1 2
(coeff. of Py). PnP + n + W nP - W 5y W Ex
2
_ h3w .
From (10.14) we get n12 = 0. Then (10.13) gives

W

From the fact that n1 is not a function of W2,

and 52 is a function of x, y only; the equation (10.16)

implies

(10.19)

1
E:Y

are now

W2, the

that 02

0.

(10.17)
(10.17)

0 and a: = 0. Similarly, the equations (10.18),

31 = 0, n: = 0 and
W

Observe that the equations (10.26) and (10.27)

“P
and (10.22) give 021 = o, n
w

satisfied. Since n1 and n3 are not functions of
I

equation (10.10) implies n1 0. From the fact

Y

and n3 are not functions of W1, the equation

Eliminating 0:, n2 from (10.10),

0.
Y

. . 2
implies nx

and (10.24), we get

115

7-2 7-2
1 7P 3 2 P 3
W - n + W - I——§—-n = 0
WI x w y
which implies that n: = n; = 0, since n3 is not a function

of W1 and W2. Then (10.10) gives n: = 0, and (10.17)

gives n2 = 0
Y D
We now have that

2 1

(10.31) 51 = 51(x). a = 62(y). n = n1(w1).

2 2 2 3
n =n(W).n3=n(P).

Divide (10.28) by P and (10.29) by w1 and substract the

results, we get

3
(10.32) (%7 - n3) + n22- flI-- E + g = O.
W

Similarly, we get from (10.25) and (10.30):
3 2
_ fl_._ 3 n _ 1 _ 2 l _
(10.33) (I’ ”P) + .7, n 1 gy + Ex - 0.
W W
The equations (10.31), (10.32) and (10.33) imply

1 _ 2 _ l _ 1
(10.34) E - alx + b1, 5 — a2y + b2, n — le ,

where a1, b1, a2, b2, k1, k2 and k3 are constants. The

equations (10.25) and (10.28) (or, (10.29) and (10.30))

imply

116

(10.35) h = k + k - a = k + k - a
which gives

(10.36) k - a = k - a

Eliminating a: and h in (10.11) and (10.15),

1
and using the values from (10.34), we find

Y - 1
l 3‘

Similarly, from (10.21) and (10.23) we have

= Y ' 1
k2 “2"k3‘
Thus,
_ _ - l _ - 1
k1 ' k2 ' I‘2'4‘3 ’ I“?“*
where we set k3 = k. Then from (10.36) we have
a1 = a2 = a, say.
From (10.35) we get
h3 = I—g—ik — a.

From (10.20) (or (10.21), or (10.23)) we find

h2 = (Y - l)k - a.

From (10.11) (or (10.12), or (10.15)) we find

117

h1 ‘= (Y - l)k - a.

We now have the required group

_ le 3L. - 1 1 a
(10.37) x — (ax + b1)3x + (ay + b2)3y + l——--—kw "I

' l 2 3 3
+$ngf+kpfi

with the properties:

X(1)F1 ((Y - l)k - a)Fl
X(1)F2 ((y - l)k - a)F2
_ y + 1 _

X(1)F3 ' (“‘2“k a)F3 '

The differential equations determining absolute invariants

of the group (10.37) are

l 2

dx _ d _ dW ,._ dW dP
ax + E - ay + 52 - 7 - k?

1 (Y - l)le/EV (Y - l)kwé/Z

 

 

or, in case a + 0

1 2

dx d dW dW dP
(10.38) ___—— = __L— = = = _—
x + c1 y + c2 mWi sz nP

where cl = bl/a, c2 = bz/a, m = (y - l)k/(2a), n = k/a. A
set of independent functions satisfying system (10.38), and

so a set of independent absolute invariants of X, is

118

y + c2 w1 w2 P

I O

(x + cl)n

 

(10.39)

 

 

X + c I I

l (x + cl)1m (x + cl)m

Thus the invariant solutions of (10.7) with respect to

X are in the form

K
W1 = (x + cl)mw1(0) , W2 = (x + cl)mw2(0)
(10.40) <
n y + c2
P = (x + C1) 0(0) . 0 = -f:--
L x cl

 

To obtain the differential equations determining wl(0),

w2(0) and 0(0); we substitute from (10.40) into (10.7):

 

r
l
l l y-l _ l _ 2 do _ y-Zdn _
mm m + yn (0w w )TRF. you 33" 0
(10 41) 1 mwlw2 -(0wl - w2)§2i,+ wY-2§£-= 0
’ d0 Y d0
1 2
1 do do 1 2 dn _
Lm + n)” n "O" d0 + "d"? ‘0‘” ‘” )—d0 "

where m = (Y - l)k/(2a), n = k/a. This is a reduced form

of the system (10.7). We now set
(10.42) 0 = n

Then (10.41) can be written as

119

 

 

 

1 2 1 2 dw Y d0
(10.43) < mm m (0w w )TE?'+ Y _ l d; — o
l dwl dwz l l 2 d6
LU“ + n)w 0 -00—a-O—- + E6— - Y _ 1(00) "' 0) )ac—I' = 0.

which is simpler than (10.41). For the consistency of

(10.43), we must have

 

 

owl - w2 0 YO
Y - l
l _ 2 Y
det 0 a“ w y - l + 0,
owl - w2
00 -0 Y _ l
or,
(10.44) owl + w2 and (0(1)1 - w2)2 + y(l + 02)0 .

From the relations (10.4), (10.40) and the first condi-

tion of (10.44), we have

y+C2

+

<24 <2“J

+
X Cl

This tells us that the direction of the flow is not along
the ray through (-c1, -c2). Thus, any set of solutions
of (10.43) gives a flow which is not a flow from a sourcecm'

a sink located at (-c -c

ll 2).

BIBLIOGRAPHY

10.

11.

BIBLIOGRAPHY

Lie and G. Scheffers, Differentialgleichungen,
Chelsea Publishing Company, Bronx, New York,1967.

 

P. Eisenhart, Continuous Groups of Transformations,
Dover Publications Inc., New York, 1961.

J. A. Morgan, The Reduction by One of the Number
of Independent Variables in Some Systems of Par-
tiaI DifferentiaIEquations, Quart. J. Math.,
Oxford Ser. 2, 3(1952), pp. 250-259.

 

D. Michal, Differential Invariants and Invariant
Partial DifferentIaIquuatio 5 Under Continuous
Transformation Groups in Normed Linear Spaces,
Proc. Nat. Acad. §oi., 37 (1951), pp. 623-627.

 

Birkhoff, Hydrodynamics, Princeton U. Press, 1950,
Revised ed. 1960.

 

Cohen, An Introduction to the Lie Theory of One-
Parameter Groups, G. E. SteChert & Co., New York,
1911.

L. Ince, Ordinary DifferentialIEquations, Dover
Publicatiohs Inc., New York, 1926.

E. Dickson, Differential Equations from the Grou
Standpoint, Ann. Math. Ser. 2, 25 (1924), pp. 28 -
378.

 

O. Friedrichs, Advanced Ordinar Differential
Equations, Gordon and Breach Solence Publishers,
New York, 1965.

 

F. Ames, Nonlinear Partial Differential Equations
in Engineethg, V01} 1, Academic Press, New York,
1965.

 

H. Wasserman, Fluid Mechanics, Unpublished Notes.

120

12.

13.

14.

15.

16.

17.

121

A. Muller and K. Matschat, fiber die Verwendung
yon Transformationsgroupen zur Gewinnung von
Ahnlichkeitsldsungen Partieller Differential-

leichun ss steme, Z. Angew. Math. Mech., 41
(I961), pp. TZI-T43.

W. Bluman and J. D. Cole, The General Similarity
Solution of the Heat Equation, J. Math. Mech.,
18 (1969), pp. 1025-1042.

 

 

 

 

 

V. Ovsjannikov, Determination of the Group of a
Linear Second Order Differential Equation,
Soviet Math. Dokl., 1 (1960), pp. 481-485.

 

 

Z. v. Krzywoblocki and H. Roth, On Reduction of
the Number of Independent VariabIes in Systems
of Partial Differential Equations, Comment.
Math. Univ., St. Paul, 13 (1964), pp. 1- 24.

 

Kucharczyk, A Geometric Method of Classifying
Solutions of Gas Dynamics Equations Using
Certain Space and the Time- Space Lie Transforma-

tion Grou s, FIuid Dynamics Transactidns,'VBI:
I, Ed. W. Fiszdon, Macmillan.

 

 

 

F. Ames, Nonlinear Partial Differential Equations
in Engineering, Vol. 2, Academic Press, New York,
1972.

APPENDIX

APPENDIX

MORGAN'S DEFINITION OF CONFORMAL INVARIANCE

Morgan's definition of conformal invariance of a
function 9 'with respect to a group G seems to lack 5%
precision because of his failure to describe the function |
H.. Actually, the basis of the problem is the failure to

describe precisely the classes of functions 9 and groups

‘Fm'u. 'n'.‘ .

G for which the definition is made. For functions 9
which never vanish, no new concept is described - for

every such 3 and for all G's we have the defined pr0p-

= 962.17....)
9(x,y,...)

are interested in making the definition only for functions

erty. We can take H(x,y,..., t) . Thus we

9 which vanish on some set in their domain and which are
defined in a neighborhood, N], of that set. Then, there

exists a function H(x,y,..., t) such that

4(§,y,...) = H(x,y,...,t)°@(x,y,...)
x,y,...€N; (t) < s
does impose a meaningful condition on 9 and G’. Further,

it is implicit in this definition that the x,y-domain of

G is contained in the domain of 6,. otherwise the defini—

tion would not make sense.

122

123

For x,y, ... such that 9(x,y,...) # O conti—
nuity and differentiability pr0perties of H are
determined by those of 9 and G, but when
9(x,y,...) = 0 no such prOperties are imposed on
H(x,y, ...).

A restricted (stronger) form of this condition is
used in this work (1) as an hypothesis in Morgan's
theorem for reducing the number of independent variables
in partial differential equations, and (2) in our method
for finding groups. The assumption that H has contin-
uous first derivatives in all its arguments will suffice
for the corresponding function h in (8.2’) to be contin-
uous which, in turn, suffices to satisfy our requirements
for (1) and (2) above.

Clearly, it is possible that imposing conditions
on H beyond those in the definition could restrict the
class of functions 9 and/or groups G) which satisfy the

definition.