THE UNI VER SITY OF MI CHI GAN Dearborn Campus Division of Engineering Thermal Engineering Laboratory Technical Report SIMILARITY ANALYSIS OF PARTIAL DIFFERENTIAL EQUATIONS BY THE MULTIPARAMETER LIE GROUP Tsung-Yen Na Arthur G. Hansen ORA Project 07)157 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GEORGE C. MARSHALL SPACE FLIGHT CENTER CONTRACT NO. NAS 8-20065 HUNTSVILLE, ALABAMA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR August 1969

FOREWORD Tsung-Yen Na is Professor of Mechanical Engineering, The University of Michigan, Dearborn Campus, Dearborn, Michigan. Arthur G. Hansen is President, Georgia Institute of Technology, Atlanta, Georgia. iii

TABLE OF CONTENTS Page ABSTRACT v 1. INTRODUCTION 1 2. THE CONCEPT OF MULTIPARAMETER INFINITESIMAL CONTACT TRANSFORMATION GROUPS AND THE GENERAL METHOD 2 2.1. The Infinitesimal Contact Transformation 2 2.1.1. Infinitesimal transformation 2 2.1.2. Notation of infinitesimal transformation 4 2.1 3. Invariant function 5 2.1.4. Invariance of a partial differential equation 6 2.1.5. Infinitesimal contact transformation 8 2.2. The General Method 9 3. APPLICATIONS OF THE METHOD 10 3.1. Application to Unsteady, Two-Dimensional, Laminar Boundary Layer Equations 10 3.2. Application to the Nonlinear Diffusion Equation 32 353. Concluding Remarks 40 REFERENCES 42 iv

ABSTRACT A systematic method using S. Lie's multiparameter infinitesimal transformation groups is developed in this report which enables the reduction of the number of variables by more than one in a single step. Details of the method are presented through two examples, namely, the unsteady boundary layer equations and the nonlinear diffusion equation.

1. INTRODUCTION The investigations to be presented in this report are an outgrowth of a continuing study of the application of Lie's continuous transformation groups to problems of physical or engineering interest, and the present report is the fourth in a seriel of reports summarizing the results of investigation. In the first report, a systematic way of reducing the number of variables was developed based on Lie's infinitesimal contact transformation groups. The second report8 treated the class of transformations from boundary value to initial value problems, and a method was introduced to search for possible groups. In the third report,9 a deductive method was developed in transforming from nonlinear to linear differential equations. In this report, we again consider the class of transformations in which the number of variables are reduced. In the original method developed in the first report,7 the number of variables that can be reduced in each step is one. If more than one variable needs to be reduced, such as in the transformation of the three-dimensional boundary layer equations to ordinary differential equations, the method has to be repeated for as many times as the number of variables to be reduced. Clearly, this is a tedius process. The question which naturally arises is whether it is possible to reduce the number of variables by more than one in a single step. Manohar4 was the first to propose such a method. Reduction of the number of variables by more than one was achieved by a one-parameter group of transformations and in another case by a two-parameter group of transformations. While the two-parameter method can be explained, by the theories given by Eisenhart,3 the one-parameter method is without foundation, since a one-parameter group can only reduce the number of variables by one. This was pointed out by Moran and Gagioli,5,6 who also discovered that the reason Manohar did get the correct transformation using the incorrect one-parameter method is that the transformations obtained, happened to be the transformations derived from another two-parameter group. Moran and Gagioli5 further illustrated by an example in showing that the one-parameter method by Manohar4 cannot, in general, reduce the number of variables by more than one. The purpose of this report is twofold. Firstly, a systematic method following the same general approach as given in the previous three reports will be developed. Secondly, the important point raised by Moran and Gagioli5 on Manohar's one-parameter method will be supported by the present method. 1

2. THE CONCEPT OF MUIJTIPARAMETER INFINITESIMAL CONTACT TRANSFORMATION GROUPS AND THE GENERAL METHOD In this section, a brief review of the concept of multiparameter infinitesimal contact transformation groups will be presented. Only those concepts closly related to the present analysis will be given. For a detailed treatment, the reader is referred to the first report in the series7 and the book by Cohen.2 Application of a r-parameter group to eliminate r-variables in a partial differential equation will be discussed at the end of this section. 2.1. THE INFINITESIMAL CONTACT TRANSFORMATION 2.1.1. Infinitesimal Transformation Consider the r-parameter group of transformations xl = O(x,y,al,...,ar) (2.1) Yl = (x,y,al,...,ar) where the a's are the parameters of transformation. If the identical transformation is o x O(x,y,al,..,a ) = x r (2.2) o 0 q.f(xya,,. ra = Y then the infinitesimal transformation o o xl = (x,y,al + 6al,...,a + 5a ) (2.3) o 0 Y1 = ((x,y,al +5al,...,a + 5a ) can be expanded. as xl = x + z (x,ya ) Ba. +... (2.4a) i=l z 2 2a

Y1 y+ i a bai +.. (2.4b) i=1 01 where the expressions 0 (x,y,a ) a + (xya ) and i 1 stand for what a(X, y. a, a2,.*,ar a6(X,y,al,a2,*., a and respectively become when a = al,a2 = a2,...,a = a, and the unexpressed term r r in Eq. (2.4) are of higher degrees than the first in bal, ba2,..,ba. Neglecting higher order terms in Eq. (2.4) and introducing the notation (x,y) a, (x,y,ay) (2.5a) Eq. (2.4) can be written as xy = X +ia.l i(x'Y)bai (2.6a) Y1 = Y + i=l ni(xy)bai (2.6b) Since bal,ba2,..,bar are any infinitesimal increments of the first order, they can be written as: 6al = e ba, ba2 = e2 a,...,ba = e ba Er r(c h o Eq. (2.6) can therefore be written as

xl = x + ~ ba (2.7a, b) Y1 = Y + ~ ba where = el 5l + e2 ~2..+ er (2.8a) r r and = el %l + e2 92 +...+ er Br (2.8b) r r 2.1.2. Notation of Infinitesimal Transformation Introducing the notation af f r Uf = 5 6c - +i (2.9) Ufax ay i=l eii ax i and similarly, U. = -+ (2.10) we have Uf = el Ulf + e2 U2f +...+ e U f (2.11) r r The above is for two variables, x and y. If n variables are involved, namely, xl,...,x, the notations become Uf = (Xl,x2,...,x ) ax +"... + sn(xl,Xn,2 —,xX) a (2.12a) n 1 ~n ~f~n or Uf = el Ulf +...+ e U f (2.12b) r r where

U.l f +.2 f n af(212c) 1 i X1 C X2 i ax n (i = 1,2,...,r) 2.1.3. Invariant Function If f(xl',x2',...,x') = f(xl,x2,...,x ), then f is invariant under the infinitesimal transformation r' =. + Z 5a (2.13) j=l J By following the same reasoning as in the one-parameter group method, the following theorem can be proved. Theorem. The necessary and sufficient condition for f(xl,x2,...,x ) to be invariant under the group of transformation represented by Uf is Uf = O; ie., =f n n Uf = (el1 + e22 +...+ e ) a +.+ (eln + e2 +.. + err a~ rr +)x = e n (2.14) To get the invariant functions, it is necessary to solve dxl dxn ba el e1 + e2 + + e el n + e2 + + er (2.5a) or, dxl =.. = dxn = ela 1j +d2 ~~+...+d ~n n n r r r r (2.15b) where d2 = e2/el,...,d = e /el. Equation (2.15) is seen to give (n - 1) inr r variants. However, in these invariant functions, there are (r - 1) arbitrary constants, namely, d2,d.3,....,d. Elimination of the (r - 1) constants from the (n - 1) invariants therefore leads to (n - r) functionally independent absolute invariants. Thus, a r-parameter group of continuous transformations has (n - r) functionally independent absolute invariants. 5

2.1.4. Invariance of a Partial Differential Equation To illustrate the manner in which the above theories can be applied to reduce the number of variables by r using a r-parameter transformation group, let us consider the case of a partial differential equation with one dependent variable and three independent variables. To reduce this equation to an ordinary differential equation, a two-parameter group is therefore needed. Consider now a function F = F (1X2x2; Y' xl) (2.16) the arguments of which, 13 in number, contain derivatives of y up to the second order. Such a function is known as a differential form of the second order in three independent variables. Designate the arguments by zl,...,z, i.e., Z1 = X1 Z2 = X2 62 p-l ax2 aX3 z = X (2.17) P ~x2 C 3 where p = 13. Thus, Eq. (2.16) can be written in a simpler form as F = F(zj,**,zp) (2.18) The function F is said to admit of a given two-parameter group represented by [cf. Eg. (2.12)] Uf = l(zl,.,zp) i+ + (z1...z) a (2.19) if it is invariant under this group of transformation. Therefore, the function, F admits of a group if

UF= (2.20a) or, (zl''' z )z +'''+ (P(Zl,..z z = 0 (2.20b) Sz,.p 6z, p 6z'Z p or, el U1F + e2 U2F = 0 (2.20c) where U =F 2 6F p 6F U.F = + F +..+ z (2.20d) 1 z aZ- + aZ2 i aZ p (i = 1,2) Since el and e2 are arbitrary constants, Eq. (2.20c) gives U1F = 0 and U2F = 0 (2.21) where U1F and U2F are given by Eq. (2.20d). Based on a theorem given by Eisenhart3 and later proved in detail by Moran, Gagioli, and Scholten,6 a function F in p variables can be expressed in terms of the (p - r) functionally independent invariants if it is invariant under the r-parameter group of transformations. For the present example in which a two-parameter group of transformations is introduced, the three independent variables can therefore be reduced to a single independent variable. To solve for the invariants, Eq. (2.15) is used. Thus, it is necessary to solve: dz dzl =..= = el ba (2.22) 1 + d2 p2 1 + d2 2 Eq. (2.22) gives (p - 1) functionally independent solutions containing the parameter d2. Elimination of d2 then leads to (p - 2) invariants which are denoted by: l~2~,.p-2 Thus, the differential equation F(Z1,Z2,...,Zp) = 0(2.2a)

can be transformed into the differential equation fr(kl~, 2 p-2) = o (2-23b) 2.1.5. Infinitesimal Contact Transformation For a given r-parameter group of transformations, the transformation functions,, are known, the above procedure is adequate in seeking the similarity transformations. However, if the group is not given, Eq. (2.14) alone will not be enough to search for possible groups. At this point, the theories developed in Reference 7 on the concept of an infinitesimal contact transformation must be introduced so that the transformation functions can be expressed in terms of r characteristic functions. The details of the derivation is given in great detail in Reference 7. Here, only the final form is summarized as follows: For the extended infinitesimal contact transformation Z. = z. + 6 c mi(z, p ) Xk = xk +c E c (z, xp) i i iv = Pi ( v' (2.24) p = p + 6 E T1 (Z,x pV p Pk - jk+ 5 E jk( v p) p s i i i v v V Pjki = Pjki +6 jk v' wZ s'P Pst the transformation functions can be expressed in terms of characterist tions W. as follows: aw V i m. = p - W. (2.25) 1 6apV[I 1 (no sum on v) aw. 1 (2.26) api (no sum on i) 8

aw. aw. =- 1 v _ 1 (2.27) 5 aZ - X Zv X a~i aIi i J s ~ apV 1lj xk aXJ bz PJ a v s (p2 *j s k + a i.t ck)ts + __ _ jk + x z P I VP av Pa bc~ Pjkt a a+ I p~ a P. bc (. (2.29) 202. THE GENERAL METHOD With the background discussed in Section 2.1.5 in mind, the partial differential equation given in Eq. (2.16) is again used to illustrate the steps necessary for reducing it to an ordinary differential equation. The method proceeds as follows: 1. An infinitesimal is defined., as in Eq. (2.24). The differential equation, F = 0, as given in (2.23a), is required to be invariant under this group of transformation, i.e., it must satisfy Eq. (2.20) or Eq. (2.21). 2. The transformation functions in U1F and. U2F, defined by Eq. (2.20d), can be expressed as functions of W and W, respectively, using Eqs. (2.25) through (2.29). Since W should not be equal to W, some properties may be imposed on W and/or W. For example, we may require W independent of Pl, while W be independent of P2, etc. W and W can then be determined by Eq. (2.21). i 3- Solve the independent invariants using Eq. (2.15). (Note that 5j are known functions since W and. W are known. ) 4. The differential equation can then be expressed in terms of the p - 2 invariants. Two examples will be given in the next section. 9

3. APPLICATIONS OF THE METHOD In this section the general theories given in Section 2 will be applied to two examples, namely, the unsteady, two-dimensional laminar boundary layer equations originally treated by Shuhl0 and the nonlinear diffusion equation discussed recently by Ames.1 Although the method is presented through two specific examples in transport processes, the general nature of the method makes it possible to be applied to equations in other fields. 3.1. APPLICATION TO UNSTEADY, TWO-DIMENSIONAL, LAMINAR BOUNDARY LAYER EQUATIONS Consider the unsteady, two-dimensional, laminar boundary layer equations, expressed in terms of the stream function A, _+ e e aPI ~ ~f,-+ U -— + (31) at 6y by ax y ax 2 at e ax subject to the boundary conditions y = 0: 0 y = = U (x,t) 2V e Equation (3.1) can be written in a shorthand form as: F = P333 + ~ - P13 P3 P23 + P2 P33 = 0 (3.2) where Pi at' P2 ax' P 3 = y' P13 t y etc and au au e e -- + U (3.3) at +Ue ax The differential equation F = 0, as given in Eq. (3.2), will be invariant under the two-parameter infinitesimal transformation 10

t' = t + bc1 li(t,x,yy,) + 6E2 al(t,x,y,*) x = x + 5e1 C12(t,X,Y,4r) + 8E2 C2(t,x,Y,Yr) Y' = y + b~l C3(tx,y,t) + 3E2 ct3(t,x,Y,4r) r = + bE1 S(t,x,y,yr) + bE2 S(txyyy) P2 = P2 + bEl Jt2(t,x,y,4lr,PP2, P3) + bE2 n2(t,x,y, P,P2,P3) (3.4) Pa = P3 + 6El J(t,x,yy,,pjp2,Pp) + bE2 Th(t,X,Y,P, Pl,P2, P3) P13 P13 + bE1 af13(t,X,Y, Y,Pl,P2,P3,Pll,...,P33) + E2 KAl3(t,x,y,r,...,P P33) P23 = P23 + E1 J123(t,x,y,fr,,...,P33) + FE2 123(t,X,Y,,...,PP33) P33 = P33 + bE1 Ta33(t,x,y,4,...,p33) + 52 t33(t,x,y,t,...,p33) P333 = P333 + b5E1 T333(t,x,y,Xr,Pl,...,p333) + 5E2 7T333(t,x,y,Y,,P1,...,p333) if UF = 0 (3or, from Eq. (2.20c), el U1F + e2 U2F = 0 (3-6) Since el and e2 are arbitrary, Eq. (3.6) means U1F = 0 and U2F = 0 (3.7a,b) simultaneously. In their expanded form, Eqs. (3.7a,b) can be written as: aF 3F aF + P 6F + 6 aF -l 6t + — 2 +a+x + C+3 1tY +i + i j +pi = 0 (3.8a) at 2ax 3y ri~ p~ij op.ij ik kp 1 ijk and aF - aF - aF aF - F F - F i a+ C2 aaF+ Ol3 - + ii 0 (3.8b) -1 r "4 I, 1 An -13 Anijk Putting F from Eq. (3.2) into Eq. (3.8), we get 11

333 + at + a 2 - T13 - p3 IT23 - P23 Jt3 + IT2 P33 + P2 T33 = (3- 9a) and a - o- -. T333 + aC + ax 2 - 113 - P3 T23 - P23 3 + 2 p33 + P2 t33 = o (3 9b) The next step is to express the transformation functions, al, C2, etc., in Eq. (3o9a) in terms of a characteristic function W; and the transformation functions, al, c2, etc., in Eq. (3.9b) be expressed in terms of a second characteristic function W. This differs from the one-parameter method in that two characteristic functions, instead of one, have to be determined. The functional form of the W's can be determined from Eqs. (3.9a) and (3.9b). Since the transformation groups in Eqs. (3.9a) and (3.9b) should be different (otherwise, the two-parameter groups will be reduced to a one-parameter group), we choose, as an example, two groups as follows: Gl: cal in Eq. (3.4) is zero. G2: a2 in Eq. (3-4) is zero. For the first group, G1, the transformation functions, a2, c3, etc., can be expressed in terms of the characteristic functions as w 1 6w1 6w, 1w, 2 = p, 3- = P2 + p3 - - W1 aw, awl awl 6w, awl awl 1 = -t - Pi, 2x - P2 r I3 P3 = P3 (3.10a-f) 2w, _2w_2W _W_ (______2w, _2w_ \ P13 = + P3 P + P P3 + P21 + 3 -t 6 3 t 6* N~ 6y 6t 2 YP2 6Y 6P2 12 ~ P31 62WJ + P3 ~2WJ + awl + P23 62We + Pi a2 I + P33 (6t2wp, + P 3 6 WP) (3 109)

H, 0 0H (0p~~ Co — >0 Co ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(r —l ~,-" o -. -- Co m' CM M Co Cor') CM~P CM6 ~6Pi)C P4 610 CY) CM~ 60 H - C~~~~~~~j~C CUj CO CO 3C CM ~ 3 Co IOP /0A 1 /0 pC Co 60 +Co tCM 6 i CoFC + HP-I OICMCCM CM 6O t6 + + Co/1 CO,~ CO~ +C "04 /10:4 P l::).4 H,-.. 60i2 O (U Co) Co Pi CO CO + C CM CM C +.-4 I 1/ r-ICU ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ N, /Co 6c Co Co: > CMO 60Co CO CMCo CO 60 pO Co ~~~~~~~~~~~~~~~~~~~~~~~~~4~~~~~~~~~~~~~~~~~~~~~~~C ~ ~ ~ P-iCo Co HCoO P1 P - 11 60 Hi> + +c CO C + C~ 411 0 l:3:: - [ CO 110 CM COC CO /01 O CO NCO CM O / 0 pq 4 O O CO 4 4 P 0 + +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -I.. P-!- + CO + -0 H Cij Ci NCMCO CM H60 C /,C P C M H /10 CM ~~~~+ ~ 3 0 H * ~ 3 + 10p CO + -.6 6+ ~+~ ~ + C O C * 60 C%~~~~~~~~~~~~~~~1 CO 60 CM 6 CO CM Co H60 H- (U P4 P H60 HP- Co CM6 6 ~~~60 ~~~~P-4 CM CMl P-i CO cm cm CM C C + CM C60 CO Co) CO 60p CM 10 Pi 610 + 60+ p4 p4 0p6 600 60 -O, Co CO + CO Co) CM H P4 H0 P4 + + P + PI /O 0]~~LIpqC~ C 3 + Co C CO + C\I Cr) C\I fOI F4, —4,,. 0 +ol, o + 6+ Co CO H60 CM H(6 CM C P-1 p CO f3: CO CY) + Y H60 Hp~~~/,4 CoCoH1 0 C C2OM 610p 6P-4 CM CM P 60 CO ~cmCM 60;1 C3 rll~~~~~~ cu~~~i cu,1 C~~ cr ~ CM CO CM 60 P CM CO 600PH,0P4 6110 A + 660 -D- 60 Co 60 60 +~~ + 60 Cr r\C M C ~~~~~- ~~~~~~~CM CY)o~~. - op 0 p C Co y060 CM CoCo Co CM Co /-O CO CO P-i CO 60 CO pq CM CO P4 ~~~Erl~~~~~~~c c~,-I~ P-4 P-4 CM 60 P-i CM Pi P4 f-\\ ~~~~~~~~~~/'" -~._~ c~o03oc ~II+ + II + I + + + + + + Co) CO CO CM CO Co Co

Now, substituting the transformation functions from Eqs. (3.10a) to (3olOj) into Eq. (3.9a) and eliminating P333 from Eq. (3.2), we get: f0 + fl Pll + f2 P21 + f3 P31 + f4 P22 + f5 P23 + f6 P33 + f7 P133 + f8 P233 + f9 P13 P33 + f10 P23 P33 +3 = 0 (3.11) where f = 3- 3 W+l 3 + 3W) +3 W + P3 + 3p2 6by3 f2y2 ~ y 6 82 i CP a~ aw, awl + (2w2 ( + /2w1 62w_ + P ++ PW + P (.12) ax aP2 a 6J, at ay at a6* Cy P3g2/ ~+ P3 aa2- + P3 a2 (2W + p3 ay a (3-12) 3 x by 82 b fl = 0 (3.13) 82 W1 6 2W1 22 + p w (314) P P2 Y + P3 6P2 f3 = - 2 a2w, - 2 52w. (3-15) f4 = (P3 6p2w, + P3 a2w )(3'16) 6P2 ~2 6P226* 63WJ 62WJ a3w, +2w, a3wW 2 P f5 = - 3 a 4p3 3 P3 -2p3 + 2W + + P33 8\ )/ + + 8WP2 +(ay + P3 aJ,,) 2 P2a + t3 aP2 at )P2 (3P17) 14

f6 6t 23 6WJ f ~ P +P3 (2W1 + 2 \ +P2 6) \t 63 6*6P3 X 6P3 /* 6P3 C)3 6* 63W, 63W, a3W, 3,2 6W 3 + 4p3; — 2p, / 3 6P 3 Ctr6P3 a2 / - 2w,31 - 2W3 + P2 (2W + )P3 62W (3.i8) ~~~~~~~~~~~~~ f7 = 0 (3.19) =- 2w_ + P3 2w) (5.20) f~ = -3 P2 C 6P2 C)* f9 = 0 (3.21) f 62W1 (5.22) ~ 2 6* f = (5.23) 11 6Pp3 Nr For Eq. (3.11) to be satisfied, all the f's should be zero since the characteristic function is only a function of t,X,y,Ir,p1,p2, and P3. Therefore, we get 11 equations: = f = f2 =...= fi = 0 (3.24) Three of these, namely, fl, f7, and f9, are zero identically. For point transformations, as in Eq. (3.4), the characteristic function is linear with respect to the p's. For the group under consideration, G1,, = 0, which means aw1 ~wJ~ =0 aPr i.e., W1 is independent of p1. Thus, we write 15

W1 - Wll(tY,x,y,Y)p2 + W12(t,x,yr)p3 + W13(t,x,y,t) (3.25) From f1o = 0 and fll = 0, we get = O and = 0 which means W11 and W12 are independent of ~. The three conditions f2 = f4 = f8 = 0 gives awl W+ p3 w 0 (3.26) Since W11 is independent of P3, Eq. (3.26) shows that W11 is independent of both y and {. Following the same reasoning, the condition f3 = 0 leads to the conclusion that W12 is independent of both y and r. The condition f5 = 0 gives aw,, aw,, uw 3awl aw,, W at + 3 + + 3 [t + x 6y Again, due to the independence of W11 and W13 on P3, the following two equations result: + aW13 = 0 (3.27) at 6Y awl + aW1_ = O (3.28) Equations (3.27) and (3.28) will be used later. From the condition f6 = 0. three equations are obtained, namely, 2W13 + aw aWl3 O0 (3.29) ay aC at ax aW1 + _aw_ = 0 (3.30) ax a =2w13 0 O(3531) 16

Equation (3530) is the same as Eq. (3.28). From Eq. (3.31), we get W13 = W13l(t,XY)\tr + W132(t,x,Y) (3.32) Substituting W13 from Eq. (3.30) into Eq. (3-29), we get w~3+ 12 + aw1_ + +W132 = o( 6y 6t 6x * +x - Since the W's in Eq. (3.33) are all independent of 4r, we get 6W,3 = 0 (3.34) awl + awl2 + W32 0 (335) ay at ax Equation (3534) shows that W131 is independent of x. Thus, the characteristic function now becomes W1 = Wll(t,x)p2 + W12(t,X)p3 + W13l(t,Y)4f + W132(t,x,Y) (3536) where the W's on the right-hand side have to satisfy Eqs. (3.27), (3.28), and (35355) From Eq. (3530), we conclude that Wll = Wlll(t)x + Wll2(t) (3.37) since W131 is independent of x. Equation (3530) now becomes Wlll(t) + W13l(t,y) = 0 (3.38) which means W131 is independent of y. From Eq. (3.27), we get dW111 dWl2 + _ = (339) _ x +__ + W0(33.9) dt dt ay which shows that W132 is linear in y, i.e., W132 = W1321(t,x)Y + W1322(t,x) (3.40) Equation (3539) now becomes 17

dWlll X + dWll2 + Wl321 = 0 (341) dt dt From Eq. (3-35), we get 6W12 + 6W1321 W1322 0 at + aw,,,, Y + uvv 1~2= o (3.42) Since the W's in Eq. (3.42) are independent of y, we get axW3 1 ~ 0(3e43) 6w1t + = o (3.44) Equation (3.43) shows that W1321 is independent of x. Equation (3.44) suggests that a function O(t,x) can be defined such that W12 = x' W1322 t (3'5) Since W1321 is independent of x, Eq. (3.41) gives: dW111 o (3.46) dt dWll2 + W1321 = ( dt Equation (3.46) shows that Will is independent of t. Equation (3.47) establishes the relation between W112 and W1321. The final form of the characteristic function is therefore: |W = [W111x + W112(t)]p2 + pX P-3 dlllL* t Y -(348) where Will is a constant, W112 is an arbitrary function of t and 0 an arbitrary function of t and x. The condition f = O has to be checked after the boundary conditions are o considered. The transformation functions, al, Q2, c3 and ~ for G1 are therefore given 18

by rl = 0 C(2 = Wlllx + W112(t) = 0 (3. 49a-d ) Ue Wll + dW11 Y a dt Y+ which are obtained by substituting the characteristic function W1, Eq. (3.48), into Eqs. (3.0la-c). For the second group, G2,'2 0= 2 (3.50) ~P2 which means the characteristic function W2 is independent of P2. The other transformations, al, Ca3, ~, etc. can be expressed in terms of the characteristic function W2 as follows: aw aw awPi w2 W2 6p1 6 P ap3 = p11 +P3~3 (3. 51a-f) = _2 = - 6 Pi t y- + P2 ay P3 at aW2 W2jw2' 2 W2 2W2 /23 a= + 2WP3 \ -13 P+iP3t +P+PP3 +2 p )l + P3d aj 1t a6y 6* w a2 1y 6w ap a2w 62W 6PW2 + 2 WP + P31 (~ )Ya + P3 + w2 + P13 aPlat apl a ) + P33 ( aW P (3-51 f P33 + — ~- (3.51g) 19

n N n CO CU f 63 CO CM~~ ri Cnj(C CM HV CM H 6 (0P4 (0O'FC PC4+ cO CO CO +l P4 +i P 4 + + CP4 + CO + 1~~~~~~~~~~~~~~f C~j "~ /i~ /j~ O C fO CM H cM CM'CC( 0 CO CM CcjCCO cu r c > q IO Pi (0(~ (0,~ (0j~P P4 (10 /6 P (0 fo~~~~~~~~~c co ~c c ~ ~~~~+ o —-- + — CM CM od~~~~~~~~~~~~~~~~~~~~~~~~~ J + CO + CM CMN COC CM C CMP CM P4 P4 H H HC~ P4 P4 Pi P4 + CQ + + CM 3- ~ ~ It)~~~~~~~~~ CMcU C3~~~~~~~~~~~ CM /10 ~~c+(0 + cu +~~~~~~ c + + P CM CM CMICM CMj CM CCcM CO CY) (0 ~~~~~~~H cMI,3 6 H CM 3 l P4 (01 P4 CO CO H H1 CM CMY C PP P4 CMCM HC P4 + CO ( Z P4 P4 CO + + + CO CM CO + CJj + P4 P4 P4 * CM i23~~~~~~~~~~~~~+ C+ CM(0 CM P4 HM CMo ~(0CMCMP4P4 0ICOM P4 CM C CC+jO +,% C CM0 /-PO, CO CMC ~ CM P4 O P4C P4 (0 cu CM CM~ P4 CM( C pq C? a) N O /10/~P-lCr) PO' CY)I OPI /-O P-, / P4 + cO CM~ ~ ~~ ~ ~P4 P4 > + CM + CM+ 41(0 + CO M CO H ~P + + c~:~ 0 ~H Pi + CC)~~~~~~~~~~~~~~~~~~~~~~~iC CO AO AR 1(0 + 4CMCM( ~O CMCMP4 CM F?-, _/P' P4~lw P(0 CM CM4 C' I CO (\J~~~~~~ (0 P4~~~~0C~ +I + ++ + O + + COCMC C~lCO O i:m C COrCM0 Cr CM O PCM CO i CO P pi P, /10 Cfj /r0 Cj PL, P, P-4 t,C CO CO C 17 — COj f

As before, the transformation functions given in Eqs. (3.51a-j) are substituted into Eq. (3.9b) and P333 is eliminated from Eq. (3.2). We then get f + fl Pll + f2 P21 + f3 P31 + f4 P22 + f5 P23 + f6 P33 + f7 P133 + f8 P233 + fs P13 P33 + flo P23 P33 + fll P23 = 0 (3.52) where fo =+ 2p3 (2~W 2 + 2 E P W22wy + 2p3 32 2 a3w o2p3 0 Ps3 2}r3 6342f2P3 5w43 (2W2 a2W2 w aw w / a(2w2 _ 2W + 30 a + P3+ + + ap 3 a r3ap3 a6* at api ax ap 2 6 at ay at a / a w2 a2w2 ( a22( 632 W2 /2 w -+ P1 a ay +3 + ( +P3 +ax ayP2 Y2 +3 a (3-53) - 2 aW) + P26 ( 354) f + a2 (3~55) f2 = P3 + P3 0W ) (3555) f3 = (3 apw P-2 + 4P3 63 a - 2p3 a3 W2_ + P3 aw 2 - a2 3W2 -/ aw1 ) a 6y 6 a~ a W2 P | Ca!2 62WZ + P3 aS at + 62W apw VP~ + P3 a2W a+2 + ) a2w P2 2 (2W2 + P3 62 ) P2 (3 56) f4 = O (3 57) f5 = 2P3 (2Wy + ~6 + P3y (3.58) 21

y = 2 + 4P3 N2 N ) - 2p3 3 + 3 f6 =-4('Y P3 2,,2 3 _ 2 fs3P = 0 ( 333 61)P fo = 0 (3.63) +fl = _ 2 aW2 +22 + +P(3i.64) For Eq. (3.52) to be satisfied, all the f's should be zero since the characteristic function W2 is only a function of t,xyt,pip2, and p3. Therefore, we get 12 equations: f= = f= f =.= f11 = 0 (3.65) Three of these equations are satisfied identically which are f4, f8, and f1o. For the point transformation under consideration, with c~2 = 0, the characteristic function can be written as W+ = WW 2(t,x,y,,)p + W22(t,x,y,)p3 + P2W(t,x,y,) (3.66) From the conditions f0 = 0, f = 0 f = 0, f = 0 we conclude that W21 is independent of y and ~r and that W22 is independent of 4. The condition f5 = 0 gives 22

2W_2j~2 +W_~23 aw_ - 2p3 P3 + by p 3 = (35.67) which gives aW23 = (3.68) by ~a~w _nnnnnnnnnn o (3.69) X =y From Eq. (3.68), W23 is independent of y. Equation (3.69) will be needed later. The condition f3 + 0 gives - 2 + + 3 2 1 (3 )y 6t +~ which can be separated into two equations: W2 = 0o (3-71) aW21 _ 2 0W22 = O (3.72) at -y Equation (3571) shows that W21 is independent of x. Since W23 is independent of y, Eq. (3.69) shows that W22 should be linear in y. Thus, we write W22 = W221(t,x)y + W222(t,x) (5.75) Substitution of W22 from Eq. (3.73) into Eqs. (3.69) and (3.72) leads to the following relations: W221 = (3.74) dW 2W221 = (3 75) dt Thus, W221 is independent of x. So, 1 dW(t 2 + W22 = 2(tx) (3.76) 23

The characteristic function W2 now becomes W2 = W21(t)pl dW21 + W222(tjx) p3 + W23(t2X),) (3577) dt From Eq. (3 74), W23 is linear in *j,i.e., W23 = W231(t,x)lr + W232(t,x) (3.78) Equations (3.74) and (3.75) then give W231-= 2 dt (3.79) 2 dt which also shows that W231 is independent of x. Thus, W23 = 1 d21 f + W232(t,x) (3.80) - 2 dt The condition f6 = 0 gives: 1 d2W2 1 + W222 W232 o ( 381) 1 dW1 + 0 (3.81) 2 dt2 at ax which can be separated into two equations as follows: d 2 dW2W = (3 82) dt aW222 + a3 = o (3.83) at ax Equation (3.82) shows that W21 is a linear function of t, i.e., W21(t) = W211t + W212 (3.84) where W211 and W212 are constants. From Eq. (3.83), a function 9-(t'x) can be introduced such that W222 =, w232 = + t (3.85) The final form of the characteristic function, W2, is therefore W2 = (W211t + W212)P1 + (1 W211Y + P + Wll + a (386) 24

where W211 and W212 are constants and e is an arbitrary function of t and x. Again, the condition fo = 0 has to be checked after the boundary conditions are considered. The transformation functions, al, C2, 3, and ~ for G2 are therefore given by 1Ca = W211t + W212 a2 = 0 (3.87a-d) 1 a a3 2W211Y + CX -: = 1 2W211* 6t 2 With the characteristic functions W1 and W2 for the two groups, G1 and G2 known, the next step is to find the absolute invariants. For the combined two-parameter group of transformations defined in Eqs. (3.4) to (3.9), the absolute invariants can be solved from the following system of equations: dt dx dy d_ l + aal (X2 + a&2 (3 + aa3 + + a: where a = e2/el. Substituting the transformation functions from Eqs. (3.49) and (3.87) into Eq. (3.88), we get dt dx a(W211t + W212) Wlllx + W112(t) dy + a 2 W211Y + a) =/Wm + dW* d0' 1 = eldE1 (3.89) dt Y at +at 2 211*) As an example, consider the case in which 0, 0, W112, and W212 are all aero. Eq. (3.89) then becomes: dt dx dy d = e aW2llt Wlllx a a W211y Wj1jy - W21r 2 2 The three independent solutions are 25

= Cl (3 91a) Will t aW211 2 = C2 (3o91b) 1 t 2 and =Cc3 (3.91c) Will 1 t aW211 2 As a final step, the parameter a has to be eliminated from Eq. (3.91). We then get - cl and -= C3 (3.92a, b) 1 1 cl 2 2 t xt where Eq. (3.92b) is obtained by eliminating a from Eqs. (3.91a) and (3.91c). The similarity variables are therefore =l =. and f(N) = (3193a,b) 1 1 2 2 t xt which are the same as the variables defined by Shuh.10 The boundary condition at the edge of the boundary layer, namely, y =: U (x,t) is then transformed to be x = = X f'(:) U e(X,t) which gives U (x,t) = (3.94) 26

The function P, as defined in Eq. (3.3), therefore becomes 4 = O. It can be shown by simple substitution that this function of $ satisfies the condition fo = 0 for both G1 and G2. Other special cases may be considered. For example, if we considered the case in which 8, Wll2, and W212 are zero; and 0 = f2(t)(x) where f2 is a constant, Eq. (3.89) becomes dt _ dx dy dr (395) aW211t Wlllx a a 2 W211Y + af2 2 W211*V 2 2 The three independent solutions are: = cl (3.96a) Will t aW2l 1 1 2 f 2 yt = c2 (3o96b) W211 and c3 (3o96c) Will 1 aW211 2 which, upon elimination of a, give 1 1 n = yt + 2 2 2 (3.97a) W211' and f(k) = 1 (5.97b) xt However, since the new form of 9 cannot transform y = 0 to ~ = O, we conclude that f2 must be zero. Equation (3.97) then becomes the transformations given in Eq. (3.93). Consider now another pair of groups, namely, 27

G3: a general point transformation G4: a general point transformation with 3 = 0. The transformation function, al, Q2, c3, 5, l1t, T2, T3, j13, 23, I{33, and 1T333 can be expressed in terms of the characteristic function, W3, as follows: aw3 O. = 1 api C = Pi 6p-[ - W3 aw3 w w3 dw3 dt i'J~2 = - dw - w2 T3 =- = a - 3 aw3 czi at a P1 +api Pii - Pi3 ( at + y a6- a a /a i i \ ~23 = 6x + a} P2 + ~p. Pi2- Pi3 -6x + P2 b~3 8a:3 /~ l'i )i* IT3 3 + d3 + T33 I dy di T p i3 + 3 P +i 613~~~~~~f1 i C By following the same steps as in the cases of groups G1 and G2, the characteristic function W3 for group G3 is found to be: W3 = (W311t + W312)p1 + [W321X + W322(t)]p2 W311Y + dx W311 - W32 ___22 Y + (3.99) 28

where W311, W312, and W321 are constants, W322 is a function of t, and B(t,x) is an arbitrary function of t and x. Ia1 addition, an equation which comes from f = 0 has to be satisfied. This equation is: o 3 W + (W311t + W312) + 6Q [W321(t)x + W322(t)] - - W31 - W 21(il - Pd3 2 d (3.100) Similarly, for G4 where a3 = 0, the transformation functions can be written as: 6 w_ aw4 =l apl, 2 ap2' 3: = Pi 6pl + P2 p - W4 ~rtaw1 a__w4 Jtj = 6 t - p ia aw4 aw4 = 2 - P2 aw4 aw4 ts3 - P3 = x + aaV3 P2 + ap3 Pi3 ~ Pi3 Xx + at P2 JT1 + 1 +( =13+ 4 Pt + 6pi Pi3 - Pi3 + - p) 1 33 = T2333 + P3 + p p (11) P3 + 1Pi3 yy + jt33 6y 6* Pa ap i i3 - Pi3 6y P, where i = 1,2 in the expressions f 33 and 333. By following the where i = 1,2 in the expressions of TtI3, n23, T(33, and n333. By following the same steps as in the cases of G1 and G2 where W1 and W2 are formed, the characteristic function W4 for group G4 can be determined, which is: 29

W4 = W41 P1 + [W421X + W422(t)]p2 - W4214r' dt Y + W4322(t) dt (3.102) where W41, W421, are constants, and W422 and W4322 are functions of t. In addition, the condition fo = 0 will lead to the following equation: W41 6 + [W421x + W422(t)] + W42 = 0 (3.103) The transformation functions for the two-parameter group are therefore: 1 aB G3: ol1 = W311t + W312, C2 = W321X + W322(t), e3 = W31 + - 2 W X [ W311 - W32] + dW322 Y at (53104) L2 dt YG4: al = W41, Q2 = W421x + W432(t), Q3 = 0, = = W421 + dW = W4322(t) (3 105) According to the theorems presented in Section 2. The absolute invariants can be solved from the following system of equations: dt dx dy (W3llt + W312) + aW41 [W321X + W322(t)] + a[W421x + W422(t)] 1 aB 2 W x d* _ dt W311 - W32 + dW y B + a W421 d Y - W 4322(t) =eld -L2 "dt at dt (3.106) As an example, we consider the case in which B, W322, W421, and W4322 are all zero and W321 and W422 are constants. For this case, Eq. (3.106) becomes: dt dx dy dre W311t +W3 + aW41 W321x + aW432 1 WllY 2 2 (3.107) 30

The three independent solutions to Eq. (3.107) are: W321x + aW422 (3 108a) (W311t + W312 + a W41) = cl(.8a) = c2 (3.108b) W311t + W312 + a W41 and andC3 -(3.108c) n - - (W31llt + W312 + aW41) where n = W321/W311. For the special case in which n = 1 and W312 = 0, elimination of a among Eqs. (3.108a), (3-108b) and (3.108c) then gives l_ - = c4 (3.109a) /W432t - W41x and 4 = c5 (3.109b) dW422t - W41x The similarity variables are therefore: al = Y, f(rk) = (3.110) "W422t - W41x W423t - W41X which is the second transformation obtained by Shuh.10 It demonstrates again that this transformation should be obtained from a two-parameter group of transformation instead of the one-parameter group of transformation given in Manohar's work. This important point was shown by Moran and Gagioli5 in a recent report, which is supported here by the analysis using the method developed by the present authors. The boundary condition at the edge of the boundary layer is transformed to the form: =: f(oo) = U (x,t) 31

which means U has to be a constant. It can be shown that this form of U e e satisfies Eqs. (3.100) and (3.103). For other values of n in Eq. (3.108), it can be shown that Eqs. (3.100) and (3.103) will not be satisfied even though transformations can be obtained. 3.2. APPLICATION TO THE NONLINEAR DIFFUSION EQUATION As a second example, the nonlinear diffusion equation is considered. This equation is: ax 6ax +y 6 nay at6 (3.111) which can be written as: n n-1 F = 4r (P22 + P33) + n n (P2 + p3) - Pi = 0 (3.112) where Pi at' P2 = x P22 2 etc. The differential equation F = 0, given in Eq. (3.112), will be invariant under the two-parameter group of transformation X = X + 6Cla2(tx,y,p,) + E2ia2(tx,Y,y,) Y = y + 6 ElC3(t,xyr) + bE2X3(t,x,y,3)'Ir ='fr + 8El(t,X,Y,i) ) + be2C(txyt) Pi = Pi + bEli(t,x,y,4rp1,p2,P 3) + E26ti(t,XYtP2,P2,P3) Pi = Pii + PJlt.ii(t,x,y,lpl,-.,p33) + 6c2tii(t,x,y,tpl,.. pP33) (5.115) if UF = O (3.114) 32

or, from Eq. (2.20c), el U1F + e2 U2F = 0 (3.115) Since el and e2 are arbitrary constants, Eq. (3.115) gives U1F = 0 and U2F = 0 (3.116) In their expanded form, Eq. (3.116) can be written as SF SF SF SF SF SF l at + 2 + C3 + + 3 a y + a (3~-117a) and - F - F - F F - F - F ll a + +2 + + +i p i 0 (3. 117b) at ax ay CNV C'V 13 6 j The second step is to introduce two groups. Let us consider two groups as follows: G1: general point transformation (whose characteristic function W1 is therefore linear with respect to the p's, G2: same as G1 except the characteristic function be independent of P3. For group G1, substitution of F from Eq. (3.112) and the transformations in terms of W1 from Eqs. (2.25) to (2.28) into Eq. (3.117a) then gives: fo + fl P12 + f2 P13 + f3 P22 + f4 P32 = 0 (3.118) where n4- Pl - nxr (P2 + P3 Pi + P2 - + PP -/ S np2+ a _ + at + a n-l /al awl n-l awl W n a2W 2W 2n4r P2 + P2 -W- + P3 a 2 + 2p2 + P2 n (p + p62 2 2+ _w + L2 \2W2 + 2p3 + ( 3.119) 33

fl ( p + P2 aP a, (3.120) =2 2 + P2 f2 2 2 aaw6 +P3 a2Wp (5.121) a2W, a2Wl a2Wl a2W, f3 = 2+ 2 a 2 W 2p3 (3.122) r = 2P ax +ape a aps ay- p3 (o f4 2 W + P2 2W + P w_ + P3 P2w / (3.123) Since the f's are independent of the f's, Eq. (3.118) is satisfied if the following equations are true: f = fl = f2 = f3 = f4 0- (3.124) Now, the characteristic function W1 can be written as: W1 = Wll(tx,y,4r)p1 + W12(t,x,y,Yr)p2 + W13(t,X,%y,0)p3 + W14(t,xy,{*) (3.125) The conditions f1 = 0 and f2 = O, Eqs. (3.120) and (3.121), show that W1l is independent of x, y, and 4r. The condition f3 = 0 gives aW u+- W13 aw___3 o (3.125) which can be separated into three equations, namely, aWls + 0 a6 xW13 - 0 (3 127a-c) ax Y W12 - 0 Equations (3.127a,b) show that both W12 and W13 are independent of $. Equation (3.127c) will be used later. Similarly, the condition f4 = O gives aw_ + aw - O (j.128) ax (~v5

Finally, the condition f = 0 gives 0 g0 + g P + 292 P2 + g3 P2 + 4 P3 = 129) Since the functions g's are independent of Pi, P2, and P3, Eq. (3.129) gives go = gl = g2 = g3 = g4 = 0 (3o130) which, in complete form, are: _W14 n (62w14 62W n (X + 2) = 0 (3.131) - nK'_W14 + dW11 + aW + -n (2 W3 + = 0 (3.132) dt (2 -1 W13 _W14 n-2 n 2W14- n-1 +_W n~- nr a 12 W + + n W4 - 2 2n ( + W =0 (3133). oy o4r /. =1 aW2 2 n-l aW14 2n 62W14 2_ n 62Wl4 0 -2nt -2 C)r -24 = 0 (35.134) n-1 W1l4 + = (3135) -2n~ 1y t (3.135) Equations (3.127c), (3.128), (3.131) through (3.135) are the conditions to be satisfied by the functions W11, W12, W13, and W14 in the characteristic function. Although the general solutions to these functions are difficult to obtain, special cases can easily be investigated. For example, if we consider the special case in which W14 is linear in 4, then W14(t,x,y,jr) = W4l1(t,x,YY)r + W142(t,x,Y) (35136) Equation (35135) gives three equations, namely, aW141 = 6W42 = awl s O0 6y 6y at from which we conclude that W141 and W142 are independent of y and W13 is independent of t.

Equation (3. 134) gives W + 2(n - )n 4 + 2n*n-1 W L= 0 6t +x 6x which can be separated into three equations (for n f 1) as follows 6ja_ = W41 = W14 = (3.137) 6t )x 6x Equation (3.137) shows that W12 is independent of t, and that W141 and W142 are independent of x. Equation (3.131) is therefore satisfied automatically and Eqs. (3 133) and (3 132) become - nt-n (2 + W142) - 2n*W + W141 ) O 2(n-2 n-l dWlj n-l 2n1 aW - n2n (W141* + W142) + n- + n W141 d+ n + (2 W141 + o dt +W (3 139) Summation of Eqs. (3.138) and (3-139) then gives n(l - n) W142 + nn- W + 2 W13 dW1l = 0 (3.140a) n(l - n)- + nlr W1414 +2 fy- dt from which, W142 = 0 nW141 + 2 dWtj 0 (3-140b) y dt Finally, conditions (3.127c) and (3.128) show that a function 01(x,y) can be introduced such that W12 =W = (3.141) and 36

2+ Y2 = (3-142) The final form of the characteristic function for group G1 is W1 = Wll(t)p! + y P2 + X P3 + - 2 I (3143) 6y 6ax n dt ax 6y! The transformation functions for group G1 are therefore al = Wll(t) a2 =a (3. 144) 0e 60,~~~~~ - n (o x by dt W For group G2, the characteristic function W2 can be found by following the same steps~ If, in addition, W2 is assumed to be linear in A, we then get 1 dW2+p2 dW W2 = W21(t)pl + W22P +n dt (3145) The transformation functions for group G2 are therefore: 1i = W21(t) U2 = W22 (3.146) U3 = 0 1 dW21 n dt The next step is to find the absolute invariants of the two-parameter group G12 by solving the following system of equations: 37

dt dx dy Wll(t) + aW21(t) al aWa..+ aW22 [11 /dW 2 2 + a dW21] = eldE (3.147) n \dt ~ax ay n dt As an illustration, consider the case in which Wll(t) = t, ~1 = xy, and W21 is a constant. Equation (3.147) then becomes dt d dx dy eldE (35148) t + aW21 x + aW22 y 1 n Solutions to Eq. (3.148) are x + aW22 t + aW21 =y c2 (3.149a-c) t + aW21 = C3 (t +aW21)n To illiminate a, Eq. (3.149a) is first solved for a as follows: a = c1t x (3.150) W22 - C1W21 The parameter "a" from Eq. (3.150) is then substituted into Eqs. (3.149b,c) and we get y C(3.151a) (W22t - W21x) c4 (.5a) 1 = c5 (3.151b) (W22t - W2lx) 38

The similarity transformation is therefore given by = (W22t - W21x) f (3 152) (W22t - W21x)n which is a form not given in the literature. Consider now a third group G3 defined as: G3: general point transformation with W3 independent of P1i By following the same steps as before, the characteristic function is found to be:.3 P2 + - n3` y3 (3.1 53) i3 =y P2 + P3-na ay where $3 is any function of x and y satisfying the Laplace equation: ax2 + y2 = (3+154) The transformation functions for group G3 are therefore: a = 0 a83 (3.155) 53 2 62Q3 n ax by For a -two-parameter group G32 whose transformation functions are given by 39

dt dx= dy dr = edE (156) aW21(t) a_+ aW22 2 ( a d )W2 ay )y ax x y n dt As an example, consider the case in which W21 = t, W22 = 0, and 43 = xyo Equation (3.156) becomes: dt dx dy d~r dt dx dy eldc (35157) at x y 2 a n njt Solutions of Eq. (3-157) are: X _ _ JL. = c3 (3.158) 1 = cl, 2 2-a a na t t Elimination of "a" then gives: C2 - = 4 x 1 2 n n t x The similarity transformation is therefore Y f(k) x 1 2 n n t x which is given in the literature.l Other groups can be defined and transformations found, by following exactly the same steps as in the above examples. 3 3. CONCLUDING REMARKS The method developed in this section is seen to be very general and, like other reports in the series,7,8,9 involves mostly algebraic manipulations. It is systematic and does not require specification of an arbitrarily defined group at the beginning. On the contrary, the group is systematically determined. 40

The method also supports the important point that reduction of r variables can only be achieved by introducing a r-parameter group.5 41

REFERENCES 1. Ames, W. F., "Similarity for the Nonlinear Diffusion Equation," I and E C Fundamentals, 4, 1965, pp. 72-76. 2. Cohen, A., Lie's Theory of Differential Equations, Heath, New York. 3. Eisenhart, L. P., Continuous Transformation Groups, Dover, 1961. 4. Manohar, R., "Some Similarity Solutions of Partial Differential Equations of Boundary Layer Equations," Math. Res. Center Rept. 375, University of Wisconsin, 1963. 5. Moran, M. J. and R. A. Gagioli, "Similarity Solutions of Compressible Boundary Layer Flows via Group Theory," MRC Rept. 838, University of Wisconsin, December 1967. 6. Moran, M. J., R. A. Gagioli, and W. B. Scholten, "A New Systematic Formalism for Similarity Analysis, with Applications to Boundary Layer Flows," MRC Rept. 918, University of Wisconsin, 1968. 7. Na, T. Y., D. E. Abott, and A. G. Hansen, "Similarity Analysis of Partial Differential Equations," Rept. 07457-17-F, University of Michigan, Dearborn Campus, March 1967. 8. Na, T. Y. and A. G. Hansen, "General Group-Theoretic Transformations from Boundary Value to Initial Value Problems," NASA CR-61218, February 1968. 9. Na, T. Y. and A. G. Hansen, "General Group-Theoretic Transformations from Nonlinear to Linear Differential Equations," Rept. 07457-19-T, University of Michigan, Dearborn Campus, April 1969. 10. Schuh, H., Uber die,,ahnlichen" Losungen der instationaren laminaren Grenzschichtgleichungen in inkompressibler Stromung. "Fifty years of boundary layer research," Braunschweig, 1955, pp. 147-152. 42

57' k4 A Co)