THE U N I V E R S I T Y OF M I C H I G A N DEARBORN CAMPUS Division of Engineering Thermal Engineering Laboratory Technical Report. GENERAL GROUP —THEORETIC TF-ANSFORMATIONS FROM BOUNDARY VAL'TE TO INITIAL VALUE PROBLEMS Tsung-Yer, Na Ar-thur GO Hansen ORA Proj ec't 07457 under cont.racet with: NATIONAL AERONAUTICS AND SPACE ADMTNISTRATION GEORGE C. MARSHALL SPACE FLIGHT CENTER CONTRACT NO. NAS 82o0065 HU'NTSVILLE) ALABAMA admiri is4 ered ih hrough: OFFICE OF RESEARCH- A3DMINISTFA'TION ANN ARBOR Febulrary 1.968

FOREWORD Tsung-Yen Na is Associate Professor of Mechanical Engineering, University of Michigan, Dearborn Campus, Dearborn, Michigan. Arthur G. Hansen is Dean of Engineering, Georgia Institute of Technology, Atlanta, Georgia. a O

TABLE OF CONTENTS Page ABSTRACT v 1. INTRODUCTION 1 2. SIMPLE GROUP-THEORETIC METHOD 3 2.0 Foreword 5 2.1 One-Parameter Methrod 2.11 Linear Group of Transformations 5 2.1.2 Spiral Group of Transformat;ions 11 2.2 Two Parame-ter Method 14 2.2.1 Transformatlion of Two Eoundary Conditions 14 2.2.2 Simutaneous Differential Equations 17 2.3 More General Types of Equations 19 2.4 Concluding Remarks 20 3. GENERAL GROUP-THEORETIC METHOD 22 3.0 General Consideration 22 3.1 The Infini'tzesimal Con-act Transformatfion 23 351.1 Infinit;esimal Transformation 23 3.1.2 No+tation for t1he Infinitesimal Transformation 24 3.1.3 Invariant Funct;ion 25 3.1.4 Extension to; n Variable 25 3.1.5 Invariance of an Ordinary Differential Equation 26 31.O6 Definit-ion of a Contact Transformation 27 3 1o.7 Inflinritesimal Contac-t~ Transformation 27 3.2 The General Method 29 4. APPLICATION OF THE GENERAL GROUP-THEORETIC METHOD 30 4.1 Applicat-ion to Falkner-Skan Solutions 30 4.2 Application to the Heat Conduction Equation with Nonlinear Heart' Generation 38 4.3 Concluding Remarks 49 REFERENCES 50

ABSTRACT A systematic method using S. Lie's continuous contact transformation groups is developed in this report which enables one to search for all possible groups of transformations under which a given differential equation can be transformed from a boundary value to an initial value problem. Examples are worked out in detail as illustrations of the procedure.

1. INTRODUCTION The present report treats one of the most important applications of the concept of continuous transformation groups: the numerical solution of boundary value problems as related to the class of transformations from boundary value to initial value problems. A completely new and general method will be developed based on S. Lie's continuous and contact transformation group. In reference 6 the same concept was applied to develop a method of searching for all possible groups of transformations in a similarity analysis of partial differential equations. A boundary value problem is characterized by the property that its boundary conditions are given at more than one point. In the absence of closed form solutions, numerical solutions must therefore be obtained by a trail-and-error procedure in which an unspecified boundary condition is assumed arbitrarily. The accuracy of the assumption is then checked by the fulfilling of the boundary condition at the other point.* It is therefore clear that the class of transformations from a boundary value to an initial value problem is of greatest importance in that it eliminates the trial-and-error procedure and simplifies considerably the process of numerical integration of the equation. The first research on this type of transformations was given by Tapfer in 19127 for the numerical solution of the Blasius steady, two-dimensional boundary layer equations with uniform mainstream velocity. After a similarity transformation is made of the governing partial differential equations, a third-order nonlinear ordinary differential equation is obtained with the boundary conditions specified at two points, namely, two at zero and one at infinity. The equation is then transormed by TPpfer's method and the problem becomes an intial value problem. There seems to be little work on this subject until 1962 when Klamkin2 published an important paper which considerably extended the range of applicability of the method, including applications to simutaneous ordinary differential equations. Both Thpfer and Klamkin's research consider the case in which boundary conditions are given at zero and infinity. No general theory, however, was given. Mostly recently, the method was reconsidered from the point of view of the theory of transformation groups in References 3 and 4. As a result, a general method was, indeed, developed for given groups of transformations. The method treated by Thpfer and Klamkin was found to be the special case of a linear group of transformations.. Introduction of a "spiral group" of transformations made it possible to extend the method to a wide class of ordinary differential equations. According to this method the boundary conditions can be specified at both finite and infinite points. Extension of the method to problems in which two boundary conditions need to be transformed was also made by using a multi-parameter group of transformations. As long as the group of transformations is initially given, *One would hope to eliminate this procedure by transforming all conditions to apply at one point. This is the method outlined here.

the method is straightforward. However, the arbitrariness in the selection of a practical group of transformations considerably limits the scope of application of the method. There is, therefore, a need to develop a met nod of searching for all possible groups of transformatio:ns for a given ordinary differential equation without resorting to an initial selection. In the present report, a systematic meti.Ld using S. Lie's contiriuous contact transofrmatioli -rerups will be developed enr-ich enables one to search for all possible groups of transformations under -Fag h the present method can be applied. The meth-od used follows closely the.c' —ral group-theoretic method given in Chapter iL of Reference 6. A s-mmmiary of te-e method given in References 3 and 4) will be given in tihe next section to s,.umnzarize the present state of the research and the general concept of the method~ c he general method will be developed next, folloowed by two examples to s.i.w t:he steps which have to be takcen to get spec i fic groups of transformati: — p iiable to a given problem. 2~~,Ab agvnp

2. SIMPLE GROUP-THEORETIC METHOD 2O FOREWORD This art icle gives a critical review and summary of the method of transformation developed in References 3 and 4 with certain modifications. In section 2.1,'the general concept of the method is illustrated by two examples, namely,'the Blasins' problem from boundary-layer theory and the heat conduction equation with power-law heat; generationo5 The boundary conditions in the first example are given over an, infinite interval. In the second example the method is appli. ed to a case where the bounrdary conditions are given over a finite interval. A disc-ussion is presented on the type of equations for which a spiral group is needed. Irn section 2,2, extension cf the method to problems in which two boundary condi'tionrs need to be transsformed is disc-ussed in detail, Extension to more general types of equations is given in Sectriocn 2.3> Finally, Section 2.4 conclues the discussion of the method wi'th an evaluation of its merits and limitations. 2o 1 ONE-PARAMETER METHOD 2olol Linear Group of Transformat;ions Consider the Blasius' equation mentioned earlier where we want to solve the equation -+- f 0 (2.1) dl 3 2 d2 subject to the boundary conditions.f (o): df(O ) f(oo) drq d~ A on.e-parameter linear group of transformation -A nf = Af (2.2):is applied too this equation, wlere A is the parameter of transformation and ~z and d2 are two constarnts'to'be determined. Under this transformation, Eq. (2.1) becomes

A%-3a, dof 22- 2a, 1 - d -fA2 d + A2c2-2 1 f = 0 (2.3) It is seen that the transformed equation, Eq. (2.3), will be independent of the parameter A if the powers of A in both terms are equal, i.e., if -2 - 3a5 2% 2a, (2.4) Equation (2o3) then becomes d3f 1 d2f ddr' 2 d -12 0 (2.5) From Eq. (2.4), we have 2 = - l (2,6) This gives one relation between a, and a2. The other equation required for the determination of a, and a2 is given by putting d2f(O) = A (2.7) from which A. df2 A (2.8) Therefore, the transformed boundary condition will be independent of A if a2 2a - 1 (2.9) which leaves d 2;(O = 1i (2.10) 4~=

The two unknown constants, ac and ~C2, can then be obtained from Eqs. (2.6) and (2.9) as a, 2 - (2.11) Finally, the parameter of transformation, A, can be obtained by using the original boundary condition at infinity which gives AU2Ce d(o) (2.12) d7 Therefore, we get A 3/2 (2*13) This example shows clearly the general concept of this technique. In general, two unknown constants (e g., cyl and a2 in this example) are to be determined if one dependent variable is involved. Two equations for their determination are therefore necessary. One of these equations is obtained by requiring that the transformed ordinary differential equation be independent of the parameter of transformation, A; the other condition is formed by setting t;he original required boundary condition at the initial point equal to the parameter A. Finally, the parameter of transformation, A, is determined from the boundary condition at the other point. The solution of the problem then consists of two steps. In the solution of the Blasius equation, for example, Eq. (2.5) is first solved with the boundary conditions daT(o) d7(o) 7(o) = ( ) d, () 1 (2.14) and the value of df (oc)/d7 is then obt4-ained from the solution which in turn gives A by Eq. (2.13). With al, 2 and A known, solution to Eq. (2.1) can be computed using Eqo (2.2). It is seen that the problem is reduced to an initial value problem. With Blasius' equation treated in this way, extension to a new class of problems involving finite intervals becomes more obvious. We now consider the equat ion deT n dX2 + B T 0 (2.15)

subject to the boundary conditions T(O) = 0, T(L) = 0 This equation can be interpreted physically as heat- conduction with power-law heat generation5 A one-parameter linear group of transformation x A= T = A2 T (2.16) is made and the two equations needed for the determination of al and a2 are obtained by requiring that: (1) the transformed equation to be independent of A; (2) by setting dT(0) = A (2.17) dx From these two equations, al and a2 are found to be 1-n = 2 = 1 (2.18) The boundary condition, (2.17), becomes dT(O) () = 1 (2.19) dx The parameter A is then found by transforming the boundary condition at x = L, which gives T 0 at Al x = L (2.20a) or, A = )(w r (2.20b) x(where T = 0J

To recapitulate, the procedure is therefore as follows: Firstly, the transformed equation is solved with the boundary conditions T(O) = 0 and the condition given in (2.19). Then, the value of x where T = 0 can be determined from the solution. The parameter of transformation, A, is finally computed from Eq. (2.20b). Again, the problem is reduced to an initial value problem. Consider next the rather general second-order differential equation N,_2ymi /d)ni ri Si i1l A x ) )x 0 (2.21) subject to the two cases of boundary conditions Case I addy(0 ) y(O) 0 d, od k Case II y(o): o dy(L) dx Consider the linear group of transformations x =, 1 y = B 2y (2.22) Under this group of transformation, Eq. (2.21) becomes N Bmi(2-25i1) + ni(52-5i) + rji2 + SiS1 i=l i i x Ai ( i y = 0 (2.23) Equation (2.23)'will be independent of the parameter of transformation, B, if the powers of B in each term are equal, i.e., ma (B- 251) + n (B2-51) + rl 52 + s1 B1 mi(52-2B1) + ni(S2-1) +ri f2 + Sil(2.24) 7

where i = 2,...,N. In general, (2.24) gives (N-1) equations with only two unknowns, 51 and 52. The method is applicable only if the (N-1) equations actually reduce to one independent equation. To illustrate the problem that may arise, the Falkner-Skan equation may be citced d3f. f dfi 3 f 2 + ( = 0 (2.25) dr3 2 dL] Under the linear group of transfcrma';i An defined by Eq. (2.22), Eq. (2.25) becomes B2-3pr1 d3f 222.' U22- 251 df 22~ B d 3 + B-2f iLl ]- o= 0 (2.26) which is independent of t;he paramete r Cf transformat ion if 2 - 351 22 - 2 0 = ~ 2P2 - 251 (2.27) Two independent equations are obtained from (2.27). As a result, we get 51 = 52 = 0 which means the method is inapplicable. Assuming for now that; such a situation does not exist, Eq. (2.23) becomes N d27mi __n.i o ZA: - 0m nl S (2.28) i=l AWith one relation between f1 and 52 obtained from Eq. (2.24), the other relation required for the solution of p1 and P2 can be obtained by putting the slope at x = 0 equal to the parameter of transformation, B, i.e,, dy(0) -B (2:.29) After transformatiorn, we have: B2 1 dY(O) B dx which is independent of B if B2 - B1 = 1 (2.50)

The transformed boundary conditions are therefore, y(O) = o0, d =( 1 (2.31) Equations (2.24) and (2.30) give solutions to p1 and P2. Finally, the value of B can be found by applying the boundary condition at the second point. Thus: Case I: dB B ~d --- = k or9 B = k l/(P2-d31) (2.32) JCase II: d - - 0 at B x L Or, aB =, <(2.33) B(where dd ( 0 Thus, Eq. (2.28) is solved with the boundary conditions given in Eq. (2o31) and the value of x where ddy/dx = 0 can be found from the solution of the transformed equation. This result Ls then substituted into Eq. (2.33) and the value of B computed. It should be noted that there are cases where additional problems may arise. Asan example, if the value of 5 in Eq. (2.15) is negative and also n = 1, then d2T d h- = o with the boundary conditions

— () ~ dT(O) T(O) =0 9 The solution is T -- ssinnh 7 which is never zero. This places anctsher limi ation on the method. The boundary conditions at, x = L in case iI need not- be homogeneous. For example, one may have dd at x L y Thus, B-d31 dd = k at B'x L (2.34) Since k, L, 51 and P2 are known constants, the value of B can be found by searching for values of x and d y/7dx in the solution of Eq. (2.28) which give the same value of B in both equations of Eq. (2~34) 0 One way of doing this is by eliminating B in Eqo (2~34) which leads i4o d y k( (2.L5 Next, (ddy/dxd) vs. x is plotted as a curve. Anot:her curve from the solution to Eqo (2.28) can be plotted with the same coordinates. The intersection of these two curves will give the required value of T and dy/dYd which in turn can be used to compute B from Eqo (2-34). One final remark about, the method:is necessary. Suppose the boundary condition at the initial point is dy(O)/'dx - Oo In this case, we merely have to put y(O) - B Thus, B2(o) B 10

and if the result is to be independent of B, 52 must be equal to 1. Under no circumstances, however, should the boundary condition at the initial point be nonhomogeneous. If it is, one more equation relating 51 and P2 will result. The method then cannot be applied. 2.o12 Spiral Group of Transformations We now consider a class of nonlinear ordinary differential equations in which a spiral group of transformation rather than a linear group is needed for the method to apply. We consider here the class of equations CN (d2_mi (d'ni i C y PiY e qi 0 (2.36) with the boundary conditions Case I: x = dy = 0; x = 1: y = 0 dx Case II: dy x = 0 -. 0; x = a: y = kl dx where Ci, mi, ni, pi and qi are constants and N is the number of terms in Eq. (2o36). Left us define the one-parameter spiral group of transformations X- eA- Y = y + 2A (2.57) where A is the parameter of transformation and cl and a2 are constants to be determined. Under this group of transformation, Eq. (2.36) becomes ae( -2m ol -n c1 +P 2z+qi l)A i=l x2\ S __ iePiY xqi = O (2.38) 11

The equation is seen to be independent of the parameter of transformation, A, if the powers of e in each term are equal, i.e., (-2mi - ni + qi)ce + + P 2 (= 1 - n( + qjl a, + p1 c2 (2.39) where i = 2,..,.N. The transformed eqaation becomes i=l ePCY q1 (2.40) Equation (2039) represents (N-1) equations. In general, the method can be applied only if one independent equation resultos from these (N-l) equations. For example, if Eq. (2~36) takes the form 2 d + eY = 0 (241) dx2 X dx Equation (2.39) then gives one independent equation for al and C2 as -2 -= a2 (2.42) Physically, Eq. (2.41) may be interpreted as the equation for heat conduction in spheres with exponential heat generation5 To determine the second relation for the solution of a1 and a2, we put y(O) - A (2.43) Upon transformation, this condition becomes: 7(o) + a2A - A which is seen to be independent of A if o2 - 1 (2.44) 12

The transformed boundary conditions become y(0) = 0 and d) = (2.45) dx For the example given in Eq. (2.41), ac and ac2 can be found from Eqs. (2.42) and (2.44): 0t = and a2 = 1 Finally, to get the parameter of transformation, A, the boundary condition at, the second point is used. The two cases are considered separately. Case I. The boundary condition at x = 1 becomes y + A = 0 at e1A = 1 (2.46) Eliminating A, we get e-cllY X = 1 (2.47) Case II. The boundary condition at x = oo becomes x -: Yc + a2A = k or A - o- [kl - (oo)] (2.48) ~2 Therefore, the method proceeds as follows: First, Eq. (2.40) is solved with the boundary conditions (2.45). In case I, the solution curve to Eq. (2.LO) can be plotted on v vs. x coordinates. Equation (2.47) is plotted on the same coordinates. The intersection of these two curves gives the values of y and x which give the same value of A from Eq. (2.46). The value of A is then determined. In case II, the value of A can be computed from Eq. (2.48). 13

The solution to the original equation, Eqo (2.36) can be obtained from Eq. (2.37) since now e,, {Y2, and A are known constants. 2.2 TWO PARAMETER METHOD 2o2.1 Transformation of Two Boundary Condit-ions The method developed i.n the preceding section can be extended to higherorder differential equamtons as long as 5only:one boundary conditxion is required to be transformed. In this section and the next section, the method will be extended to higher-order differential equations in whsich more than one boundary condition need to be +transformedo For such cases, multiparameter groups of transformations are requiredo Consider now +the third order different7ial equation i Ai ( d3Yi m S i y y51 sx- 0i O (2.49) subject to the boundary conditions Case Io dda y() dd2y(o) y(O) = 0 kl k dx dx Case II. y(L, dd2Sy(L) y(O) O (, dL - k2 dx1 dxd2 We now define a two-parame4-er group of iransformat;ion x 2 x y -y (2.50) Under this group of:transformation) Eq. (2.49)j becomes N A1 Bm:i( 3 1)'ni(2 41) (+r 2 1S2 i *'1 ('d y (i Si (2.1) 14

The method can be applied if, for all i's, mi( 32-3 )+ni( f2-2p1 )+ri( 2-l )+SiP2+ti 1 = C1 (2.52a) mi(72-37Y1)+ni(7Y2-2 1)+ri (72-71)+Si 2+ti 71 = C2 (2.52b) where C1 and C2 are two arbitrary constants. Equation (2.51) then becomes N d3ymi / dni _i Si ti = O (2.3)0 Z Al i~~i) yY) S1 ~i = o (2.53) i=l dx/ \dx/ y For example, the equation d3y 1 dy d2y ( 2 = 3' + v 3'2 =7 (2.54) belongs to this class. The boundary condition at the initial point, y(O) = 0, can be transformed to 7(o) = o (2.55) To get the other boundary conditions at the initial point, let us put dy(O) d2B Y( dx = andx 2 C (2.56) Upon transformation, (2.56) becomes BP2-Ad CY2-71 d7(O) = B (2 57a) dx and B 22~C1221d2( = C (2.57b)

which are seen to be independent of B and C if, from (2.5'7a), 2 -: 1. Y 2 1 0 O and) from (2. 57b), 32 Y 22- 0, 27 Y 1. (2.5y[a) and (2.57b) are t;hen transformed to dt(O)'..2 y = 1 (2.58) dx Urx and the values of p1, P2) y1 and Y2 are l = 1i 52 = 2 1 Y2. To get the parameters of transformation, B and C) the boundary conditions at; the second point are used. Case I 2~d- d., i _k_ ~B Cd l kC (2.59a) B2-ds2 Cd21 k2 (2.59b) d 7(dO) Case II =i L BC = (2.60a) x (where iddy 0) B2d2 C+2 k2 (2.60b) d2y d 2 16 -i

Therefore, B and C can be solved from Eqs. (2.59) or (2.60). The method can easily be extended to equations of the type: N ~.dy(d mi (dYri Z A1'i dx i (dx esi x O (2.61) The only difference here is that one assumes a transformation group defined by x = e x, Y = 7 + 52B + 72 C (2.62) Other steps remain the same. 2.2.2 Simutaneous Differential Equations Application of the method to simutaneous differential equations again involves mutli-parameter groups. Consider now the following system of two simutaneous equations: N ym1 (~$~ni P d Q2zr (dzf zi ix = (2.6a) J Q + dx zi x = 0 (2.63a) M 2 M (dym i2 2 S 4 B. - "Y) 1 j Jy~~ d 2r;j c z'~ X (2.63b) j=l J'W Y dx subject to the boundary conditions Case I. ddy(oo) - ddz(o) y(O) = o, z(O) - o, d = k, k2 Case II. ddy(L) ddz(L) y(0) = 0, z(0) = 0, xd - o k Let us now define a two-parameter transformation group X = -, y = X2y, z = X3 5 (2.64) 17

Again, the differential equations, (2. 653), are independent of the parameters of transformation, X and t, if the powers of X and 4 in each term are respectively the same. This leads to the following system of equations for the solution of 51, 52, 53 and 5: (S2-25.)ml + (P2-Pl)ni + 2 Pi + + Sz(53(3-:) + tS3-3 + qi (Pfi2-: jm2 L + (P2-iAnl + P2( Pi _:13-2p) + Sl(P3~ 3) + tlql P (2.65) ri+ S + S i = r + Si + i1 (2.66) (P2f-2f1i)j + ( 2=5l) j + 2 P + ) + SJ (3-) + j + qj 1 (P2-2 +I)m1 + (02-1)K1 + +2 PL f (+3 2i) 3 Sl(:2cI) + 7 L3 + ql + 1 (2.67) Tj + Si + tj - l + +(2. 68) where i - 2,. o o, N and j - 2, oo,Mo Substitution of Eqs. (2.66) and (2.68) in-tco Eqso (2.65) and (2.67), resepctlvely, gives (mi +n1i+Pi ) 2 ( 2mi+ni -+2r i+S i qi ) 5:. (ml+n +P ) 2 - ( 2m +n +2r S 1 q ) 1 ( 269) (MJ.jPj)2 s (2 ~j+j+2+S j-qj) + (r l -tC1 ) t 2 2 (r1i;+ + S1 i ql)1 (2.70) The method is applicable.if Eqso (2.69) and (2.7'0) each represent only one independent- equatxion and that+ bosth give -;,he same ratio of P2/Pl. If these conditions are satisfied) the ratio of p2/~;. is known. Next, the required boundary con!ti.ons are definrled to be equal to X and p respectively, i eo, 18

y'(O) = X and z'(O) = Upon transformat ion, Bkei7T' (0) = \ and k1 \03~1Z(O) = which then give 2 - 5. =i 81 =_ 1 and 33 8 P 1 = 0 (2.71) The ratio of 5/i1 obtained from Eqso (2e69) and (2070), together with Eqo (2071), gives solutions of p1, 32, 53 and 6o To get the parameters of transformation, the same method discussed in previous paragraphs can be applied~ It will not be repeated here. The method can be easily generalized to include cases with exponentials of y or z or both in Eqso (2U63 ) 203 MORE GENERAL TYPES OF EQUATIONS The method developed above can be extended to more general types of equation0 Two cases are considered here. Consider now the general sce.... crder differential equations: b({ b (U )Si L y'x = (2072) where G represents an arbitrary function of the argument indicated. Case Io The method can be applied if., under the linear transformation group x A x, y A y, only one relation between ani and a2 is obtained from the condition that Eq. (2~72) is invariant under thhis group of transformation. As an exsmple, the equation 19

3 -y + sin1y + = 0 dx + sin will give one relation between al and a2, namely, 2a2 - 1 = 0 Case IIo If y is absent in all terms in Eqo (2.72), the spiral group of transformation can always be applied for any arbitrary function in Eqo (2072). As an example, consider d2+y (dy>+ 1 O (2o73) where fl is any arbitrary function of dy/dx. Under the spiral group of transformat ion x =e e x, y y + a2a Eqo (2o73) becomes -2aa d2- -aa dVy ~e dxa + fl(e ) +1 = 0 dx dx which is independent of a if aC 0 for any arbitrary function fl. The remaining steps remain the same. 20 4 CONCLUDING REMARKS In this section, the application of a linear or spiral group of transformation to the class of transformation from a boundary value to an initial value problem is treated~ The method consists of three basic stepso First, a transformation group is defined and the given dcifferenrtial equation is required to be invariant, i eo, independent of tihe parameter of transformation, under this group of transformation. In step 2, the required boundary condition is set to be equal to the parameter of transformation~ Finally, the parameter of transformation is found by using the boundary condition at the second point. Knowing this general concept, the method treated in this article can be applied to higher-order equations or other types of equationso It is simple to apply and only algebraic solutions are required to get the transformation~ The main disadvantage of this method lies5 however, on the arbitrariness in the selction

of a proper group for a given differential equation. In the next two sections, a very general method will be developed which makes it possible to search for all possible groups of transformation under which the given differential equation can be reduced to an initial value problem. 21

3. GENERAL GROUP-THEORETIC METHOD 3o0 GENERAL CONSIDERATION In order to introduce the general group-theoretic method, the method developed in section 2 will be summarized by considering a second-order ordinary differential equattion as follows: Consider the ordinary differential equation T; I Y; = ~ (3-1) with the boundary conditions y(O) =, y(i) = a. The differential equation is transformed by introducing a one-parameter group of transformation, vizo, x = f(X, A, aCl 02) y = g(ye A, al a02) (3.2) where an and a2 are conrstant;s to be determined before the transformed equation is solved, and A is the parameter of transformatiorn to be determined after it is solvedo To determine cl; and a2, two conditions are imposed: i. the given differential equation is to be invariant; i.e., it should be independent of the parameter of transformation, A; and iil the boundary condition dy(0)/d is t(o be independent of A for some choice of dy(O)/dx as a function of Ao If /v and 2~ can be found satisfying the above conditions, the method can proceed The transformed differential equavion can now be solved as an initial value problem with th&e initial conditions y(O) - O and dy(O)/dx = b, where 22

b is the value resulting from condition ii. If the solution of the initial value problem is denoted by 7 = b(x), the value of A needed for the completion of the solution of the original equation is sought by solving the following system of equations: y = h(x) R ~ - f(x, A, aC, Y2), a = g(y, A, cl, C2) (3.3) The last two equations come from the boundary condition at x = ~. The method fails if no values of A can be found from Eq. (3.3)~ The key steps in the above scheme are the selection of a specific group of transformations and the requirement that the given differential equation be invariant, under this group of transformations. For a given differential equation, the equation may not be invariant under a specific preassigned group of transformations. This does not rule out that it will always so if other groups are introduced. It is therefore clear that a method of searching for possible groups under which the given differential equation be invariant is of great importance0 To achieve this goal, the two steps mentioned above are reversed. One starts by requiring that the given differential equation be invariant under ar "infinitesimal transformation". The resulting equation is then used to search for the possible groups of transfdrmation which satisfy this requirement. This necessitates a brief review of those concepts given in Reference 6 which are related to this method. 3o1 THE INFINITESIMAL CONTACT TRANSFORMATION* 3lo 1 Infinitesimal Transformation Let the identical transformation be g(x,y,a ) = x *(x,y,ao) = y (3.4) then the transformation X1 = X(x,y, ao+8C) = ~(x,y'a~ ) + 1 ( a 2' a S ao +. (3.5) *For detail, the reader is referred to Reference 60 23

y = g(x,y,ao+St),(x Y, ~). — -. (3.6) assuming 5c is infinitesimalg neglecting higher order terms of 6E and using the relation for the identical. transformation9 we put xi -- x + S(x,yj) 5 Yi = y + r(x,y) 5c (3.7) 3olo2 Notation for the Infinitesimal Transforma+tion The employment of the'infinitesimal -transformation xl = x + g 3- and Y1 - y + T EE (3.8) in conjunction with the function f(x,y) will be'to transform f(x,y) into f(xl,yl) which upon expanding in. Taylor series9 becomes f(xl,yl) = f(x+X6S, y + rSEC) = f(xn) + -- ( + ) 5E2 22f 2f + 2f + ( X2 II + 2 2xy ~a n anf n n-! an f n+ ( a- c. rj n-ly nn anlf xn wherf(xy) + U + U wh ere 24

Uf = 1- (so)+ 6 Uf = X e f y (3 10) is called the group representation and LPf means repeating the operator U for n times. 3.1.3. Invariant Function If f(xl,Yl) = f(x,y), then f is invariant under the infintesimal transformation. Theorem. The necesary and sufficient condition that f(x,y) be invariant under the group represented by Uf is Uf = 0; i.e., S d + n - 0 (.3-11) To solve for the invariant function, we solve the related differential equation dx = dy (3.12) If the solution is 2(x,y) = constant (3.13) this function is the invariant function for the infintesimal transformation represented by Uf. Since Eq. (3.12) has only one independent solution depending on a simple arbitrary constant, a one-parameter group in two variables has one and only one independent invariant. 3.1.4. Extension to n variable The condition for f(xl,...,xn) to be invariant under an infinitesimal transformation is Uf = tl(xl,... n) -x l. + n(xi,.'',xn) -x= 0 (3.14) To get invariant fmnctions, we solve

dx1 d-= xn ~l0~~ ~t~n (3.15) Since there exist (n-l) independent solutions, a one-parameter group in n variables has (n-l) indpendent invariants. 3olo 5 Invariance of an Ordinary Diffcrential Equation Consider a kth-order ordinary differential equation F(x, y, y?' Yl" P *,y(k)) O (3.16) This equation is invariant under the infitesimal transformation defined by x = x - + bE (x, y, y') y = y + (x, y, y') y' = y' + be It1 (x, y, y') (k) = y(k) + be'k (x, y, y.,9,',y(k)) (3.17) if the following condition is satisfied: UF = 0 (3.18a) or, in expanded form, x F + I yF + T C + k k - (3.18b) For a given group of transformation, the functions r, 1, *l,.., ltk are known. Equation (3.18) gives the condition which the given differential equation, Eq. (3516), must satisfy if it can be transformed to an initial value problem. However, if the group of transformation is not given, Eq. (3.18) alone will not be enough to search for possible groups. At this point, the theories

developed in Reference 6 on the concept of an infinitesimal contact transformation must be introduced which will ultimately makes it possible to express these functions in terms of the so-called "characteristic function". 3lo.6 Definition of a Contact Transformation* When Z, X2o, XI,Xn, Pl n~ Pn are 2n+l independent functions of the 2n+l independent quantities zl, xl,.., xn, Pi,..., Pn such that the relation dZ - Pi dXi p(dz - pidxi) (3.19) (where p does not vanish) is identically satisfied, then the transformation defined by the equations' = Z, x' X, p' = P (3.20) is called a contact transfoirmation. 3.1o 7. Infinitesimal Contact Transformation From Eq. (3.19), dz + x dxi + --- dp P i xi dz + i + dpr) p(dz - Pi dxi) (3.21) For the infinitesimal transformation Z - z + bc~; Xi = x + c i~, Pi Pi + ic (3.22) we get ~So Lie, Matho Ann., to viii, po 220 27

z - P i z )Pr Pi r = rax P OrT (3.23) If a characteristic funrction, W) is d9 efined as W = Pii -, then r -- ~Pr ) Pi ~pi ~' — ax,. " aZ (3.24) Higher order transformation functions, nTij, rrijk, etc., can also be expressed in terms of W. However, due to the complexity in their derivation, they will not be included here. For detail, the reader is refered to Reference 6o However, a special case with one independent and one dependent variable will be given here since it will be needed in the next section. For this case, we consider an infinitesimal transformattion xt x + (bE) (x, y, P) y Y: y + (E) O(X2 y, P) p t-p + (6b) nt(X, y, p) q' = q+ (tb) k(x, y, p, q) r r + (be) p(x, y, p, q, r) (3.25) where p = dy/dx, q = d2y/dx2 and r = d3y/dx3o The;transformation functions can be expressed in terms of a characterist;ic function W as6: aw p w - g - paw-p W ~:=- XW e8

-k = (x + 2qX + 2 )W -p = (X3 q + 3q3q2X + q3 + 3qX32 )W 6p 6P2 6P3 6 9x f 3q~p )f _ + r(q 2 + 3Xf (3a 6) 67" 2P -f) (36) where 4the operator X =/~ax + p 6/6y, 352 THE GENERAL METHOD With the background discussed in Sections 3,0 and 3o1 in mind, the secondorder ordinary different;ial equaticn is again used to illustrate the steps for the transformation from a boundary value to an initial value problem and the search of possible groups to achieve the transformation. Consider again Eq. (3.1), the method proceeds as follows: i. An infitesimal transformation is defined, as in Eq. (3.25) except the transformation for r is not needed here. The given differential equation, Eq. (3.1), is required to be invariant under this group of transformation, i.e., it must satisfy Eq. (3518). iio The transformation functions can be expressed as a function of the characteristic function, W, as given in Eq. (3.26! Eq. (3 18) now becomes an equation with an unknown function W- The functional form of W can be predicted. iiio After W is known, the transformation functions become known functions and the finite form of the transformation can be derived by Eqo (3G9). Two examples will be given in the next section. 29

4. APPLICATION OF THE GENERAL GROUP —THEORETIC METHOD In this section, the general theories given in section 3 will be applied to two examples, namely, the Falkner-Skan problem and the heat conduction equation with non-linear heat generation. These examples serve only as illustrations of the method~ 4.1 APPLICATION TO FALKNER-SKAN SOLUTIONS Consider the well-known Falkner-Skan differential equation from boundary layer theory1: f"1+ f f"t + (1 _ f,2) = 0 (4.1) The boundary conditions are f(O) = f'(O) = 0; f'(~) = 1 We now use the notations p = f', q f", r f"' (4.2) then Eq. (4[1) and its boundary conditions become r + f q + 5 (1 - p2) = 0 (4.3) with boundary conditions f(O) = p(O) = 0; p(o) = 1 Next, an infinitesimal transformation is defined as6 Tf' = f + (bc) (rf,p) 30

p = p + (bE) t(r, f, p) q' q + (E) k( \, f, p, q) r = r + (bE) p(q,, f, p, q, r) (4~ 4) where, in term,; of the characteristic function, W, paw - XW -k = (X2 + 2q x + q2 + q -P = (X3 + 3q X2 + 3q + q3 a+ 3q x + )W 6e;f 3q~ 6ap + r(3q y-p + 3X ap + )W (45 The operator X in Eqo (4.5) is defined as X = + P (4o6) According to the theory discussion in the previous article, the condition imposed on the differential equation is that it is independent of the. parameter of transformation, 6, under the transformation defined by Eq. (4-4), i.e., a6 0 a 4o?7) where F represents the differential equation, (4o3). Equation (4~.() can be written in its expanded form as 31

6F + O 6F + I tF + k-F + p- = 0 (4.8) Replacing F by the left side of (4.3) gives: q - 21pPT + f k + p = O (4.9) The functions Q, A, k, and p, given by Eqso (4.5), are substituted into Eq. (4.9) and we get A0 + Al q + A2 q2 + A3 q3 = 0 (4.10) where the variable r in the function p, Eq. (4~5), was eliminated by using the differential equation, (4o3)9 and the A's are given by A0 = 2PpXW - fXW - X3W + B(1 - p2) (3X + - )W (4.11a),A1 - p = _ w - 3x _ _ 3X a+ f ap + 3( p2) (4.11b) p —w - X -2w A 2f 2f ~2 A2 = 2f p2 - X p2 3 (4,11c) A3 = 3 (4.lld) Since the characteristic function, W, is independent of q, Eq. (4.10) is satisfied if the coefficients are all equal to zero identically., T'hus Ao = A1 = A2 = A3 = o (4.12 The equation A3 = 0 gives 3 = ~ (4.13) which means W is quadratic with p, ioe, 32

W(r,f,p) = WIn,f)p2 + W2 (h,f)p + W3(%,f) (4)i44 This form of W can now be substituted into the equation A2 = O, Eq. (-.12), and the result is (4fWj - 6W - 3 ) 212 p =0 (4.15) Since both W1 and W2 are independent of p, Eq. (4o15) leads to PO af = 0 (4.16) p ~ 4fW1 - 6 - 3 f 2 = (4.17) f Eq. (4.16) shows that W1 is independent of f, i.e., W1 = Wl(r). Thus, W2 can be!'ound from Eqo (4.17) as W2(l,f) = - t2f2W1 - 6 W'f + C1(r) (4.18) The characteristic function, W, now takes the form W(1,f=,p) = W5.()p2 + 5[2f2Wir-,B ) - bWl'(r)f + Cl(rj)p + W3(rL,f) (419) This new form of W is now Iubstituted into the condition A1 = 0 which than gives B0 + B1 p + B2p2 = 0 (4.20) where B0 = -W3 - 4f2 W + 6W:f' f - 3 "f3Wl + - 61 (4.21a) B1 = (6 - 8f)W1' + 6W1" - 5 f221b) 35

Since both W1 and W3 are independent of p, Eq. (4.21) gives B0 = B1 = B2 = 0 (4.22a) or, -W3 - If2!W1' + 6Wbl"f - Cl -3 f + 6W1 (4.22b) (6-8f)W,' + 6i"T-T 3 2= 0 (4.22c) 3Wl(l + 25) = 0 (4.22d) Thus, from Eq. (4.22d), J.l =0 (4.23) From Eq. (4.22c), 27f = o(4.24) which gives W3(q, f) = W31 (r) + w32( n) (4.25) Eq. (4.22b), then becomes -W31 - W32f - C1 - 3W32' + 3C1' C (4.26) hnich gives -W31 - C1 - 3W32' = 0 (4.27a)

-W32 + C3 (4.27b) since W31, W32 and C are functions of 9 alone. Eqs. (4.27a) and (4.27b) give W31 = -2C1, (4.28a) W32 = C1' (4.28b) Thus, the characteristic function, W, becomes W(r,f,p) =3 C;()p + 3 Cl'f- 2C1' (4.29) Finally, the characteristic function, W, is substituted into the last condition in Eqo (4.12), namely A0 = O, which leads to the following equation: Do + Df + D2p + D3f2 + D4 f P = O (4 30) where o0 2C1(v) + 4 cl'~ 2D 2 D3 = - C1' D4 = (3 - 1)C (4.31) Since C1 is a function of ~ only, Eq. (4.30) gives DO D1 = - D2 3 = D4 = O (4.32) 35

If 5 f O, then Eq. (4.31) shows that C1 must be a constant which then leads to the result that W(l,f,p): Clp (4.33) For the case in which 5 = O, C1j is zero, i.e., C1(T) = C11T + C12 (4.34) The characteristic function becomes - 1 W(, f,p) =3(Cll + C12)p + 3 C1lf (4.35) As a last step, the finite form of the infinitesimal transformation must be sought. This can be done by using the equation 6c (bE)2 O(il,fl) = ~(,f) + U + u2 +....... (4.36) 11 2. where arl af Consider first the case in which 0f 0 with the characteristic function given in Eqo (4L33). For this case, using Eq. (4o5), the operator U is U = C By taking a to be ~ and f respectively, we get: TI1 = + 3 C1 bE fl - f (4.37)

Although this group exists under which Eq. (4.1) is invariant, we will not be able to trarnsform the boundary conditions. Thus, the problem cannot be transformed to an initial value problem, unless 5 = 0. For the special case in which = O, the characteristic function is given by Eq. (4o35) which gives the operator U as U =.(Cii + C12) a7 Clf a- f (4.38) Again, by putting into Eq. (4.36) ~ = ~ and ~ = f, respectively, we get = 1cl C2) 2 1 2 C12 + 2 or, C12 Ci bcl bc 1 (rll +-) = + C )[l + - CI +21 C +C11 C11 1' 3 2) 3 (n + C1) e3 (4.39) Cii and fl = f+ ()f + 5c (- Ci2)f + ( C i) 1' 53 2 = f[1 - 1: 5 +2 f e -T- (4.40) If we put A - e, Eqso (4.39) and (4.40) give + C12 ) ll (4.41a) Ci 11 C11) A C11 fl = f A 3 (4.41b) This is seen to be the linear group of transformation. For the present case in which the initial conditlion is given at zero, C12 = 0. 57

To finish the analysis, we can follow exactly the same steps as given between Eqs. (2.7) and (2.14). 4.2 APPLICATION TO THE HEAT CONDUCTION EQUATION WITH NONLINEAR HEAT GENERATION We now consider the heat conduction equation with nonlinear heat generation as follows: d2T k+l dT dx2 X x + f(T) = O (4.42) dx X dx The boundary conditions are dT(O) - 0 and T(1) = TO dx The value of k can be -1, 0 or 1 which correpsonds to plate, cylinder or sphere cases, respectively. If the following notations are defined: dT d2T p = dx q = (4.43) Equation (4042) then becomes k+l q + p x + f(T) = 0 (4.44) with the boundary conditions p(O) = o, T(1) = To It is the purpose of this example to find the function f(T) which enables the problem to be transformed to an initial value problem. Again, an infitesimal transformation is defined as x' = x + (Sc) S(xT,p) T' = T + (6c) Q(x,T,p) Q8

p' = p + (be) n(x,T,p) q' = q + (be),(xT,pq) (4.45) where5 in terms of the charactleristic function, W, ap = -6p - = XW - (X + 2qX a+q2 a + q )W (4.6) The operator X in Eq. (4.46) is defined as: X =x: + p (4.47) Next, the condition that the differential equation under investigation be independent of the parameter of transformation (i.e., invariant under the transformation) is introduced. Under the infinitesimal transformation defined in Eq. (4 45), - -OF (4.48) = C where F represents the left side of the differential equation defined by Eqo (4.44). Or, in expanded form, SuF + t F + dJ fi ea = i o (4.49) Substituting the differential eq-uation, Eq. (4.44), into condition (4.49), we get (q+f)~ t xf'G + (k*l)j + x4 - O (4.50) The functions, @, in, and, are now substituted into Eq. (4.50) and we get 39

((f+xf'p) - xf'W - (k+l) (W + p -) - x(- 2 + 2p xTP aT2 6w _2W_ 2W, 2 w + a ap a j q2 2x 0 (4.51) The variable q can now be ei im.navted by usfing Eqo (4.44) which gives (~.f'W- ( TT3w __j xf w 23 {xfw -W (k+l) X 2- x + 2xf ax-a + xf -T - xf2 f+ vtk(xf'. 2x 2(kf ) p + 2xf - 2f(k+l) 2 + p((xf2 (k 2 p {x aT2 + 2(k+l) aTp x 0 (4.52) For point transformation* under consideration, the characteristic function, x Opes =: 0 W, is linear in p, i.eo. W(xT,p) - W-(x,T) + p W2(x,T) (4.53) Substltuting W from Eq. (4.53) into Eq. (4, 52), we get a0 + a2 P2 a3 p3 = (4.54) where aO = -xfW1z - (k+) x 2 + 2xf - + xf a = (k+l) W2 = 43x W2 - x =J,2 _ _..2 3 2.. k+W2 ae 2 -2 x 3xT x T2' + 2(k+1) 3T 83= -x aT2 (4.55) *A point transformati. on is one whose transf-;ormat- ion functions, 5 and G, in Eq. (4.45) i.s i tependent, of p. 4o

Since the a's are functions of x and T only, it follows that a0 = a1 = a a3 = 0 (4.56) The condition a3 = 0 gives T2 = (4.57) which implies that W2 is linear in T, i.e., W2(x,T) = W21(x) + W22(x)T (4.58) Substituting this form of W2 into the condition a2 = O, we get -2xW22 - x 2 + 2(k+l)W22 = (4.59) Integration of Eq. (4.59) with respect to T yields wi = (-W22 + -W22) T2+ WT + T Wi2 (4.60o) With W1 and W2 given by Eqso (4.60) and (4.58), the condition al = 0 leads to the following equation: 3xfW22 + {(k+l)w2l x WX 1 - 2-x121 }T ixW~ (k+l)W+ (k+l) + 3T{xW22 - (k+l)22 + W22 = (4.61) Equation (4.61) shows that f must be a linear function of T which is impossible under the assumption made at the outset that f is a nonlinear function. The only possibility remaining is W22 = 0 (4.62) With this, Eqo (4o61) is reduced to: 41

- k+l - (k+l)W21 - x W2 "- W21 2xW11 = 0 (4.63) Finally, the condition aO 0 gives f - - _=rf +. (4.64) W:T + W-2 WilT + W12 Where bo 2W1 + w: (4.65) b: (X-j W:1 + WI 9) (4.66) k+i, b2 -a -( X W 2 +..2) (4.67) Equation (4.63) and (4.64) can now be used to determine the functional forms of f(T) for which Eqo (4 42) can be reduced t4o an initial value problem. We now consider the following case: Case 1 W1i A O, bO / 0 Using an int~egrading factor ef pdx (Wi1T + Wlbo/Wl (4.68) the solution to Eq. (4064) can be wri t.,jen as bo/1W 1l {2 b __+; T+W dT f(b2-b'- ),T+ -) {-(bO/iWi +)T + C2 (4.69) For the case in which bo/W; does not equal,to unily, Eq. (4.69) gives b2 b;(.Ty24b~-) + c 1T+w2bi~/ (470) bo bb(W1:-bo)

Recalling that the heat generation function, f, is defined as a function of T only, the Coefficients must be independent of x, i.e., constants. Thus, we put b2 - 02,C 23bl W12 CbO2 bo(W-bi -:0) W12 = C5, = (4.71) WI' The conditions W11 = C4 and WI2 = C5 indicate that bl = b = 0, based on Eqs. (4o66) and (4o67)o This in turn means C2 = C3 = 0. The last condition gives b0 = C C6 C4 From EqO (4I 65), we get C6 C4 2W2, + C4 Thus, W21 C42 x + C7 (4.72) Substituting W21 from Eq. (4o72) into the condition given by Eq. (4.63), we get C7 = 0. As a summary, the following functions are obtained: b0 = C6 C 4 bl b2 0 O, W11 = C4 W12 = C5 W 4(c6-1)X,W = (4.75) 2 Thus, the heat generation function f(T), takes the form f(T) o C(C4T + Cs) C (4.74) and the characteristic funct;ion becomes W(x,T,p) = (C4T + Cs) + p [4(-1) x] (4.75) The transformation functions$ ~ and Q in Eq. (4.45) become 4-3

-a C4(C6-1) c-(6~- x (4.76) ~p 2 0 = p - w (C4T + C5) (4.77) ap The infinitesimal. itransformation.. EqJ (4, 45-) can thus be represented by (see Reference 6) C C I e) if __ Uf x - (C4T + C5) (478) As a final step9, he fini;e form of t;he -nfinitesimal transformation, (4.45), is sought. This can be done by- usin.s.g 4the equation (cf. Reference 6, pO 44-46): f(xlyi ) = f(xxy) + (, Uf, U2f + (4.79) ~ 2C By putting f - x, Eq. (4.79) becomes ~ c4(c6-i) (~c)2 c4(c6-i) X1 4 + ]2 X + ----- 1 + 1 C4(C61)5 1 C4 2(C6-1)2(5C)2 + ---— ] x[l1' p 22 2 where A - exp(bE). By putning f T, Eq. (4.79) gives T1 = T [ - (c4+c5)4 -c) + —Ti = T + [-(C4T+Cs)] + (- --- Lr -c+ C4 4(T 4-4

C5 -T + C_~) sE (T + =C) 1 ) ()(C4) + (TE) (C4,) C4 C4 1' 2' or, T + C = (T + -) exp[-C45E) = (T + C5) A-C4 (4.81) C4 C4 C4 Thus, the finite transformation is given by X1 = x A4( T, + - (T + 5)AC (4.82) C4 C4 This is seen to be linear group of transformation discussed in Section 2.1 for the case of a power law heat generation. Consider next the case in which the ratio bo/Wl7 = 1. The solution to f in Eq. (4.69) than becomes b, n(-WTW) + Cz(wzzT~z)(4.8 3) f = -(W nlT+W) 2 T+12) + (b2-b 12)(bl.83) Again, since the heat generation function, f, is only a function of the temperature T, the coefficients must all be constants, i.e., ~W1 = C2, b = C3 W12 = C4 b2 = C (4.84) The conditions W11 = C2 and W12 = C4 indicate that b, = b2 = 0, based on Eqs. (4.66) and (4 67), which in turn means C3 = C5 = 0. Eq. (4.65) now gives W21 = 0 so, we can write W21 = C6. Substituting W21 = C6 into the last condition, Eqo (4.63), we conclude that C6 0. O The heat generation function is thus: f = Cn(C2T+C4) (4.85) and the characteristic function become

W = C2T + C4 (4.86) By following the same steps as before, the finite transformation is found to be C4 x, - x, n (T1 + C) - (T + -) AC2 (4.87) C2 C2 This case is a special case of "power-law heat generation" discussed above for a power of unity. ILt is to be noted that the same form of W can be obtained by putting C6 equal to 1 in Eq. (4.82). Case 2 bo - 0 For this case, integration of Eqo (4.69) gives Wlf - -T n(W11T + W12) + C1 (4.88) Again, the coefficients in (4.88) should be constants, ie.,, b -= C2, b2 C3, Wil = C4 J W12 = C5 The conditions W11 = C1 and W12 Cs again lead to C2 = C3 = 0 (or bl = b2 = 0) Equation (4o65) gives 2W'1 + C4 O0 and thus W21 = xx + C6 The condition, (4.63), then gives C6 = 0. The heat generation function and the charact eristic function are therefore as follows: f - C= (4.89) 46

W = C4T + C5 + p[- x] (4.9o) This is seen to be a special case of case 1, and can be obtained by putting C6 = 1 into Eqs. (4.74) and (4.75). Case 3 Wll = 0 The differential equation for f becomes f' - C2f = C3 (4.91) in conformity with the requirement that f be independent of x. The constants, C2 and C3, are = C2 and b = C3 (4.92) W12 W12 From Eqs. (4.65) and (4.67), Eq. (4.92) can be written as C 2W12 = 22. (4.93) and k+l _ C3W12 = - X'12 - Wi2 (4.94) Also, from condition (4063), another relation is obtained as: k+l - ~~(k+l)W~l~RW21 - (4.95) Eqo (4.95) now gives -- C4k+l W21 = C4x + CsX (4.96) Substitution of W21 into Eqo (4~93) yields, after integration, the solution of W12: 47

W12 = 2(k+l)4 xk + 2Cs(497) C2 C2 Finally, W12 from Eq. (4.97) is substitut;ed into Eqo (4.94) and we get C3 C4(k+l)xk + 2k(k+l)C~X + C3 C O= (4.98) To satisfy this condiLtion, six possibilities exist, namely, 1 k = 0 la: C3 = 0 lb: C,(k+l) + C, = 0 20 k = -1 2a; C3 0 2b: Cs 0 3. k / O, k -1 3a: C3 = O C 4 0 3b: C =09 C4 0 O Using the same technique as before, the final form of the heat generation function f, the characteristic function, W. and the finite transformation are found to be as follows: 1. k = 0 la: C3 O0 f = C6 e2T (4.99) C4+C5 W = 2 + p(C4+C5)x (4.100) C2 (C+C+5)bc 2 X1 = xe and T1 = T - (C4+C5)E (4.101) C2 lb. C4 + C5 = 0 f 3 C eC2T f = C —+Ce C2 W 0, which means only the identical transformation will be applicable transformat-ion. 20 k = 1 2a: C3 O f= Cg e[02T (4.102) 48

W = 2 + p(C4 + C5X) (4.103) C2 (xl + (x + ) = (x + and T1 = T C- s (4.104) C5 C5 C 2b: f C + C6 eC2T (4.105) C2 W = C4p (4.106) x - x + C45c, T1 = T (4.107) 3. k O 0 k - 1. C3 = O C4 = 0 C2T f - C6 e (4.108) W = 2 + p(C5x) (4.log9) C2 xl = xe 5, T T - 2 C5 e (4.110) C2 3b. C5 = O, C4 = 0 Again, the result is the identical transformation, as in case lb. Details from this poirnt on and the limitations to the method are the same as in Section 2. They will not be repeated here. 4.3 CONCLUDING REMARKS The method developed in this section is seen to be very general and, like the simple group-theoretic method, only algebraic equations need to be solved. The general method is, however, considerably longer than the simple grouptheoretic method. For a given ordinary differential equation, therefore, it is prefered to try the simple group-theoretic method first. If this method fails, the general method is then applied and the possible group of transformations searched. For certain problems, e.g.,, the example given in Section 4~2, only the general method can provide the answer. In case both methods fail, we can conclude that the problem cannot be transformed to an initial value problem. 49

REFERENCES 1. Falkner, VO M., and S. W. Skan, "Some Approximate Solutions of the Boundary Layer Equations," Philo Mag,, 12 (1.931)o 20 Klamkin, Mo S., "On the Transformati.con of a Class of Boundary Value Problems into Initial Value Problems for Ordi.ary Differential Equations, " SIAM Review, 4 (1962), pp. 43-47. 3o Na, T. Y,, "Transforming Boundary Conditions to Initial Conditions for Ordinary Differential Equations," SIAM Review, 9 (1967), ppo 204-210. 4o Na9 T. Y., "F-urther Extension on Trarsforming from Boundary Value to Initial Value Problems) " SIAM Rev.iew, 10 (1968). 5. Na, T. Y. and S. C. Tang, "A Method for the Solut;:on of Heat Conduction Equation with non-linear Heat Generation," iln Press. 6. Na, T. Y., D. E. Abbott, and A. G. Hansen, "Similarity Analysis of Partial Differential Equations," Technical Report. NASA contract NAS8-20065, Univ. of Mich., Dearborn Campus, Dearborn, Mich., March 1967. 7. Topper, Zeit; Math. U. Phys., 60 (1912), ppo 397-398. 5(

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