CHAPTER-III GENERAL GROUP THEORETIC TRANSFORMATIONS FROM BOUNDARY VALUE TO INITIAL VALUE PROBLEMS

Transcription

1 CHAPTER-III GENERAL GROUP THEORETIC TRANSFORMATIONS FROM BOUNDARY VALUE TO INITIAL VALUE PROBLEMS 3.1 Introduction: The present chapter 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 transformation from boundary value to initial value problems. A completely new and general method will be developed based on S. Lie s continuous and compact transformation group. The concept that already discussed in detailed with some counter examples in Chapter 2 of searching for all possible groups of transformations in a similarity analysis of partial differential equations is applied here too to develop method of transformation of Boundary Value Problem to Initial Value Problem. The advancement in the analytic theories of fluid mechanics have contributed greatly to our knowledge of non-linear differential equations, both ordinary and partial. One of the classes of differential equations resulted from fluid mechanics problem is the nonlinear two point ordinary differential equations. As in other types of nonlinear equations, analytical solutions to these equations generally do not exist except for some very special cases and numerical integration methods, such as the Runge-Kutta Method, usually have to be used. The main difficulty in the numerical integration of this type of equation is due to the fact that the boundary conditions are given at two points. Thus the missing boundary condition at the initial point has to be assumed and the equation is then integrated as an initial value problem. If the assumed initial condition is correct, the solution will satisfy the boundary condition at the second point. Otherwise, another value has to be chosen and the process is repeated. This trial and error method usually takes a long computing time and the solution frequency is very sensitive to even a very small change of the assumed initial condition. A major simplification can be achieved if a transformation can be reduced to an initial value problem with all boundary conditions specified at a single point Runge-Kutta method. 70

2 A well-known example of this class of transformation is found in the solution of Blasius boundary layer equations. After a similarity transformation is made, the boundary layer equations are reduced to nonlinear third order ordinary differential equation with two boundary conditions given at the surface of the plate and third at infinity. A method of transformation was introduced by Toepfer (1912) and the problem was transformed to an initial value problem. The successful application of this type of transformations in Blasius equation makes one wonder whether it can be extended to other equations. A start in this direction was made by Klamkin (1962) in which Topfer s method was extended to many similar type of equations. The method was further extended [Na (1968)] by using the theory of transformation groups to problems whose boundary conditions are given at finite intervals. From this point of view, Toepfer (1912) and Klamkin s (1962) transformations belong to the so-called linear group of transformation. By introducing a spiral group of transformation, the method was seen to be applicable to many new types of equations. One example is found in the solution of conduction heat transfer with nonlinear heat generation, Na and Tang (1969). Although the method developed so far has been successful, the class of equations, which can be treated in this manner, is still limited. Many attempts have been made recently to extend the method and present work represents one successful result. The extended method to be given here can be applied to problems in which certain physical parameters appear in the differential equation and/or in the boundary conditions. The well-known example of fluid mechanics namely, Blasius equation is used to illustrate the entire development of present technique. As pointed out earlier, the initial value method was first introduced by Toepfer (1912), in his attempt to solve Balsius s equation in the boundary layer theory by a series expansion method. No progress was made for half a century, until 1962 when Klamkin (1962), following the same reasoning, extended the method to a wider class of problems. Major extensions were made possible only when the concept was reexamined and interpreted by Na ( ) in terms of transformation groups. 71

3 Even though the method is sometimes referred to as the method of transformation groups, perhaps the only group concept used in the method. Attempts have been made to derive groups of transformations, which can reduce boundary value problems to initial value problems by using the invariant properties of a differential equation under a socalled infinitesimal transformation group. The theory is, however, incomplete at this time and considerable work is needed before the method can be established. Until that happens, it is much simpler to view the method as a straightforward transformation. For a complete review of the various aspects of the group-theoretic method, one has to refer Ames (1972), Na and Hansen (1971), Bluman and Cole (1974), Ovsjannikov (1967), and Na (1974). Basically, method starts by defining a group of transformations. The particular transformation within this group of transformations which can reduce the boundary value problem to an initial value problem is identified by the requirement of invariance of the given differential equation under the applied group of transformation at the missing boundary conditions. The method is simple and its application is straight forward Blasius Solution in the Boundary Layer Theory: To introduce the background, which led to the method discussed in this chapter, let us consider the flow of an incompressible fluid over a semi-infinite flat plate, as shown in figure 3.1. Such a flow is usually called the boundary layer flow since the viscous effects are limited to a thin layer near the surface, across which the velocity changes from zero at the wall to the main-stream velocity U at the edge of this layer. The analysis of such a flow requires the application of the Navier-Stokes equations, which are derived from the laws of conservation of mass and of momentum. By introducing the boundary layer assumptions (i.e. the existence of a very thin layer and that u>> v), the Navier-Stokes equation become, u v x + y 0 (3.1) u u x + v u y ν 2 u y 2 (3.2) 72

4 Figure 3.1 Boundary layer over a plate 73

5 subject to the boundary conditions y 0: u v 0 y : u U Where u and v are the velocity components in the x and y direction, respectively, and ν is the viscosity of the fluid. The boundary conditions at the wall (y 0) are based o the physical condition that neither slip nor mass transfer occurs on the surface. The boundary condition at the edge of the boundary layer simply means that, mathematically, the velocity component u approaches the mainstream velocity U asymptotically. Equations (3.1) and (3.2) were solved by Blasius in As the next step, a stream function ψ is introduced such that u ψ y, v - ψ (3.3) x Besides the physical reason of introducing this function (i.e. constant ψ lines are stream lines), the mathematical significance is that the equation of continuity, Equation (3.1), is satisfied identically and the momentum equation becomes, ψ y 2 ψ - ψ 2 ψ x y x y 2 ν 3 ψ y 3 (3.4) subject to boundary conditions, y 0: ψ x ψ y 0 y : ψ y U In other words, the solution of two equations with two variables (u, v) is reduced to the solution of one equation with one variable (ψ). Blasius introduced the following transformations: η y U νx (3.5) 74

6 and f(η) ψ νx.u (3.6) This transforms equation (3.4) to the following equation: d 3 f 3 d 2 f + 1/2 f 2 0 (3.7) subject to the boundary conditions f(0) df(0) 0, df( ) 1 The class of transformations in which a nonlinear partial differential equation is transformed to a nonlinear ordinary differential equation is known as the class of similarity transformations. The foundation of the method is contained in the general theories of continuous transformation groups that were introduced and related extensively by Lie in the latter part of the last century (1884). Equation (3.7) is a boundary value problem. In order to follow the reasoning which led Toepfer (1912) to introduce a transformation of variables and transform the problem to an initial value problem, solution of equation (3.7) by series expansion should be given in detail. Let us assume the solution of equation (3.7) to be f(η) C n η n (3.8) n0 Before substituting the solution in to the differential equation, some of the coefficients can be evaluated from the boundary conditions. From the two boundary conditions on the surface, η0: f(0) 0, df(0) 0 75

7 We get C 0 C 1 0 (3.9) Next, let us introduce the notation d 2 f(0) 2 λ (3.10) Where, of course the constant λ is unknown quantity. In terms of λ, the coefficient C 2 is given by C 2 λ 2! (3.11) Equation (3.8) can now be substituted in to equation (3.7), and we get n(n-1)(n-2) C n η n-3 +1/2 C n η n n (n-1) C n η n-2 0 n3 n0 n2 or, in expanded form, [(3)(2)(1) C 3 +1/2 C 0 (2)(1) C 2 ] +[(4)(3)(2) C 4 +1/2 C 0 (3)(2) C 3 +1/2 (2)(1) C 2 C 1 ] η +[(5)(4)(3) C 5 +1/2 C 0 (4)(3) C 4 +1/2 (2)(1) C 2 C 2 + 1/2 C 1(3)(2) C 3 ] η 2 +[(6)(5)(4) C 6 +1/2 C 0 (5)(4) C 5 +1/2 (2)(1) C 2 C 3 + 1/2 C 1(4)(3) C 4 + 1/2 C 2 (3)(2) C 3 ] η (3.13) By setting the coefficients of equation (3.13) equal to zero and employing equations (3.9) and (3.11), we get C 3 C 4 0, C 7 C 6 0, C 5 λ2 5!(2) 11λ 3 C 8 8!(2) 2 (3.14) 76

8 The solution, equation (3.8), can be therefore be written as, f(η) λη2 λ 2 η 5 11λ 3 η λ 4 η 11 2! + 5!(2) 8!(2) 2 11!(2) 3 (3.15) Which still involves the unknown constant λ. In principle, the boundary condition at the second point should be applied to determine the constantλ. However, this can not be done directly here due to the fact that boundary condition at the second point is given at η. Blasius (1950) used a matching method, in which the solution given by equation (3.15) applied to the region close to the wall is matched to an asymptotic expansion for largeη. The process is tedious and is of no interest here, By arranging the variables, equation (3.15) can be written as, (λ -1/3 f) (λ 1/3 η) 2 (λ 1/3 η) 5 11(λ 1/3 η) 8 375(λ 1/3 η) 11 2! 5!(2) + 8!(2) 2 11!(2) (3.16) This suggests a transformation of the form F λ -1/3 f, ξλ 1/3 η (3.17) Equation (3.16) then becomes, F ξ 5 11ξ ξ 11 2! +.. 5!(2) 8!(2) 2 11!(2) 3 ξ 2 (3.18) The significance of the transformation of variables will become obvious of the differential equation (3.7) and its boundary conditions are transformed. From equation (3.7), we get d 3 F dξ 3 + 1/2 F d 2 F dξ 2 0 (3.19) The boundary conditions at the wall are transformed to ξ 0: F(0) 0, df(0) dξ 0 (3.20) 77

9 Using the transformation, equation (2.20) gives d 2 F(0) dξ 2 1 (3.21) The transformed equation (3.19), subject to the boundary conditions (3.20) and (3.21), is an initial value problem. The solution of equation (3.19) can be obtained by a forward integration scheme. The relation between (η, f) and (ξ, F) is defined in equation (3.17). If the constant λ can be found, the solution of (η, f) can be calculated from the solution of (ξ,f). To this end, the boundary condition at infinity df( ) 1 is transformed according to equation (3.17) and we get, λ -2/3 df( ) dξ 1 or λ df( ) dξ - 3/2 (3.22) Thus, the solution of equation (3.7) consists of three steps. 1. Solve equation (3.19), subject to the boundary conditions (3.20) and (3.21), as an initial value problem by forward integration. In particular, we get the value of df( ) /dξ. 2. Compute λ by equation (3.22) 3. Compute f(η) by using equation (3.17). The above demonstrates an important idea of solving boundary value problems, that of solving a complementary initial value problem. In this way the solution becomes non iterative. 78

10 While the value of such a transformation was obvious, no further work was done to extend the idea to other types of problems until 1962, when Klamkin (1962) published an extension of this method by following the same reasoning as given in Toepfer s method. For example, it was found that for a general second order ordinary differential equation of the form N A i i 1 d 2 m f i 2 df n i f r i η s i 0 (3.23) Subject to the boundary conditions f(0) 0, df( ) k a transformation F λ α -1 f(η), ξ λ α η (3.24) enables the solution of equation(3.23) to be obtained by first solving the initial value problem, N A i i 1 d 2 F dξ 2 m i df dξ n i ξ s i F r i 0. (3.25) subject to the boundary conditions, df(0) F(0) 0, 1 dξ and then calculating the constant λ from λ k / F ' ( ) (3.26) Finally, the solution of equation (3.23) can be calculated by using equation (3.24). 79

11 The constant α in the transformation (3.24) is determined by requiring that, upon transformation, equation (3.25) should be independent of A. Thus, from the above theory, we can conclude that although Klamkin s method represents an important extension of the technique, the method is still quite limited for the following three reasons. First, only the types of transformations similar to those given above are known to be applicable. Second, the boundary conditions at the second point have to be given at infinity. Third, the boundary conditions at the initial point must be homogeneous. To remove some of the limitations, the present method has to be interpreted in terms of the concepts of transformation groups. All these limitations are still severe. T.Y. Na ( ) has tried to remove above limitations by introducing group transformations. He found partly successful. This is because he has explained his views through same particular examples and he couldn t derive same general mathematical theory. Actually, the method of Klamkin (1962) can be applicable to ordinary differential equations or system of ordinary differential equations, which were invariant under certain groups of homogeneous linear transformations. Additionally, the boundary conditions were specified at origin and at infinity were found homogeneous at origin. Subsequently in two papers by Na ( ), it was shown that the method was applicable to finite intervals and also to equations, which were invariant under other groups of transformations. It was also stated by Na (1967) that the boundary conditions at the initial point must be homogeneous at the initial point for the method to be applicable. Although all the boundary conditions considered by Klamkin (1962), Na ( ) were homogeneous at the initial point, it will be shown subsequently that this condition is not always necessary. 80

12 In all the boundary value problems and initial value problems to be considered, we are tacitly assuming the existence and uniqueness of the latter, which in turn will have implications for the former. In the next section, the general theory of reduction of Boundary Value Problem to Initial Value Problem is given Second-Order Boundary Value Problems: First we consider general second order differential equation: A mnrs y'' m y' n y r x s 0 (3.27) m,n,r,s subject to the boundary conditions, y ' (0) ay(0) + b, y e ( ) k (3.28) m, n, s,r are arbitrary indices, A mnrs are arbitrary constants and (e) is an arbitrary integer. If (3.27) is multi valued for y'', we assume a particular branch is specified. We now assume that y can be expressed in the form y λ F(μ x) (3.29) Where F(x) also satisfies (3.27) for arbitrary constants λ, μ, but subject to the initial conditions F(0) F (0) 1 In order that both y and F(x) satisfy (3.27), the D.E must be invariant under the twoparameter group of homogeneous linear transformations. x 1 μ x, y 1 y λ The condition that this implies on the indices m,n,r,s, got by substituting (3.29) in to (3.27) i.e. A mnrs F n (μ x) m F (μ x) n F (μ x) r (μ x) s λ c μ d 0 m,n,r,s where, c m + n + r d 2m + n s (3.30) 81

13 Consequently, c & d must be constant for all sets of indices m, n, r, s and then (3.27) reduces to A mn y m y n y r x s 0 (3.31) m,n Where, r and s are given by (3.30) It now follows that y(0) λ, y (0) aλ + b λμ (3.32) K λ μ e F (e) ( ) We now solve the initial value problem for F and determine F (e) ( ). Then λ and μ are determined from the simultaneous equations in (3.32) which then give y(0) and y (0), Thus we have converted the original Boundary value problem in to similar Initial value problems or else as noted earlier, y can be determined from y λf (μ x). Having tacitly assumed the existence and uniqueness of F(x) in (0, ), the existence of y depends on the existence and uniqueness of μ and λ. Eliminating λ in (3.32), we have μ e k' (μ -a) Thus, depending on the relative values of e, k and a, there can be zero, one, two or three solutions for μ and the same corresponding for y. The case for a finite interval instead of an infinite internal for the preceding problem can be treated in a similar fashion. If the boundary condition was given by y (e) (L) k We would then have y ' (0) aλ + b λ μ, K λ μ e F (e) (μl) again there are two equations for λ and μ but not quite so simple as before. Although these equations can be easily solved numerically, it may be easier to start out with the original system, pick a trial value for λ and then refine it by interpolation techniques. For higher order equations, the method will more useful. 82

14 If instead of boundary conditions (3.28), we had y(0) a, y (e) ( ) k, We would assume that y could be expressed in the form y F (μ x), where, F(x) also satisfy (3.27) for arbitraryμ. This entails that in (3.27) 2m + n s Constant. Since, y(0) a, F(0) a If we now let F (0) 1, then y (0) k, which is determined from K μ e F (e) ( ) Provided e 0. For the case e0, we have an anomaly that could be due to impossible boundary conditions. An example, consider the differential equation (xd-1) (xd +1) y 0 Whose solution is yax + (B/x) For a finite interval, with boundary conditions y (e) (L) k instead of y (e) ( ) k, we can proceed as before. We could also replace boundary condition at the terminal point by either of the more general ones or A i y ( i ) (L) K i lim x A i x (e i ) ( i ) y (x) K i 3.4 Transformations of Boundary Conditions: There are Boundary Value Problems where the previous method will not apply directly unless the Boundary Conditions are first transformed into a suitable form. As an example, consider the Boundary Value Problems: x n / (n+1) y y (2n+1) / (n+1) y (0) 1 (3.33) y ( ) 0 83

15 For n1, we get the Thomas-Fermi equation (1960) which had arisen in the determination of the effective nuclear change in heavy atoms. It follows that if F(x) is a solution of the above Differential Equation, so also is y λ (n+2)/n F(λ x) even though the equation has a one-parameter group of transformations, the method does not work. It would work, if the boundary conditions were interchanged to y(0) 0, y ( ) 1 This can be done by letting x 1/x. Since y '' x 4 y '' + 2xy ' It might appear as if we will lose the one-parameter group of transformations. Fortunately, if for any differential equation where F(x) and λ α F(λx) are both solutions, one makes the transformation x x -r, then if G(x) is a solution so also is λ α G(x/λ 1 /r ). Similar results apply if we transform y y s. Carrying out the transformation x 1/x on (3.33), We get x -n/ (n+1) { x 4 y '' + 2x 3 y ' } y (2n+1)/ (n+1) (3.34) y(0) 0, y( ) 1 Now, letting y λ (n+2)/ n F(x/λ) with F(0) 0, F ' (0)1, We get λ {F( )} -n/(n+2) It should be noted that the differential equation is singular at the origin. Consequently, in order to start the numerical solution for the Initial Value Problem, first one has to find the asymptotic solution in the neighbourhood of the origin. Now in the next section using simple group transformation, we show thoroughly how to transform above boundary value problem in to initial value problem. 84

16 3.5 Initial Value method by group of Transformation: The Blasius Equation in Boundary Layer Flow: The Blasius Equation, equation (3.7), which Toepfer (1912) treated, will now be reconsidered from the point of view of transformation groups. The equation is d 3 f 3 + 1/2 f d 2 f 2 0 (3.35) Subject to the boundary conditions, f(0) df(0) 0, df( ) 1 Let the given group of transformations be the linear group defined by η A α 1 η, f A α 2 f (3.36) Where, A is the parameter of transformation and α 1 and α 2 are constants to be determined. Under this group of transformations, equation (3.1) becomes d 3 f 3 + 1/2 f d 2 f 2 + 1/2 A(2α 2-2 α A α 2-3α 1 d 3 f 1) 3 d 2 f f 2 0 (3.37) We first require that the transformed differential equation be independent of the parameter of transformation A. This requires that the powers of A in the two terms be equal, i.e. α 2-3α 1 2α 2-2α 1 (3.38) Equation (3.3) gives d 3 f + f d 2 f d 3 f ½ /2 f d 2 f Aα2-3α From equation (3.39), we get d 3 f + 1/2 0 3 f d 2 f 2 (3.39) (3.40) 85

17 Fig. 3.2 Solution of equation

18 In the second step, the missing boundary condition is set equal to the parameter of transformation, i.e., d 2 f(0) (3.41) 2 A This step is somewhat arbitrary, but it does not place any limitation on the method since A is an unknown constant to begin with. Under the linear group of transformations, this condition becomes A α 2-2α 1 A d 2 f(0) 2 (3.42) and we further require that the transformed boundary condition be independent of A. This is possible if α 2-2α 1 1 (3.43) Equation (3.42) then gives, d 2 f(0) 1 2 Equations (3.38) and (3.43) give (3.44) α 1 - α 2-1/3 (3.45) Finally, the parameter A can be evaluated from the boundary condition at infinity, which gives df( ) A α 2-α 1 1 (3.46) Or, using equation (3.45), -3/2 A df( ) 1 (3.47) The other boundary conditions are transformed simply to 87

19 f (0) df(0) 0 (3.48) Equation (3.40), together with the boundary conditions (3.44) and (3.48), constitutes an initial value problem, the numerical solution of which is shown in figure 3.2. df It is seen that as η approaches infinity approaches the value of , which is taken as df( ). Substituting in to equation (3.47), we get -3/2 A df( ) Which is the missing boundary condition at η 0, as defined by equation (3.41). Since α 1, α 2 and A are all known, the solution of the original equation, equation(3.35) can be completed by using equation(3.36). It will be of interest to give a physical interpretation of fig-3.3, which shows df/ as a function of η. From equation (3.3) and (3.6), we get u ψ y y νxu f νxu df η y ν xu U df νx U df or u U df (3.49) 88

20 Fig. 3.3 Solution of equation

21 Thus, the curve given in figure 3.3 is a plot of the velocity distribution as a function of η. Referring to figure 3.1, if the fluid and the mainstream velocity U are given and the velocity distribution at a distance x x 0 is needed, equation (3.5) then gives the relation between η and y, namely η y U νx 0 (3.50) Since, U, v and x 0 are known; equation (3.50) gives the corresponding value of η for any y. The curve in fig.3.3 can then be used to read u/ U at this value of y. The method can now be summarized as consisting of the following steps. 1. A group of transformations is defined. From the point of view of transformation groups, it is immediately recognized that the class of transformation treated by Toepfer (1912) and Klamkin (1962) is the familiar linear group of transformations. The immediate conclusion is that there must be other groups. Questions as to whether there is a deductive way to derive the particular group of transformations may also arise (1968). At the present time, these theories are still incomplete and we have to proceed with the method based on a given group of transformations. 2. The differential equation is required to be independent of the parameter of transformation, which should lead to only one relation between α 1 and α 2. This step offers the first test as to whether or not the method is applicable. For example, the Falkner-Skan equation, d 3 f 3 + 1/2 f d 2 f + 2 β 1 - df 2 0 is independent of the parameter of transformation in the linear group defined in equation (3.36) only for α 1 α 2 0. The method thus fails to apply. 90

22 3. The unknown boundary condition is set equal to the parameter of transformation. Another relation connecting α 1 and α 2 can be written from which α 1 and α 2 can be solved, leaving A as the only unknown in the transformation. 4. The parameter of transformation is determined from the boundary condition at the second point such as in equation (3.47). One interesting case will be when the boundary condition at infinity is homogeneous, i.e. when the right hand side of equation (3.46) is zero. One example is found in the flow of a laminar Non- Newtonian jet (1968). Another case is when the boundary condition at the second point is given at a finite distance instead of at infinity. 5. As a last step, the boundary conditions at the initial point are transformed. Here, it is required that they be homogeneous. Otherwise, the method can not be applied. For example, if f(0) k, then upon transformation. A α2 f (0) k Which is independent of A only if α 2 is equal to zero. At this point, a work by Klamkin (1970) should be cited. In his work, Klamkin tried to remove the requirement that the boundary condition at the initial point be homogeneous. To illustrate his reasoning, let us consider one example given in Klamkin (1970). d 2 f 2 + 1/η df α + β 1 f 2 We will, however, write the boundary conditions as df(0) af(0) + b, f( ) 1 After a linear transformation, equation (3.36), the differential becomes A α 2-2α 1 d 2 f 2 + 1/ η df α + β A -2α 2 1 f 2 We will consider three cases. 91

23 Case:1 If neither α nor β is zero, invariance of the differential equation gives two equations, namely α 2-2α 1 0, 2α 2 0 Which means the problem can not be transformed to an initial value problem since now both α 1 and α 2 are zero. Case:2 If α equals to zero (i.e., the differential equation does not contain constant term), then one equation results from the invariance of the differential equation, namely α 2-2α 1-2α 2 and the method presented above can be applied. Since we now have one equation for the solution of α 1 and α 2, only one equation relating α 1 and α 2 should come from the boundary conditions. This is the reason why the boundary condition at the initial point should be homogeneous, since the boundary condition at the second point always gives one such relation. Case:3 If both α and β are zero (i.e. for differential equations whose transformed form, like the two terms on the left-side of the transformed equation above, has exactly the same powers of A, which can then be factored out for all values of α 1 and α 2 without the necessity of imposing any relation between α 1 and α 2 ), then no relation between α 1 and α 2 will emerge from the differential equation. This leaves the two boundary conditions to provide the two equations necessary for the solution of α 1 and α 2. We now set f (0) A The first boundary condition gives By outing A α 2-α 1 df(0) aa + b f(0) 1 and df(0) 1 We get two equations, α 2 1, A α2-α1 Aa + b 92

24 Which together with the transformed boundary condition at the second point, A α 2 1 f( ) are the equations required for the solution of α 1, α 2 and A. The above cases show clearly that by placing limitations on the form of differential equation, some of the restrictions imposed on the method can be removed. 3.6 Development of General Procedure: Consider general second-order differential equation N i 1 A i d 2 y 2 m i dy n i y r i x s i 0 (3.51) Subject to the two cases of boundary conditions Case I d d y( ) y(0) 0, k d Case II y(0) 0, d d y(l) d 0 Consider the linear group of transformations x B β 1 x ; y B β 2 y (3.52) Under this group of transformation, Eq. (3.51) becomes N B m i (β 2-2β 1 ) + n i (β 2-2β 1 ) + r i β 2 + s i β 1 i 1 N d x m y i dy A i 2 n i x s i y r i 0 (3.53) i 1 93

25 Equation (3.53) will be independent of the parameter of transformation, B, if the powers of B in each term are equal, i.e. m 1 (β 2-2β 1 ) + n 1 (β 2 - β 1 ) + r 1 β 2 + s 1 β 1 m i (β 2-2β 1 ) + n i (β 2 - β 1 ) + r i β 2 + s i β 1 (3.54) Where i 2,..., N. In general, (3.54) gives (N-1) equations with only two unknowns, β 1 and β 2. The method is applicable only if the (N-1) equations actually reduce to one independent equation. To illustrate the problem that may arise, the Falker-Skan equation may be cited. d 3 f 3 + 1/2 f d 2 f + 2 β 1 - df 2 0 (3.55) Under the linear group of transformation defined by Eq. (3.52), Eq. (3.55) becomes d 3 f d 2 f B β β 0 2-3β B 2β 2-2β 1 1/2 f B 2β 2-2β 1 df 2 (3.56) Which is independent of the parameter of transformation if β 2-3β 1 2 β 2-2β 1 0 2β 2-2β 1 (3.57) Two independent equations are obtained from (3.57). As a result, we get β 1 β 2 0 which means the method is inapplicable. Assuming for now that such a situation does not exist, Eq. (3.53) becomes N i 1 A i d 2 y 2 m i dy n i x s i y r i 0 (3.58) With one relation between β 1 and β 2 obtained from Eq. (3.54), the other relation required for the solution of β 1 and β 2 can be obtained by putting the slope at x 0 equal to the parameter of transformation, B, i.e., dy(0) B After transformation, we have: (3.59) β2- β1 B dy (0) B 94

26 Which is independent of B if β 2 - β 1 1 (3.60) The transformed boundary conditions are therefore, y (0) 0, dy(0) 1 (3.61) Equations (3.54) and (3.60) give solutions to β 1 and β 2. Finally, the value of B can be found by applying the boundary condition at the second point. Thus: Case I: or, B β 2 - dβ 1 d d y ( ) d k B k d d y ( ) d 1/ (β 2 - dβ 1 ) (3.62) Case II: d d y d 0 at B β1 x L Or, L 1/ β 1 B x where d d y d 0 (3.63) Thus, Eq. (3.58) is solved with the boundary conditions given in Eq. (3.61) and the value of x where d d y 0 can be found from the solution of the transformed equation. This d result is then substituted in to Eq. (3.63) and the value of B computed. It should be noted that there are cases where additional problems may arise. 95

27 To illustrate the practical application of the above theory, we now consider the following boundary value problem: d 2 T 2 - λ 2 0 T With the boundary conditions T (0) 0, The solution is dt(0) 1 T 1/λ sin h λ x This is never zero. This places another limitation on the method. The boundary conditions at x L in case II need not be homogeneous. For example, one may have at x L, d d y d K Thus, B β 2 - dβ 1 d d y d K at B β 1 x L (3.64) Since, K, L, β 1 and β 2 are known constants, the value of B can be found by searching for d d y values of x and in the solution of Eq. (3.58) which give the same value of B in d both equations of Eq (3.64). One way of doing this is by eliminating B in Eq. (3.64) which leads to d d y K d x d L x d- β 2 /β 2 (3.65) Next, (d d y/d x d ) Vs. x can be plotted as a curve. Another curve from the solution to Eq. (3.58) can be plotted with the same co-ordinates. The intersection of these two curves will give the required value of x and d d y/d x d which in turn can be used to compute B from Eq. (3.64). 96

28 One final remark about the method is necessary. Suppose the boundary condition at the initial point is dy(0)/ 0. In this case, we merely have to put Thus, y(0) B B β2 y (0) B, And if the result is to be independent of B, β 2 must be equal to 1. Under no circumstances, however, should the boundary condition at the initial point be non homogeneous. If it is, one more equation relating β 1 and β 2 will result. The method then cannot be applied 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 N i 1 C i d 2 y 2 m i dy n i x q i e p iy 0 (3.66) With the boundary conditions Case I: x 0 : Case II: x 0 : dy dy 0 ; x 1 : y 0 0 ; x : y k 1 Where C i, m i, n i, p i and q i are constants and N is the number of terms in Eq. (3.66) Let us define the one-parameter spiral group of transformations x e β 1A x ; y y + α2 A (3.67) 97

29 where A is the parameter of transformation and α 1 and α 2 are constants to be determined. Under this group of transformation, Eq. (3.66) becomes N i 1 C i e (-2m iα 1 n i α 1 + p i α 2 + qiα 1 )A x d 2 y 2 m i dy n i e piy i x qi 0 (3.68) 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., (-2m i n i + q i ) α 1 + p i α 2 (-2m 1 n 1 + q 1 ) α 1 + p 1 α 2 (3.69) where i 2,..., N. The transformed equation becomes N i 1 C i d2 y 2 m i dy n i e piy x qi 0 (3.70) Equation (3.69) represents (N-1) equations. In general, the method can be applied only if one independent equation results from these (N-1) equations. For example, if Eq. (3.66) takes the form d 2 y x dy + e y 0 (3.71) Equation (3.69) then gives one independent equation for α 1 and α 2 as -2α 1 α 2 (3.72) Physically, Eq. (3.71) may be interpreted as the equation for heat conduction in spheres with exponential heat generation. To determine the second relation for the solution of α 1 and α 2, we put y(0) A (3.73) Upon transformation, this condition becomes: y (0) + α 2 A A 98

30 Which is seen to be independent of A if α 2 1 (3.74) The transformed boundary conditions become dy (0) y (0) 0 and 0 (3.75) For the example given in Eq. (3.71), α 1 and α 2 can be found from Eqs. (3.72) and (3.74): α 1-1/2 and α 2 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 e α 1 A x 1 (3.76) Eliminating A, we get e - α1 y x 1 (3.77) Case II. The boundary condition at x becomes Or x : y + α 2 A k 1 A 1/α 2 [ k 1 - y ( ) ] (3.78) Therefore, the method proceeds as follows: First, Eq. (3.70) is solved with the boundary conditions (3.75). In case I, the solution curve to Eq. (3.70) can be plotted on y vs. x co-ordinates. Equation (3.77) is plotted on the same co-ordinates. The intersection of these two curves gives the values of y and x which give the same value of A from Eq. (3.76). The value of A is then determined. In case II, the value of A can be computed from Eq. (3.78). 99

31 The solution to the original equation, Eq. (3.66) can be obtained from Eq. (3.67) since now α 1, α 2 and A are known constants Two Parameter Method: Transformation of Two Boundary Conditions: The method developed in the preceding section can be extended to higher order differential equations as long as only one boundary condition is required to be transformed. In this section, the method will be extended to higher-order differential equation in which more than one boundary condition need to be transformed. For such cases, multi parameter groups of transformations are required. Consider now the third order differential equation N i 1 Subject to the boundary conditions Case I. Case II. A i d 3 y 3 m i d 2 y 2 y s i x t i We now define a two-parameter group of transformation n i dy d d 1 y( ) k 1, d d d 2 y( ) y(0) 0, k 1 2 d 2 d d 1 y(l) 0, d d d 2 y(l) y(0) 0, k 1 2 d 2 r i 0 (3.79) x B β 1 C γ1 x ; y B β2 C γ2 y (3.80) Under this group of transformation, Eq. (3.79) becomes N i 1 A i B m i (β-3β 1) + n i (β 2-2β 1 ) + r (β i 2 -β 1 ) + s β i 2 + t β i 1 x C m i (γ 2-3γ 1 ) + n i (γ 2-2γ 1 ) + r i (γ 2-γ 1 ) + s i γ 2 + t i γ 1 X d 3 y 3 m n i d 2 r y i dy i y s i t x i 0 2 (3.81) 100

32 The method can be applied if, for all i s, m i (β 2-3β 1 ) + n i (β 2-2β 1 ) + r i (β 2 -β 1 ) + s i β 2 + t i β 1 C 1 m i (γ 2-3γ 1 ) + n i (γ 2-2γ 1 ) + r i (γ 2 -γ 1 ) + s i γ 2 + t i γ 1 C 2 (3.82a) (3.82b) Where C 1 and C 2 are two arbitrary constants. Equation (3.81) then becomes N i 1 A i d 3 y 3 For example, the equation m n i d 2 r y i dy i y s i t x i 0 2 (3.83) d 3 y x 3 dy d 2 y 2 + y x (3.84) belongs to this class. The boundary condition at the initial point y(0) 0, can be transformed to y (0) 0 (3.85) To get the other boundary conditions at the initial point, let us put dy(0) B, and d 2 y(0) C 2 (3.86) Upon transformation, (3.86) becomes B β 2- β 1 C γ 2- γ 1 dy (0) B (3.87a) and B β 2-2β 1 C γ 2-2γ 1 d 2 y (0) 2 C (3.87b) Which are seen to be independent of B and C if, from (3.87a), 101

33 β 2 -β 1 1, γ 2 - γ 1 0 and, from (3.87b), β 2-2β 1 0, γ 2-2γ 1 1 (3.87a) and (3.87b) are then transformed to dy (0) 1, d 2 y (0) 1 2 (3.88) and the values of β 1, β 2, γ 1 and γ 2 β 1 1, β 2 2, γ 1-1, γ 2-1 To get the parameters of transformation, B and C, the boundary conditions at the second point are used. Case I B 2-d 1 C d 1-1 k 1 d d 1 y ( ) d 1 (3.89a) B 2-d 2 C d 2-1 k 2 d d 2 y ( ) d 2 (3.89b) Case II BC -1 L x where d d 1 y d 1 0 (3.90a) B 2-d 2 C -1+d 2 k 2 d d 2 y (3.90b) d 2 102

34 Therefore, B and C can be solved from Eqs. (3.89) or (3.90). The method can easily be extended to equation of the type: N i 1 A i d 3 y 3 m i d 2 y 2 n i e s iy x t i The only difference here is that one assumes a transformation group defined by dy r i 0 (3.91) x e β 1B+γ1C x ; y y + β 2 B + γ 2 C (3.92) Other steps remain the same Simutaneous Differential Equations: Application of the method to simutaneous differential equations again involves multi-parameter groups. Consider now the following system of two simutaneous equations: N i 1 A i d 2 y 2 m i dy n i y p i d 2 z 2 r s i dz i 0 z t i x q i (3.93a) M i 1 Bj d 2 y 2 m j dy n j y p j d 2 z 2 r s j dz j z t j x q j 0 (3.93b) Subject to the boundary conditions Case I. y(0) 0, z(0) 0, d d y( ) d k 1, d d z( ) d k 2 Case II. y(0) 0, z(0) 0, d d y(l) d k 1, d d z(l) d k 2 103

35 Let us now define a two-parameter transformation group x λ β 1 x, y λ β2 y, z λ β 3 μ δ z (3.94) Again, the differential equation, (3.93), are independent of the parameters of transformation, λ and μ, if the powers of λ and μin each term are respectively the same. This leads to the following system of equations for the solution of β 1, β 2, β 3 and δ: (β 2-2β 1 ) m i + (β 2 - β 1 ) n i + β 2 p i + r i (β 3-2β 1 ) + s i (β 3 -β 1 ) + t i β 3 + q i β 1 (β 2-2β 1 ) m 1 + (β 2 - β 1 ) n 1 + β 2 p 1 + r 1 (β 3-2β 1 ) + s 1 (β 3 -β 1 ) + t 1 β 3 + q 1 β 1 (3.95) r i + s i + t i r 1 + s 1 + t 1 (3.96) (β 2-2β 1 ) m j + (β 2 - β 1 ) n j + β 2 p j + r j (β 3-2β 1 ) + s j (β 3 -β 1 ) + t j β 3 + q j β 1 (β 2-2β 1 ) m 1 + (β 2 - β 1 ) n 1 + β 2 p 1 + r 1 (β 3-2β 1 ) + s 1 (β 3 -β 1 ) + t 1 β 3 + q 1 β 1 (3.97) r j + s j + t j r 1 + s 1 + t 1 (3.98) where i 2,..., N and j 2,...,M. Substitution of Eqs. (3.96) and (3.98) into Eq. (3.95) and (3.97), respectively, gives (m i + n i + p i ) β 2 - (2m i + n i + 2r i + s i - q i ) β 1 (m 1 + n 1 + p 1 ) β 2 - (2m 1 + n 1 + 2r 1 + s 1 - q 1 ) β 1 (3.99) ( m j + n j + p j ) β 2 - ( 2 m j + n j + 2 r j + s j - q j ) β 1 ( m 1 + n 1 + p 1 ) β 2 - (2 m 1 + n r 1 + s 1 - q 1 ) β 1 (3.100) 104

36 The method is applicable if Eqs. (3.99) and (3.100) each represent only one independent equation and that both give the same ratio of β 2 /β 1. If these conditions are satisfied, the ratio of β 2 /β 1 is known. Next, the required boundary conditions are defined to be equal to λ and μ respectively, i.e., y ' (0) λ and z ' (0) μ upon transformation, λ β2- β1 y ' (0) λ and μ δ λ β3- β1 z ' (0) μ Which then give, β 2 -β 1 1, δ 1 and β 3 -β 1 0 (3.101) The ratio of β 2 /β 1 obtained from Eqs. (3.99) and (3.100), together with Eq. (3.101), gives solutions of β 1, β 2, β 3 and δ. To get the parameters of transformation, the same method discussed in previous paragraphs can be applied. We will not repeat here. The method can be easily generalized to include cases with exponentials of y or z or both in Eqs. (3.93). 3.7 More General Types of Equations: The method developed above can be extended to more general types of equation. Two cases are considered here. Consider now the general second order differential equations: G M j 1 b j d 2 y 2 t j dy s j y r j x u j 0 (3.102) Where G represents an arbitrary function of the argument indicated. Case I. The method can be applied if, under the linear transformation group x A α 1 x, y A α 2 y, 105

37 only one relation between α 1 and α 2 is obtained from the condition that Eq. (3.102) is invariant under this group of transformation. As an example, the equation y 3 d 2 y 2 + sin y dy will give one relation between α 1 and α 2, namely 2α 2 - α 1 0 Case II. If y is absent in all terms in Eq. (3.102), the spiral group of transformation can always be applied for any arbitrary function in Eq. (3.102). As an example, consider d 2 y 2 + f 1 dy (3.103) where f 1 is any arbitrary function of dy/. Under the spiral group of transformation. x e α 1 a x, y y + α 2 a Eq. (3.103) becomes e -2α1 a d 2 y 2 + f 1 e -α1 a dy which is independent of a if α 1 0 for any arbitrary function f 1. The remaining steps remain the same. 3.8 Conclusion : 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 steps. First, a transformation group is defined and the given differential equation is required to be invariant, i.e. independent of the parameter of transformation, under this group of transformation. In step 2, the required boundary 106

38 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 can be applied to higher-order equations or other types of equations. It is simple to apply and only algebraic solutions are required to get the transformation. The main disadvantage of this method lies, however, on the arbitrariness in the selection of a proper group for a given differential equation. 107