Singularities of systems of arbitrary equations

Chapter 6 Singularities of systems of arbitrary equations 1.

Truncated systems

Let an n-vector X be split into three parts: parameters X1, independent variables X2 and dependent variables X3 with dimensions nl, n2 and n3 respectively, i.e. X = (X1,X2,X3), n = nl + n 2 4-n3, ni >_ O. We denote as a differential monomial a(X) the product of a constant, of powers of coordinates xi of the vector X, and of derivatives of the form

oIIg211Xj/OX g2,

(1.1)

where/(2 = ( k n l + l , . . . ,kn2) >_O, ILK211 = kn~+l + . . . 4-k,~2, and j > nl 4-n2, i.e. the coordinate xj belongs to X3. To each differential monomial a(X) we put in correspondence its vector power Q = Q(a) E ]~n by the following rule. To a constant there corresponds Q = 0. To a monomial X Q there corresponds its vector power Q. To the derivative (1.1) there corresponds the vector power exponent Q = (Q1, Q2, Q3), where Q1 = O, Q2 - - K 2 , Q3 - Ej. Here Ek denotes the kth unit vector. When the monomials are multiplied, their vector powers Q are summed, i.e. Q(ala2) = Q(al) + Q(a2). We denote as a differential polynomial f ( X ) the finite sum of differential monomials:

f(X) - ~

a~(X).

(1.2)

r=l

The set of vector powers of its monomials (Q(ar), r = 1 , . . . , s } is called the support of the polynomial (1.2), and it is denoted as S ( f ) or s u p p f . Here to different monomials ar there may correspond one point Q, but to different points Q there correspond different monomials. In the same way as 277

Chapter 6. Systemsof arbitrary equations

278

in Chapter 1, according to the set S - S(f), we determine its polyhedron r r(s), its faces r~ d), and the boundary subsets S~d) in lI(~, and also their normal cones U~d) in the dual space I~.n. To each boundary subset S~d) we put in correspondence its truncated polynomial (or truncation)

a,.(X) over r: Q(a,.) e S k(d) .

]~d) (X) - ~

(1.3)

This truncation is the first approximation to the polynomial (1.2) if log IX[ -~ oo near the normal cone U~d). Now we consider the aggregate of differential polynomials f~,...,fm.

(1.4)

To each fj there corresponds its support and all the accompanying objects r j ~ ~r(dJ) TT(dJ)~ *~(dj) i(dj) " Besides, to each non-empty intersection jkj ~ "jkj "jkj ~ Jjkj

u lkl ( dNl ). . .

N U (din)

mkm C Ii~.n

(1.5)

there corresponds the aggregate of truncations of the form (1.3)

]1r

kl

(1.6)

~ " " " ~ J mkm

which is the first approximation to the aggregate (1.4), when log IX I --+ oo near the intersection (1.5); and it is named the truncation of the aggregate (1.4). We consider now the system of equations

fj = O,

j = l...,m,

(1.7)

corresponding to the aggregate (1.4). To System (1.7) there correspond all objects indicated for the aggregate (1.4), and also the truncated systems of equations ]j(d~) (1.8) kj --0, j - - 1 . . . , m , each of which corresponds to one aggregate of truncations (1.6). We say that the truncated system (1.8) is the truncation of System (1.7) with respect to the order P ~ 0 if the vector P lies in the cone (1.5). Every truncated system (1.8) is the first approximation to the complete system (1.7). Let us specify this statement. Let Z1 = ( z l , . . . , z i , ) , where ll <__nl, Z2 = X2 and Z = (Zl, Z2) are/-vectors where 12 = n2 and l = l l + 12. Let the system of equations (1.7) have a solution of the form XI=GI(Z1),

X2=Z2,

X3=G3(Z).

(1.9)

1. Truncated systems

279

We assume that the components of vectors G1 and G3 expand in power series in their arguments: gi = gi(Z) - ~ 9 i R Z n Over R E Gi C I~l,

(1.10) i = 1 , . . . , nl, nl + n2 + 1 , . . . , n, where Gi are supports of series 9i lying in I~z. To them there correspond their polyhedrons, their faces, boundary subsets etc. Let truncations of series 9i with respect to the vector order T de._f(tl,..., tl) de__f(T1, T2) E I~l, according to Chapter 2 be ~(~') 9 (1.11) [h - ~ gil~ZR over R E "~ik~ That means that substituting

zi - bit t' (1 + o(1)),

bi

dej const r 0,

i - 1 , . . . , l,

(1.12)

in the series (1.7), we obtain 9 =

+ o(1)),

i = 1,...,

1 ,'" . ,

nl , nl + n2

n,

where B = (bl, . . . , bt) and Pi

-

(T, R>,

R

(~) E ~"-"ik,

,

i-

+ 1, "" . , n ,

(1.13)

Pi = ti, i = nl + 1 , . . . , nl + n2,

We form the vector P = (pl,... ,pn) e IRn.

(1.14)

Thus, for the system of equations (1.7) and for its solution (1.9), (1.10), to each vector order T of coordinates Z there uniquely correspond: (a) the first approximation of the solution X1 - (~1 (Z1),

X2 = Z2,

X3 - (~3(Z)

(1.15)

determined by (1.11); (b) its order (1.14) determined by (1.13); (c) the truncated with respect to the order P system of equations (1.8). T h e o r e m 1.1 [Bruno 1994] Let the system of equations (1.7) have solution (1.9), (1.10). Let the truncation of the solution with respect to the order T e I~t. be (1.15), and let it have the order (1.14), (1.13). Let S y s t e m (1.S) be the truncation of S y s t e m (1.7) with respect to the order (1.14). Then the truncation (1.15) of solution (1.9) is a solution to the truncated system (1.8).

Chapter 6. Systems of arbitrary equations

280

Proof. We make it under the additional assumption that coordinates X1 and X3 are only in integer powers in the polynomials (1.4). Then we may assume that their powers are integer and non-negative, since we can attain this multiplying polynomials (1.4) by corresponding powers of X1 and of X3. According to (1.10)-(1.13), the support of the series gi(Z) lies in the halfspace (T, R) <_pi of the space ~t = {R}, i.e. S(gi) C {R: (T,R) <_pi}, besides in the boundary hyperplane (T, R) = Pi there lies the support S(~i). For an integer non-negative q the power gq is the power series in Z, and its support is also placed in the half-space It~: S(g q) c {R: (T,R) <_piq}, besides in the boundary hyperplane (T, R) - piq there lies the support s(~q). Consequently, for Q: Q1 E Z~_1, Q3 E z~ 3, we have S(G Q) c (R:
S

g211gj) C {R:
since T2 = P2 according to (1.12). Besides in the boundary hyperplane
-

P,Q) +

where e > 0. By the statement of Theorem f ( G ( Z ) ) - O, hence ](G(Z)) - O. And it is valid for every differential polynomial from the aggregate (1.4). [:] Theorem 1.1 means that for some class of solutions, the first approximation to a solution to the system of equations is the solution to the corresponding first approximation to the system of equations. Apparently, this property

1. Truncated systems

281

should be considered as characteristic with the definition of the first approximation. The reverse statement, generally speaking, is not correct, i.e. not every solution to the first approximation (the truncation) of a system of equations is the first approximation to some solution to the system. R e m a r k 1.1. In the extraction of truncations only the local and the asymptotic coordinates xi are essential, which tend either to zero or to infinity. If differential polynomials also depend on the neutral coordinates Y in such a way that the coefficients in monomials are functions of Y, which remain bounded away from 0 and c~, then the described above procedure of extraction of truncations is applicable in this case as well. Here the coefficients 9iR and his in series (1.10) are also bounded functions of Y. The neutral coordinates Y are also split into parameters, independent variables and dependent variables. In the derivatives only the contribution of asymptotic variables is taken into account, i.e. to the derivatives

and

oIIK211-FIILIIxJ

ox '-oY

C~IIK211+IILIlyJ

ox '.oY

there correspond points Q = ( 0 , - K 2 , R) with R = Ej and R = 0 respectively. R e m a r k 1.2. If the coordinate xi occurs in System (1.7) only as the derivative d x i (or O x i ) , then the asymptotic property (tending to zero) is assumed for Axi, i.e. for the displacement along the coordinate xi, and not for the coordinate itself. E x a m p l e 1.1 [Bruno 1995b, c]. Let us find asymptotics of solutions to the equation y. de_.fd2y / d x 2 = 3y2 + x whenx--+0orx--+oc. Herenl =0, n2-n3-1, n - 2 , X2 = x , X 3 - y . The set S - {Q1 - ( - 2 , 1 ) , Q2 - (0,2), Q3 = (1,0)}. The polyhedron r is the triangle with the vertices r~ ~ - Q k , k = 1, 2, 3 (see Fig. 6.1). To the edge r~ 1) there corresponds the truncated equation y" = 3y 2. Its solutions with P E U~ 1) have the form y - bx - 2 , i.e. 6bx - 4 - 3b2x - 4 , whence b - 2. To the edge r~ 1) there corresponds the truncated equation 3y 2 + x - 0, whence y - = t = x / - x / 3 . To the edge r~ 1) there corresponds the truncated equation y" - x. Its solutions with P E U~ 1) have the form y - bx 3, i.e. 6 b x - x. Consequently, b - 1/6. To the vertex r~ ~ there corresponds the truncated equation y" - 0. Its solutions y - co + clx when T-1 -- X --+ 0 have Pl - - 1 and

P2

_~ [

0,

if

c0~0,

-1,

if

c0-0.

Chapter 6. Systems of arbitrary equations

282

Both vectors P - ( P l , p 2 ) l i e in the cone U~~ To the vertex ~ there corresponds the truncated equation y2 _ 0, the solution y - 0 of which cannot be represented in the form (1.11). The same is true for the vertex r~ ~ I-1 q2

Q

)

1

3 :2

-'1

0

q~

1

Figure 6.1: The support and the triangle of Example 1.1. E x a m p l e 1.2. Let us find asymptotics of solutions to the Emden-Fowler equation (compare with Examples 2.1, 5.1, 6.4 and 7.1 of Chapter 3) d 2 y / d x 2 - ax'~y ~ - 0,

a-~l,

(1.15')

when x--+ co. H e r e n l - 0, n2 - n3 - 1, X2 - x, X3 -- y. The support of the equation consists of two points Q = ( q l , q 2 ) : Q 1 = ( - 2 , 1 ) and Q2 = (a, #) if Q1 ~ Q2 (i.e. # r 1 and a ~ - 2 ) , which we assume to be fulfilled. Then r is the segment [Q1, Q2]. Since the normal vector to the segment r is N - (1 - # , a + 2), then for # ~ 1 the equation has a solution of the form y = bx a, where a = (a + 2)/(1 - #), and the coefficient b is found from the equation a ( a - 1)b = ab ~. Here we have to exclude the values a = 0, 1, when the equation has only the trivial solution b = 0. We consider now truncated equations corresponding to ends of the segment r . To the point Q2 there corresponds the truncated equation x ~ y ~' - 0, the solutions to which are trivial. To the point Q1 there corresponds the truncated equation d 2 y / d x 2 = O. Its solutions have the form yl - const 9x ~,

u - 0, 1;

(1.16)

their order is P - (1, v). The equation d 2 y / d x 2 - 0 is the truncation of the original equation with respect to the order P if (P, Q1) > (P, Q2), i.e. v ( # - 1) + a + 2 < 0. With such values of parameters, we obtain the first approximations yl of the form (1.16) for solutions y. The second approximation Y2 is obtained by the integration from the equation d 2 y 2 / d x 2 = a x ' ~ y [ . These results were obtained in Chapter 3 by transition from the original equation to the system of three equations resolved with respect to

I. Truncated systems

283

derivatives, and by the consideration of the support of that system in the three-dimensional space (compare Formulae (5.14')-(5.16) of Chapter 3). l-] E x a m p l e 1.3. In the singularly perturbed system

~dxldt - ~oi (x, y) = O,

dyldt - ~o2(x, y) = 0

(1.17)

with the small parameter 6 the differential dt may also be a small value. According to Remarks 1.1 and 1.2, here ni = n2 = 1, n3 = 0, n = 2, Xi = ~, X2 = t; coordinates x, y are neutral. The support Si of the first equation (1.17) consists of two points Qi = ( 1 , - 1 ) and Q2 = (0, 0). The support $2 of the second equation (1.17) consists of two points Q3 = ( 0 , - 1 ) and Q2. For P = ( - 1 , - 1 ) , we have (P, Qi) = 0, (P, Q2> = 0, (P, Q3) = 1. Hence the corresponding truncation of the first equation coincides with the equation itself, and the truncation of the second equation is dy/dt = 0, i.e. we obtain the truncated system

r

= ~oi (x, y),

(1.18)

dyldt = O.

For P = ( - 1 , 0 ) , we have (P, Qi) = - 1 , (P, Q2) = 0 = (P, Q3). The corresponding truncation of the first equation is ~oi = 0, and the truncation of the second equation is the equation itself, i.e. we have the truncated system

~oi (x, y) = O,

(1.19)

dy/dt = ~o2(x, y).

The solutions to System (1.18) are horizontal segments y = const, along which x moves with great velocity ~oi (x, y)/~, and stationary points (x ~ y0) satisfying the equation ~oi (x ~ y0) = 0. Along these points, solutions to System (1.19) move with moderate velocity. When ~ --+ 0, solutions to the complete system tend to the limit positions, which are composed from parts of solutions to truncated systems (1.18) and (1.19). In particular, such is the limit cycle, which describes the relaxational oscillations [Mishchenko and Rozov 1975, Mishchenko et al. 1994]. O E x a m p l e 1.4. Let the mass center of a satellite move along an elliptic orbit of eccentricity e. In the plane of the orbit the motion of the satellite around its mass center is described by the Beletskii [1959] equation (1 + e cos u) d2 (~ - 2e sin v ~d~ + # sin ~ - 4e sin u,

(1.20)

where # is the inertia parameter, v is the independent variable (the true anomaly on the orbit). Equation (1.20) is singular when e = 1, v = r, since in the point the coefficient at the highest derivative turns into zero. We introduce new (local) coordinates x = v - ~r, r = 1 - e. Then Equation (1.20) takes the form

e + -~

+ o ( e , x 2) -~-~x2+2[x+o(e,x)]-~x+#Sin~-

-4[x+o(e,x)].

(1.21)

Chapter 5. Systemsof arbitrary equations

284

H e r e n l = n2 = 1, n3 - 0, n = 2, X1 = 6, X2 = x. Coordinates# and are neutral. The corresponding set S is shown in Fig. 6.2, a in coordinates ql = ord~, q2 - ordx. The boundary 0 r of the set r consists of three edges: r~ 1) , r~ 1), r~ 1), and two vertices. Each element of the boundary gives its own first approximation to Equation (1.21). On the edge r~ 1) the unit vector is (0, 1), which corresponds to the variable x. On the edge r~ 1) the unit vector is ( - 1 / 2 , 1 ) , which corresponds to the variable x/v/~. Using the variable with such type of behavior (the eccentric anomaly for example), we can regularize Equation (1.20) and compute its solutions as the relaxational oscillations [Bruno and Petrovich 1994a, b, 1997; Varin 1996, 1997, 1999a, b]. In [Bruno and Varin 1995, 1996, 1997a, b, 1999], the limit equations were studied corresponding to the vertex r~ ~ and edges r~ 1), r~ 1). From solutions to these equations the limits of solutions to Equation (1.20) can be matched when e -~ 1. The theory of singular perturbations for solutions to Equation (1.20) with small # is constructed in [Sadov 1995, 1996a, b, 1997a, b, 1998, 1999]. It also uses Power Geometry. V1

P2

r~l r~0)/~ i -I

_217

r~~)

]1 "

u~)u/CP)///

u~ ~)

r (1)

2

u(~~)

Pl

b

Figure 6.2: The polyhedron r (a) and the normal cones (b) of Example 1.4. E x a m p l e 1.5. In Chapter 3 the system of ordinary differential equations of the form

dyi Yi dt

fi(Y) - O, i - 1 , . . . , m ,

Y E I~m or Y E C m

(1.22)

was considered. Here nl = 0, n2 -- 1, n3 -- m, n = m + l , X2 = t, X3 = Y. Let F - ( f l , . . . , fro)= ~ FQ3YQ3 be the vector polynomial. Let F - (]1,...,]m) be its truncation with respect to the order P3 E ~m. Then ]~ ~ 0 as vector, but some of its components ]i may be identical zeros. If the unique support S(F) E ll~TM is put in correspondence to System (1.22), as it was done in Chapter 3, then according to the definition of this Section, the support $1 of System (1.22) in I~m+l consists of the point Q0 = ( - 1 , 0 , . . . , 0 ) and of the support S(F) placed in the coordinate hyperplane ql = 0. Hence System

1. Truncated systems

285

(1.22) has three truncated systems corresponding to the one vector order

P3e

, -]i(Y)-O, d

i-1,...,m;

_ L(Y)

ad~ , _ 0 ,

-

0,

i -

if

1,...,m;

i-1,...,m;

if

-po<(P3,Q3); if

-po

-

(P3,Q3);

(1.23)

-po>(Pa,Q3).

Here the vector P - (Po, P3) E ~.m is the normal vector to the hyperplane Hp, which is the supporting one to the mentioned support Sl E Rm+l; and Q3 is an arbitrary point in S(F). Among the truncated systems (1.22), the first one corresponds to the case when the point Qo does not lie in the hyperplane Hp; for the second one the point Qo E n p and S(F) E Hp; for the third one the point Qo E Hp, but S(F) does not lie in HR. If ]i ~ 0 for all fi ~ 0, then by definition of the present Section, only these three systems are the meaningful truncated systems with respect to the vector order P3. But if ]i - 0 for some fi ~ 0, then by definition of the present Section, the truncated systems for - p l < (P3, Q3) are also the systems of the form ^

- ] i ( Y ) = O, over i" ci > -p0; dy. ~ - ]j(Y) - O, over j" cj - -po; ~j

dyk _ 0 , Yk dt -

over

k" Ck < --po

where c ~ - (P3,Q~)

for

Q~ e S(f~),

l- 1,...,m

and fi is the truncation of the scalar polynomial fi with respect to the order P3, which is always different from the identical zero if fi ~ O. Hence, these truncated systems are more meaningful than the first System (1.23). El E x a m p l e 1.6. Let us consider the differential polynomial of the form f(D)u, where f(D) is the polynomial in the differential operator D (O/Oxl,... ,O/Oxm). Here nl - 0, n2 - m, n3 - 1, n - m 4- 1, X2 - X, X3 - u. The support $1 C ~l+m of the polynomial f ( D ) u consists of points of the form Q - (Q2,1), where the point -Q2 E ~'~ belongs to the support S(f) C I~m of the polynomial f(.=.). Since all points of the support $1 have the same last coordinate, so it does not participate in the extraction of truncations. Hence truncations of the polynomial f ( D ) u with respect to the order P - ( - T , pm+l)have the form ]Jd)(D)u, where /~d)(~)is the truncation of the polynomial f(S) with respect to the order T. In [Mikhailov 1963, 1965, 1967a, b; Volevich and Gindikin 1968, 1985; Gindikin 1973, 1974; Volevich 1974; Gindikin and Volevich 1992], the first approximations to the

Chapter 6. Systemsof arbitrary equations

286

differential polynomial f ( D ) u were determined exactly in the same manner. D E x a m p l e 1.7. Near the point t -- x - 0, we consider the equation E2 0u _ ~2 02u -~ Ox 2

02u = ~f(t, x, u) coy2

(1.24)

with the small parameter e and regular function f. Here nl - 1, n2 - 2, n3 = 0, n = 3, X1 = e, X2 = (t,x); coordinates y and u are neutral. Correspondingly the support S of the equation consists of the points Q1 = ( 2 , - 1 , 0 ) , Q2 = ( 2 , 0 , - 2 ) , Q3 = 0, Q4 = (1,0,0), and points of the form (ql,q2,q3), where ql _> 1, q2,q3 >__ 0 representing the support of the expansion for el. The polyhedron r includes the face r~ 2) spanned on vertices Q1, Q2, Q3. For P < 0 all truncations of Equation (1.23) correspond to faces lying in that face r~ 2). The directing vectors of its edges [Q1, Q3] and [Q2, Q3] are ( - 2 , 1 , 0) and (-1, 0,1) respectively. They form the basis in the face r~ 2). To them there correspond "variables of the boundary layer" e-2t and e - i x . In Section 15 in [Vasil'eva and Butusov 1990] it was proved that near the point t = x = 0, the solution u to Equation (1.24) is the regular function in these variables and in variables e, t, x, y. 2.

Power

transformations

Let an n-vector X be split into three parts: parameters X1, independent variables X2 and dependent variables X3, with dimensions nl, n2 and n3 respectively: X = (X1,X2,X3), n = nl + n2 + n3, ni >_ 0. We consider the power transformation log X1 = Wll log ]I1, log X2 = W21 log ]I1 + W22 log ]I2,

(2.1)

log X3 = W31 log ]I1 + W32 log ]I2 + W33 log ]I3. Here log X1 = (log X l , . . . , log xnl)*, Wii are non-degenerate square matrices, Wij are rectangular real matrices, W = (Wij) is the square block matrix, and the asterisk means the transposition. Let the change of coordinates (2.1) transform the differential polynomial f ( X ) in g(Y) = f ( X ) . We want to study the relation of their supports S(/) = {Qj} and S ( g ) = {Sj}. T h e o r e m 2.1 [Bruno 19965] Under the power transformation (2.1), the differential polynomial f (Z) transforms into the differential polynomial g(Y),

2. Power transformations

287

their supports S(g) = {Sj} and S(f) = {Qj) are related by the linear transformation S = W'Q, (2.2) and vectors of the dual space ~n. are transformed as R = W-tP.

(2.3)

Proof. We write the power transformation (2.1) in the form log X = W log Y. The reverse transformation is log Y - A log X,

(2.4)

where A - (aij) - W -1. For the monomial X Q - y S the equality (2.2) was proved in Chapter 3. Let us prove it for the derivative (1.1). From (2.4) it follows that Ologyi/Ologxj - aij, i.e.

Oyi/Oxj = aijyilxj.

(2.5)

1~ Let b(Y) be a differential monomial, S(b) - S, nl < i <_ nt + n2. Then Ob/Oyi is a differential polynomial with the support consisting of one point S - El. Indeed, it is true for b = y S , for b - oIIK211yj/OYg2, and for the product of such monomials, i.e. for every monomial. 2~ Now we prove that the derivative (1.1) is the differential polynomial in Y, i.e. h(Y) - E b~(Y), (2.6) where br are differential monomials, and its support consists of one point

(2.7)

S(h) - S de_f W* (Zj - g 2 ) .

We make the proof by the induction on the increasing ILK211. When 11/s - 0 the derivative (1.1) is xj = yW*Ej, i.e. Formula (2.6) is valid. Let Formulae (2.6) and (2.7) be valid for all/s with ILK211_ k. We prove it for IIK2[I = k + 1. Obviously, it is sufficient to prove this for one monomial br. So, let S(br) = S, IIS211 = k and nl < j <_ n2. According to (2.5)

Obr _OXj

~n2 + i=

Obr aijxj Y_L Obr cqyi _ En2 -~yi 1 0 y i Oxj i= nl'~f-1 1

n2 E

Xj i : nlq-1

Obr def -~yia i j y i - l Z ( Y ) "

(2.8)

Xj

According to Statement 1~ f~(Y) is a differential polynomial, the support of which consists of one point S. Consequently, the derivative (2.8) is also

Chapter 6. Systems of arbitrary equations

288

the differential polynomial in Y, and its support is S - W*Ej. That proves Statement 20 and Formulae (2.6), (2.7). Thus, each differential monomial a(X) of the differential polynomial f (X) is the product of some elementary monomials of the form X Q and (1.1), for which the statement (2.2) of Theorem is proved. Since when the monomials are multiplied, their supports are summed, hence it follows the statement (2.2) of Theorem for every monomial and for every differential polynomial. Since (2.2) is the transformation in the main space ~n, then the transformation in the dual space ll~,~ is (2.3), because the scalar product (P, Q) - (R, S) must be preserved. [2] According to Theorem 2.1, all sets in I~n and II~, n are also changed by the corresponding linear transformations (2.2) and (2.3). Consequently, the extraction of a truncation commutes with any power transformation. Besides, the power transformation can be used for the simplification of the truncated equation making its support parallel to the coordinate subspace. T h e o r e m 2.2 [Bruno 1996b] Let d = dim D(f) < n. Then there exists such a matrix W that after the transformation (2.1) the values of n - d coordinates sj are constant for all S = ( S i , . . . , S n ) E S(g), where g(Y) = f ( X ) . Let --Sj sj - const. In yj g(Y), the coordinate yj is absent if j <_ ni, and it is present only in the form 0 log yj if j > ni. P r o o f . For the first statement of Theorem it follows from Theorem 2.1 and the results of Section 11 of Chapter 1. The second statement for j _< n i follows from the results of Section 3 of Chapter 2. Let us prove the second statement for j > n i. Let j _ ni + n2. The statement of Theorem follows from the identity k

Ok

_

Ok

(2.9)

YJ oy] - O(1og y~)k" Let n i + n2 < j. We prove by induction on k that in this case

Ok Yj -- Yj Pk (0 log yj, 02 log y j , . . . , Ok log yj),

(2.10)

where Pk(~l,..., ~k) is a polynomial with constant coefficients. We put v = log yj. Then yj = exp v and Oyj = yjOv. Let Formula (2.10) be true for some k. We prove it for k + 1. Indeed, from Equality (2.10) follows the equality

OPk 0 i+i log YJ' ok+ly i -- yj(Ologyi)Pk + yj ~, i -~i which has the form (2.10) with

OP~

Pk+l(~l,...,~k+l) -- ~lPk(~l,'",~k) T i ~ "~/~i+1.

2. Power transformations

289

According to (2.10), we have yj c9 yj - P k ( O l o g y j , . . . ,

logyj),

whence it follows the second statement of Theorem for n l + n2 < j. [-'] We note that the multiplication of the polynomial g(Y) by the factor y T means the parallel translation of the set S(g) by the vector T: S ( g y T ) = S(g) + T. In this way, if dim r ( f ) < n, then by the power transformation (2.1) and by the multiplication by y T one can place the support of the polynomial f in the d-dimensional coordinate subspace. That allows to lower the dimension of the truncated problem. The original equation is the regular perturbation of its truncation in the corresponding domain of the X-space, where the truncation is the first approximation. It remains valid after the application of the power transformation as well. But then the domain may be made close to the coordinate subspace in the Y-space. From Section 11 of Chapter 1, it follows T h e o r e m 2.3 Let ]~d)(X) be a truncation of the differential polynomial f ( X ) . There exist such a matrix W and a vector T that after the transformation (2.1) there are n - d coordinates yj, which either absent in :the polynomial yT[?~d) ( y ) or present only in the form 0 log yj, where 9~d) ( y ) _ ]~d) ( X ) . Besides s ( y T g ) C I ~ , where g(Y) - f ( X ) and I ~ - {S >_ 0}. E x a m p l e 2.1 (the continuation of Example 1.2). transformation z = x - ~ y , we have y' = z'x ~ + ( a -

Under the power

1)zx ~-1, y" = z " x ~ + 2 a z ' x ~-1 + a ( a -

1)zx ~-2, (2.11)

where ' de_f d/dx. According to the definition, a - (a + 2)/(1 - #), and we have a + # a = a - 2. Hence Equation (1.15') takes the form x~-2[x 2z'' + 2axz' + flz - az ~] = O,

(2.12)

where fl = a ( a - 1). Cancelling x ~-2, we obtain the equation x2z '' + 2axz' + f l z - az ~ = O.

(2.13)

Its support consists of two points: (0, 1) and (0, #) (see Fig. 4.3). If we put t = log x, then z' - x -1~, z" - x - 2 ( ~ - ~), where" de_fd/dt, and Equation (2.12) takes the form + (2a - 1)~ + flz - az ~ = 0.[:3

(2.14)

Chapter 6. Systems of arbitrary equations

290 q3 #

,q3

q3

q3

Q4

Q2

-2

ql -'I 0 ql ,~ g

-'1

ql

0

b

.

. Q2

:2

51 c

T 1ql

-

1

-Q~

ql

1 0 ~

-2

Ll d

C

Figure 6.3: Supports and polygons for Equations (1.15') (a), (2.13) (b), (2.14) with fl ~ 0 (c), and (2.14) with/~ - 0 (d) The co-dimension of the intersection (1.5) is called the dimension o] the truncation (1.6) of the aggregate (1.4). So Theorem 11.1 of Chapter 1 gives 2.4 Let the aggregate of differential polynomials (1.4) have the truncation (1.6) o] the dimension d. There exist such a matrix W and such vectors/"1,..., Tm that after the transformation (2.1) there are n - d coordinates yj, which either absent in polynomials y T ~ ( y ) , i -- 1 , . . . , m ~(d,)(X) or present only in the form O logyj, where gi(Y) JO~ " Besides s(yT~gi) C I ~ , where gi(Y) - f i ( Z ) , i = 1 , . . . , m. There are two types of the power transformation (2.1): with an arbitrary real matrix W, and with an unimodular matrix W (i.e. det W = +1) with integer elements. The power transformation of the second type gives the oneto-one correspondence between X and Y outside the coordinate subspaces. In order to find the matrix W, it is necessary to use algorithms described in Section 11 of Chapter 1. Theorem

_

3.

The logarithmic transformation

Let a differential polynomial g(Y) be such that for some j the coordinate sj = 0 for all S E S(g). If yj is the algebraic coordinate (i.e. a parameter, j _ nl), then g does not depend on yj. If yj is the differential coordinate (i.e. variable, j > nl), then g depends only on log yj. Let J be the set of such indices j > nl that sj = 0 for every S E S(g). We introduce new variables by means of the logarithmic transformation vj-logyj

for j E J ,

v k - - Y k for k ~ J . We denote h(V) - g(Y). If g really depends on yj, j E J, then the support S(h) has points Q with qj ~ 0. Hence we can find the first approximations for h(V) using its polyhedron r . Solutions to the corresponding truncated

3. The logarithmic transformation

291

equations may give logarithmic asymptotics of solutions to the original problem. We note that vj --+ c~ when yj --+ 0, or yj ---4 cx3 and j E J. E x a m p l e 3.1 (the continuation of Example 2.1). After the transformation t - log x, Equation (2.12) takes the form (2.14). If/3 def_.a ( a - 1) ~ 0, then its support consists of 4 points: Q1 = ( - 2 , 1 ) , Q2 = ( - 1 , 1 ) , Q3 = (0,1) and Q4 = (0, #), where ql = ord t, q2 = ord z. Since t -4 c~, then for the studied solutions the vector P = (pl, p2) has pl ___0. The polyhedron r for Equation (2.5) has only one edge r~ 1), the normal vector P of which has pl > 0 (see Fig. 6.3, c). The corresponding boundary subset S~l) consists of two points: Q3, Q4, and the truncated equation is ~ z - a z ~ = O. Its solutions are the values b cited in Example 1.2. If/3 = 0, i.e. a = 0, 1, then the support of Equation (2.14) consists of three points: Q1, Q2, Q4. Again the polyhedron r has only one edge, whose normal vector P has the component pl > 0 (see Fig. 6.3, d). But now the corresponding boundary subset consists of points Q2, Q4, and the truncated equation is -y~ - a z ~ = 0, where -y = 2a - 1. Its solutions are z - [a(1 - #)7 -1 (log x + c)] 1/(1-u), Consequently, for a asymptotics

-

0,1,

c - const

Equation (1.15') has solutions with the

y ,,~ x-a[a(1 - #)7 -1 log x] l/el-u). That coincides with Formulae (6.33), (6.34) in Section 6 of Chapter 3. We note that the found in Example 6.4 of Chapter 3 logarithmic asymptotics (6.32) cannot be the asymptotics of a solution to Equation (1.15'). It relates only to solutions to the system of equations (2.12) of Chapter 3. But not every solution to that system is a solution to Equation (1.15'). K] E x a m p l e 3.2 (the continuation of Example 1.2). We consider Equation (1.15') when # = 1 and a ~ - 2 . Now the points Q1 and Q2 have the same second coordinates q2 - 1. Hence we have to cancel y and make logarithmic transformation v = logy in Equation (1.15'), i.e. y = expv. Since y' = y v ' , y " - y(v') 2 + y v " , where ' de__fd / d x , then Equation (1.15') with # - 1 takes the form (v') 2 + v " - a x " = 0. (3.1)

Its support consists of three points, and the convex hull is the triangle. Only one edge of the triangle has the normal P = (Pl,p2) with the component P2 > 0. To this edge there corresponds the truncation (v') 2 = a x " of Equation (3.1). Its solutions are -

•

+ 2),

Chapter 6. Systems of arbitrary equations

292

where c - const. To these solutions there correspond asymptotics of solutions to Equation (1.15') Y1,2 = cexp[+2(2 + a) -1 v/'ax("+2)/2].V1

Example

3.3. We consider the equation

02r Ox 2

I

02r Oy 2

-- f ( x , y ) ,

where f is a homogeneous polynomial in x, y of the degree m - 2 . Here nl - 0, n2 - 2, n3 - 1, n = 3, X2 - (x,y), X3 - r The support of the equation consists of points ( - 2 , 0,1), ( 0 , - 2 , 1), (k, m - 2 - k, 0), where 0 < k < m - 2. All the points lie in the plane ql + q2 + mq3 - m - 2, where ql - ord x, q2 - ord y, q3 - ord r The power transformation u - x,

and the cancellation of

Um-2

z - x-ly,

~o - x - m e

reduce the equation to the form

0~o u2 02~o

02~o

z2 02~o

0~o 02~o + 2 z ~ + o z ~ = / ( 1 , z). (3.2) Its support lies in the plane sl - 0, where sl = ord u, s2 - ord z, s3 - ord ~o. Hence we assume t = log u. Then the equation takes the form

m(m-1)~+

2mu~

m ( m - 1) ~o + z 2 02 ~o

+

~

- 2uz o ~ o z +

~

0~o 02 ~o 0~o 02 ~o 02 ~o + 2Z~z + ~ + ( 2 m - 1) -~- - 2z OtOz + ~ O2t = ](1, z). (3.3) Its support consists of points (0, 0, 1), (0, - 2 , 1), ( - 1 , 0, 1), ( - 2 , 0, 1), (0, k, 0), and it has three-dimensionM convex hull r . Now ql = ord t, q2 = ord z, q3 = ord ~o. Fig. 6.4 shows the support and its polyhedron r for m = 4. Since t -+ c~, then here we need only faces with the normal P = ( p l , p 2 , p 2 ) , which has Pl > 0. Such a face here is unique. To it there corresponds the truncation of Equation (3.3), where the derivatives with respect to t are absent, i.e. for ~o we obtain the ordinary differential equation. The same equation is obtained from (3.2) if ~o does not depend on u. The corresponding solution r to the initial equation is the so called "self-similar solution" (see Chapter 7). [:3

4. A big example

293 q3

4 5 6

ql 1

q2

Figure 6.4: The support and the polyhedron for Equation (3.3) of Example 3.3. 4.

A big example

Let us find asymptotics of solutions y ( x ) to the equation y'" - x " y ~' = O,

a, # E IR,

(4.1)

when x -~ oo. The equation is always real when x, y > 0. If a or # are irrational with an odd denominator, then the equation is also real when x < 0 or y < 0 respectively. In notation of Sections 1 and 2, in Equation (4.1) nt - 0, n2 = n3 = 1, n = 2, X2 = x, )(3 = y. The left hand side of Equation (4.1) consists of two monomials with the vector powers: Q = ( - 3 , 1 ) , Q2 = (a, #). Firstly, we consider the case # # 1.

(4.2)

Then Q1 ~ Q2, and the polyhedron r is the oblique segment [Q1, Q2] (see Fig. 6.5, a). The vector R de.._f Q 2 - Q 1 - ( q + 3, # - 1) is directed along the segment. We put a = - ( a + 3 ) / ( # - I). (4.3) Then the normal to the segment I' is N = (1, a). The segment r has three ces: r l - r, ~ = ~ W e consider the corresponding truncations one by one. To the edge I'~ t) there corresponds truncation, which coincides with the

f

def

original equation (4.1). We denote x - sgnx - =kl. We are going to find the solution to Equation (4.1) corresponding to the normal vector N, i.e. y = blxl ~,

b = const

(4.4)

Chapter 6. Systems of arbitrary equations

294

Substituting this expression in (4.1), we obtain _

where/3 - a ( a - 1)(a - 2). By virtue of (4.3), we have a - 3 - a + a#. Reducing the obtained equation by b{xla-3>g, we obtain the equation (4.5)

1~/~--1 __ ~ X - - ( a ~ - 3 ) .

If

def r

_

1)(a - 2) r O,

then for each value x = 4-1, Equation (4.5) solutions b. Thus, when/3 > 0 there are ~< = =t:l. So, when conditions (4.2), (4.6) solutions (4.4), (4.5), which coincide with Section 8 of Chapter 3 under the condition

Qlj

J

~~Q2

$2

-1

S1

-2

-1

i

01

Q5

2

81

-1

a

o0[

-3

-2

b

-1

q2

ql

01

c

q2

Q5

1 oy_

-1

ql -3

may have no more than two real always the real solution b when are satisfied, Equation (4.1) has solutions (8.9), (8.14) found in (8.15).

82 -#

q2

(4.6)

-3

1

Q3

"Q2

-2

-1

ql 0

d

Figure 6.5: Supports and polygons for Equations (4.1), (4.2) (a), (4.9) (b), (4.11) with ~ ~t 0 (c), (4.11) with/3 - 0 (d). To the vertex r~ ~ de._fQ1 there corresponds the truncated equation y " - 0. Its solutions have the form y-cx

V,

c-

def

const ~t0,

u-0,1,2.

(4.7)

The vector order P of such a solution is P - (1, u). It lies in the normal cone U~ ~ of the vertex Q1 if (P, R / < 0, i.e. ~+3+v(#-

1) < 0.

(4.8)

For u = 0, 1, 2 the asymptotics (4.7) under the condition (4.8) coincides with asymptotics (8.13), (8.12), (8.11) respectively, which were found in Section 8 of Chapter 3. In fact, asymptotics (8.13) and (8.12) from Section 8 of Chapter 3 include also the second terms of the expansion y - yl + y2 + . . . over decreasing powers of x. The second terms y2 of the expansion are obtained

4. A big example

295

by integration from the equation y~" - x ~ y ~ , where yl is the corresponding first approximation (4.7). To the vertex r~ ~ de_f Q2 there corresponds the truncated equation - x ~ y ~' - O, which has no non-trivial solutions. When # ~ 1, Equation (4.1) has dimension d - 1 < n = 2. According to Theorem 2.2, we simplify it by means of the power transformation y '- zlx] '~. According to the rule of differentiation of a product y'" - z ' " I x l ~ + 3 ~ x z " I x l ~ - ~ + 3c~(a - 1)z'lxl '~-2 + ~ x z I x [ ~ - 3 .

Substituting this expression in Equation (4.1), we obtain the equation z'"lxl" + 3 , ~ " l x l "-~ + 3 , ( ~ -

1)z'lxl " - : + ~xzlx[

~-3

-

~l~l~+."z .,

(4.9)

the support of which is shown in Fig. 6.5, b. Cancelling Ixla-3x in it and taking into account the equality (4.3), we obtain the equation x 3 z ''' + 3 a x 2 z '' + 3 a ( a - 1 ) x z ' + ~ z - x ~ - l z ~ ' .

(4.10)

According to Section 3, we make now the logarithmic transformation t log x, x - e t. Denoting the differentiation with respect to t by dot, we obtain z'- x-l~,

z"-

x - 2 ( / / - ~), z ' " - x-3(~" - 3 5 + 2~).

Substituting these expressions in (4.10) and collecting similar terms, we obtain the equation "z" + 3 ( a - 1)~ + -),~ + f~z - ~ - Z z ~, (4.11) def

where "7 - 3a 2 - 6(~ + 1. For f~ ~ 0 the support of this equation consists of five points, and it is shown in Fig. 6.5, c. Since t -+ c~, we are interested only in truncations, the normal cones of which contain the vector P - (Pl, P2) with Pl > 0. It is clear from Fig. 6.5, c, that there is only one edge (vertical) with that property. To it there corresponds the truncated equation b - ~-Zz'. Its solutions (z0, xo) are the values (b0, x0) satisfying Equation (4.5). To the vertices of the vertical edge there correspond truncated equations, which have only trivial solutions. Consequently, when fl ~ 0 we obtain only asymptotics (4.4), (4.5) found earlier. Let (b0, x0) be a solution to Equation (4.5). When x = x0, then z = bo is the stationary solution to Equation (4.11). In order to find the equation in variations for that solution, we put z = b0 + w and isolate the linear part in w. We obtain the equation iii +3(c~ - 1)~ + 7~b + f l w - ~ o - Z # b ~ w .

Chapter 6. Systems of arbitrary equations

296

According to (4.5), ~0 +~ b~-1 - ~, hence that equation is iii + 3 ( a - 1)5 + 7~b + ~(1 - #)w = 0. Its eigenvalues Ai are roots of the equation A3 + 3(a - 1)A2 § "),A+/~(1 - #) = 0. The discriminant of the equation is D

de_f _

27~2(1 _ #)2 + 5 4 ( a - 1)7/~(1 - # ) +

+9(a-

1)2~ 2 -4") '3 - 1 0 8 ( a - 1)3f~(1- #).

If D > 0, then all eigenvalues Ai are real, and near the solution z = bo there are no oscillations. If D < 0, then there are two complex eigenvalues, i.e. near the solution z = b0 there are oscillatory solutions. When ~ = 0, the support of Equation (4.11) consists of four points, their convex hull is the triangle (Fig. 6.5, d). Again only one its edge (the right one) has the normal vector P = (pz, P2) with pl > 0. To the edge there corresponds the truncated equation 72 - ~0 -1Z tt. Its solution t - ~ 0 - 1 ( 1 - #)-z z~-~ or z - [ ( 1 - #)~f-ZXoa+Zt]l/(z-u)

(4.12)

gives the logarithmic asymptotics y = Ixl'~z

(4.13)

of solutions to Equation (4.1). Since ~ = 0 only when a = 0, 1, 2, and for these values a we have 3' = 1 , - 2 , 1 respectively, then the asymptotics (4.12), (4.13) coincides with asymptotics (8.22), (8.21), (8.20) of Chapter 3 respectively. We note that to the ends of the right edge of the triangle in Fig. 6.5, d there correspond truncated equations -y~ - 0,

~0+lz u - 0.

(4.14)

The first has solutions z = const Their vector order P = (1, 0) does not lie in the cone of truncation. The second equation (4.14) has no non-zero solutions. We consider now the case # = 1. Here Equation (4.1) is linear

y"'=

(4.15)

If a + 3 ~ 0, then Q1 r Q2. In that case the segment I' is placed horizontally (see Fig. 6.6, a). According to Section 3, we make the logarithmic transformation u = log y, i.e. y = e u. We compute derivatives

4. A big example

297

Substituting the last expression in Equation (4.15) and cancelling y, we obtain the equation + 3u" ' + (u') = (4.16) Its support consists of four points, their convex hull is the triangle A (see Fig. 6.6, b). Only at the upper edge of the triangle A the normal P = (pl, P2) has p2 > 0. To the edge there corresponds the truncation (u') 3 = x ~

(4.17)

of Equation (4.16). Equation (4.17) falls apart in three differential equations: u }I - e k Xa/3 ,

k=0,1,2,

(4.18)

def

where ek = e x p ( 2 r i k / 3 ) are roots of the third power from unit: c 3 Solutions to Equations (4.18) are u k -- ~'k(1 + a / 3 ) - l x

1+~

k - 0, 1, 2.

1.

(4.19)

Consequently, solutions to the linear equation (4.15) have asymptotics of the form y = co exp u0 + Cl exp Ul § c2 exp u2 composed of solutions (4.19). Since Im Cl = - I m e2 = vf3/2 ~- 0, then among these asymptotics there are oscillatory ones. q2

9q2

8

ql -3

-2

-I

C

-3

-2

-1

a

0

1

2

b

Figure 6.6: Supports and polygons for Equations (4.15) (a) and (4.16) (b). If # - 1 and a + 3 - 0, then Equation (4.1) is y " ' - x - a y . Its support consists of one point Q1 - Q2 - ( - 3 , 1 ) . After the multiplication by x 3, we obtain the Euler equation x 3 y ''' - y. Its solutions are y -- clxOL1 -~- C2 xOe2 ~ C3XOe3~

where Ck = const, and ak are roots of the equation c~(c~- 1)(c~ - 2) - 1. R e m a r k 4.1. Using the procedure expounded in this Section for the equation y(m) _ x~ y~,

(4.20)

Chapter 6. Systems of arbitrary equations

298

with arbitrary integer m > 0, one can find asymptotics of solutions when x --+ c~. Here a = (a + m)/(1 - #),/~ = c~(c~ - 1 ) . . . (~ - m + 1). If/3 = 0, then c~ = k < m and 7 = ( - 1 ) m - k - l k ! ( m - k - 1)! ~ 0. Many formulae of this Section either preserved or changed insignificantly. Since the system of equations/~ = 0 = 7 has no solutions, solutions to Equation (4.20) have no asymptotics with multiple logarithms.

5.

One partial differential equation

The general concepts of Section 1 are described for a partial differential equation with two independent variables x, y and one unknown function r in the following way. We define the differential monomial a(x,y, r as the product of powers of coordinates x, y, r and derivatives ok+tr To the monomial a(x, y, r there corresponds the point (its vector power) Q = Q(a) E /~3: the vector Q = (ql, qz, q3) corresponds to the product const xqlyq2r q3, the vector Q = ( - k , - l , 1) corresponds to the derivative ok+tr the sum of vectors Q corresponds to the product of their monomials. A sum of differential monomials is called the differential polynomial f ( x , y , r to a polynomial f there corresponds the set S = S(f) of powers Q of its monomials in ~3. The set S is called the support of the polynomial f. Using the set S, one can build in I~3: the polyhedron r ( f ) as the convex hull of the set S(f), faces r~ d) of its boundary 0 r ( f ) (here d is the dimension of the face, and j is its number), and the boundary subsets S~d ) - S M r~.d). To each face

d) in

the dual space ~3. there corresponds its normal cone U~d). It consists of such P E ~3., for which H p f i r - r~ d), where n p is the plane, which is supporting to the polyhedron r and orthogonal to the vector P. The truncation ]Jd) of the differential polynomial f ( x , y , r is the sum of all such monomials a(x, y,z), for which Q(a) E S~d). The truncation ]~d) is the first approximation to the polynomial f in curves of the form

x - blTPl(1 + o(1)),

y -- b2TP2(1 + o(1)),

r -- b3TP3(1 + o(1))

(5.1)

with bi # 0 and T -~ C~ if P - (Pl, P2, P3) E U~-d). Let a function h(~) be defined when 0 < ~ < c~. We put s - lim l~ Ih(~)l log ~

-

for

~ -+ O;

~ _ limlog Ih(~)l log

for

~ --+ ~ .

The asymptotic support of the functions h is denoted as supp h, and it is defined as the following set in ~: supph-[s,~],

if

s<__~;

supph-I~\(s,~),

if

s>~.

5. One partial differential equation

299

We define the asymptotic support of the function xPh(y/x~)(log x) k as such a part of the straight line L' - {Q' = (qz,q2)" q~ - p - ca, q2 = c, c E I~} in the plane (ql,q2), for which q2 E supph. The straight line L' is given by the equation qz + a q2 - P, and its normal vector is P ' - (1, a). Let hk(~), k - 0, 1 , . . . , m be functions of one variable. The function m

gl - E

xPhk(y/x'~)(logx)k

k=O

is called pseudo-homogeneous. The asymptotic support of this function represents the union of supports of all addends, and it lies in the straight line L'. The pseudo-homogeneous function gl = xOh(y/x ~) is called the selfsimilar one. If g(x,y) = gl + O(x p-~) when x -+ oo, y / x ~ = const and > 0, then the pseudo-homogeneous function gl is the first approximation to the function g with respect to the vector P', i.e. in curves (5.1) with P' = (pl,p2) = (1, a). To the expression r = g(x, y) there correspond the support of the function g in I~2 and the point E3 = (0, 0, 1) in l~3. Let Lp, denote the straight line in the plane ll~2 = {q3 - 0} with the normal vector P ' - (pl,p2) E 1~2. -- {P3 -- 0}. Let M p be the plane in ~ 3 passing through the point E3 and orthogonal to the vector P = (pl,p2,p3). If the plane MR passes through the straight line L p,, then P-

(P',p3),

P3 - (P', Q')

when

Q' E Lp,.

(5.2)

T h e o r e m 5.1 Let r = g(x,y) be a solution to the equation f ( x , y , r = 0. Let the pseudo-homogeneous function OR, be the first approximation to the function g with respect to the vector P~, and the support o/ the function OR, be placed in the straight line Lp,, and let the vector P defined in (5.2) belong to the normal cone U~ d) . Then r - OR, is the solution to the truncated equation

y, r - 0. Further we assume that the solution r to the equation f ( x , y , r = 0 expands in the asymptotic series in pseudo-homogeneous functions with respect to each P ' e ~2. \ {0}. Let for the equation / ( x , y, r

= 0

(5.3)

we need to find a solution r - r y) satisfying the boundary conditions of the form r - gi(x,y) when (x,y) E A/[i, i - 1 , . . . , m , (5.4) where A/[i is the set of points in infinity or in coordinate axes, and gi are polynomials. In general, we consider in ~3 the support of Equation (5.3) and the asymptotic support of the solution r - g(x, y) as well. From conditions

Chapter 6. Systems of arbitrary equations

300

(5.4) one can find its separate parts. We are going to consider Equation (5.3) and boundary conditions (5.4) as a mixed system of differential and algebraic equations. Then to each vector P - (Pl, P2, P3) e II~3.\ {0} there correspond: (a) the truncated equation

y, r - 0 ,

(5.5)

(b) the subsets A~4i of sets AA~, which may be reached in curves (5.1) with the vector P (some of subsets J~74~may be empty), and (c) truncations gip' of functions gi with respect to the vector P' = (pl,pe). By Theorem 1.1, the first approximation r - hp, (x, y) to the solution r - h(x, y) must satisfy the truncated equation (5.5) and truncated boundary conditions ^

r

when (x,y) eA)/i,

i-1,...,m.

(5.6)

Here, if the support of the function ~0i is placed in the straight line Lp, in ll~2, then vectors P' and P = (P',P3) are related by the relation (5.2). In order to consider algorithmically the system of equations (5.3), (5.4), we need for each of the equation to build its own polyhedron, to find its faces and their normal cones. To each non-empty intersection of these cones there corresponds its own truncated problem, which consists of the truncated equation and truncated boundary conditions. The solutions to these truncated problem may serve as asymptotics to the solution to the original problem. In the considered three-dimensional case, in order to find the intersection of the normal cones of Equations (5.3), (5.4), one may use the graphical method of Section 3 in Chapter 1, i.e. to consider the intersection of the cones with the planes P3 = 1 and p3 - - 1 . Not every boundary condition has the form (5.4), but the general principle remains valid: one needs to seek such solutions to the truncated equation, which satisfy the corresponding "truncated" boundary conditions. This approach is realized in the following section. 6.

The

viscous

fluid flow around

a plate

1. The first approximations to the Helmholtz equation. We consider the system of the Navier-Stokes equations describing the stationary flow of the viscous incompressible fluid [Navier 1827, Stokes 1849]:

Ou Ov

Ou Ov

lop

(02u

02u~

p Oz

\ Oz .

- y2 ]

10p

(O'v

+

02v~

,

5. The viscous fluid flow around a plate

301

Ou Ou + : O--x Uy

-

o.

Here x, y are rectangular coordinates, u and v are components of the vector of the velocity of the flow with respect to axes x and y, p is the pressure, p is the density, u is the kinematic coefficient of the viscosity. Q1

Q5

-4

1[:6

r ~ 1)

I3

-2

q3

r ~ 1)

-1

--% Q2

Q7

2

3

ql -- q2

4 q2

-4

-3,

-2,

-1J

ql

0

Q5

--1

Q1

--2

Q6 --3

7

~-4

Figure 6.7: The support and the trapezoid F of Equation (6.3) in coordinates ql - q 2 , q3 (above), and its projection on the plane ql, q2 (below). If we exclude the pressure p from the first two equations, and using the third equation we introduce the stream function r by formulae

0r u-

Oy'

v-

0r

(6.2)

Ox'

then for the function r we obtain the Helmholtz [1858] equation"

0r 3r Oy Ox3

{

0r 03r

0r 03r

0r 3r

Oy OxOy2

Ox Ox20y

Ox Oy3 ~, \ ox 4 + 2 o-~-Oy 2 + ~

.

(6.3)

Chapter 6. Systemsof arbitrary equations

302

Let us count the powers of all monomials as they appear in the equation: Q1-(-3,-1,2),

Q2-(-1,-3,2),

Q5 = ( - 4 , 0 , 1 ) ,

Q3-Q1,

Q6 = ( - 2 , - 2 , 1 ) ,

Q4-Q2,

QT- (0,-4,1).

The support S consists of five points: Q1, Q2, Q5, Q6, Q7. They are all placed in the vertical plane ql + q2 - - 4 . Their convex hull r is the trapezoid placed in the same plane. In the upper Fig. 6.7, it is shown in coordinates ql -q2, q3. Its boundary OF consists of four edges F~1)- F~1). The lower Fig. 6.7 shows its projection on the plane ql, q2. For each edge r!. 1) we isolate its boundary subset S~1), write down the a~

corresponding truncated equation, and compute its normal cone U~ 1) by formulae of Chapter 1. 1) The edge F~1). The boundary subset S~1) - {Q1,Q2,Q3,Q4}. The truncated equation Oy

~

+ OxOy2

- ~

Ox20y + ~

-0"

(6.4)

The normal cone U~ 1) - {P: (P, Q1) - (P, Q2) > (P, Q6)} - {P:pl = p2, p3 > 0}. 2) The edge r~ 1). truncated equation

The boundary subset S~1) -

Oy OxOy 2

cox Oy3

=

{Q2, Q4, QT}.

The

(6.5)

Oy4.

The normal cone U~ 1) - {P: (P, Q2) - (P, QT) > (P, Qs)} - {P: pl - p2 + p3, pl > p2}. 3) The edge

Here S~1) - {Qs, Q6, Q7}. The truncated equation

04r 0- u

04r + 04r

~x 4 + 20x2Oy.--------------~

~

.

(6.6)

The normal cone U~ 1) - {P: (P, Q~) - (P, QT) > (P, Q1)} - {P:pl - p2, p3 < 0}. 4) The edge F~1) 9Here S (1) - {Q1, Q3, Qs}. The truncated equation Oy Ox 3

Ox Ox20y

"--V--

Ox4"

(6.7)

6. The viscous fluid flow around a plate

303

If x and y change places, and the sign of r is changed, then Equation (6.7) is transformed in Equation (6.5), and all sets related to it are transformed into the corresponding sets for r~ 1). The points Q1, Q2, Q5 and Q7 are vertices of the trapezoid r. In order to simplify notation, we are going to denote their normal cones as U~~ U~~ U~~ and U~~ respectively. Fig. 6.8 shows intersections of the normal cones U(_. d)" 3 with planes p 3 - 1 and p 3 - - 1 .

2I p2 /

P2 .(o) U~//u(O)/

u~ ~

2 1

u~/~

/

/

u (~

i~ Ui(1)U~I) 11 ._./"1 a

u~i' Y

i

"7

Pl

v

u(~o) U(31)

b

Figure 6.8" Intersections of the normal cones with planes P3 - 1 (a) and p3 - - 1 (b) for the Helmholtz equation. We note that Equation (6.4) is obtained by the substitution (6.2) from the Euler equations for the non-viscous flow [Euler 1755]" Ou Ux

~=-+~+

Ou Uy

10p pO~

=o

'

~

Ov ~

+~

Ov

+

10p pay =0 '

Ou Ov -0. ~ +

Equation (6.5) is obtained by the substitution (6.2) from the Prandtl equations for boundary layer [Prandtl 1904]" Ou Ou 10p ~+~+---=. pox

02u Oy2'

Ou Ov --+ =o, Ox -~y

Op -- = 0 . Oy

Equation (6.7) is also obtained from such equations by substitution of (y,x, v, u) instead of (x,y, u, v). Equation (6.6) is obtained by substitution (6.2) from the Stokes equations for the creeping flow [Stokes 1851; Schlichting 1965, Chapter 6]: lop 02u Ou o o~ = ~ -5-~ + ~

'

lop = u + p oy \-5-~ -5~y~ '

+

~

-0

N

Therefore for the sake of brevity, we are going to call Equations (6.4), (6.5) and (6.6) as Euler, Prandtl and Stokes equations respectively.

Chapter 6. Systems of arbitrary equations

304

2. T h e first a p p r o x i m a t i o n s of a solution in infinity. Let the plane semi-infinite plate be placed in the half-line {x, y: x >_ 0, y = 0}. We consider the stationary flow of the viscous incompressible fluid around the plate in the positive direction of the x-axis. Such a flow is described by the Navier-Stokes system of equations (6.1) with the boundary conditions u-uoo, u-0,

v-0

v-0

when x - - > - o o ;

when y - 0 ,

x_>0.

That flow is also described by the Helmholtz equation (6.3) with two boundary conditions Or - O, Or - uoo when x ~ -oo; (6.8) Or

Or

when x_>0, y - 0 .

(6.9)

Let us make the substitution

r

y-

Omitting tildes at new variables, we obtain Equation (6.3) with ~ - 1, and the boundary conditions (6.8), (6.9) with u o o - 1. The boundary condition (6.8) takes the form r - y when (x, y) - (-oo, y).

(6.10)

We consider Equation (6.3) and the condition (6.10) as the system of two equations: the differential and the incomplete algebraic ones. We are going to study this system by methods of Sections 3 and 9 of Chapter 1 (see the previous Section). To the equation r -- y there corresponds the support $1 consisting of two points: E3 = (0, 0, 1) and E2 = (0, 1, 0). Their convex hull is the segment - [E3, E2]. Its normal plane is U~11) de_f {p: p2 - p3} C ]I~3.. It intersect the plane P3 = 1 by the straight line p2 = 1 (see Fig. 6.8, a). We consider at first the problem for x < 0, where the condition (6.9) is absent. In Fig. 6.8, b to the boundary condition (6.8) there corresponds the point at infinity ( p l , p 2 ) - (+oo, 1)lying in the intersection U~~ U~11). To the vertex Q7 there corresponds the truncation 0 = 0 4 r 4 of Equation (6.3). The function r = y satisfies the equation. Consequently, the function is suitable as the first approximation in the whole intersection U~~ N U~11) , i.e. for the whole half-line pl > 3/2, p2 = 1 of the plane P3 -- 1 in Fig. 6.8. The point (pl,p2) - (3/2, 1)lies in the normal cone U~1). To the edge there corresponds the truncated equation (6.5), which is satisfied by the function r - y. The interval pl E (1,3/2), p2 - 1 lies in the normal cone U~~ To the vertex Q2 = Q4 there corresponds the truncated equation (6.5) with zero instead of the right hand side; it is satisfied by the function r = y. The

6. The viscous fluid flow around a plate

305

point (Pl, P2, P3) - (1, 1, 1) E U~ 1). To the edge there corresponds the truncated equation (6.4), which is satisfied by the function r - y. Moving further left along the straight line p2 - 1, we pass the points belonging to normal cones of the vertex Q1 - Q3, of the edge r (1), and of the vertex Qs. The function r - y satisfies the corresponding truncated equations. To the movement along the straight line p2 - 1 from p~ - +c~ to pl - - c ~ there corresponds the shift along Ix I + [y[ = c~ from x = - c ~ to x - 0 in the plane (x, y). Our analysis shows that here the function r - y is the first approximation to the solution r y) everywhere. Now we consider the problem when x _> 0 moving from x - 0 to x - + ~ . In the plane P3 - 1 to this movement there corresponds the return along the straight line p2 - 1 towards the increasing p~. When x > 0 there is the boundary condition (6.9). For each P~ - (Pl, 1) we will seek the solution to the corresponding truncated problem jR-O,

r - y when Ixl + lyl - oc,

(6.11) when x > 0 - y .

Or162

For the pseudo-homogeneous solution r = g p , of the problem (6.11), the support of the function t}P' (x, y) lies in the straight line passing through the point (q~,q2) - (0, 1) and orthogonal to the vector P' - (pl, 1), i.e. in the straight line Pl ql § 1. It intersects the axis ql at the point ql 1/pl If the function ~p, (x, y) is a self-similar one, then it has the form r

--

xah(~),

~ def -- y/x

~,

a

def --

1/pl.

def

(6.12)

The boundary conditions in (6.11) for it are h - ~ + o(1) when ~ --+ co, h-dh/d~-O

when ~ - 0 .

(6.13)

L e m m a 6.1 For pl < 2, the problem (6.11) ]or the corresponding truncated equations has no self-similar solution (6.12), (6.13). P r o o f . Increasing pl from - c o to 2, we consider the corresponding truncated problems (6.11). When pl < 0, the point (pl, 1, 1) E U~~ The corresponding truncation of Equation (6.3) is 0 4 r 4 = O. All its solutions are polynomials of the third power in x. If they are self-similar, i.e. have the form (6.12), then r -- boy + blxy 1-pl + b2x2y 1-2pl § b3x3y 1-3pl .

Chapter 6. Systems of arbitrary equations

306

Since - p l > 0, from the first condition (6.13) it follows that bl = b2 - b3 - 0, and from the second it follows that b0 = 0. Consequently, for pl < 0 the problem (6.11)-(6.13) has no solution. When pl - 0, the point (0, 1, 1) E U (1). The corresponding truncated equation is (6.7), and the solution (6.12) has the form r = y h ( x ) . If this function satisfies the first boundary condition in (6.11), then h ( x ) - O. Again there is no solution. For the further reasoning we compute partial derivatives of the function (6.12) with respect to x and y through the derivatives of the function h with respect to ~, which we denote by prime. We obtain c9r

- h',

c92r

2 - x - ' ~ h '',

04d2/Oy 4 - x - 3 a h (4),

c9r

03r

3 - x - 2 a h ''',

- axa-l(h-

~h'),

02r

2 - a x ~ - 2 [ ( a - 1)(h - ~h') + a~2h"],

c93r

3 - c~xa-3[(a - 1)(a - 2)(h - ~h') - 3a~2h ' ' -

Or

- -ax-'~-~[h

O3r

- x-2[(a 2 + a)~h"' + a2~2h'"].

(6.14) c~2~2h"'],

'' + ~h'"],

When pl E (0, 1) the point (pl, 1, 1) E U~~ The corresponding truncated equation is (6.7) with zero instead of the right hand side. After the substitution there expressions (6.14), cancelling ax a-3 and collecting similar terms, we obtain the equation (a- 1)(a- 2)hh'-(o~-

1 ) ( a - 2)~h': -(c~ 2 + o L ) ~ h h " + + (a 2 - 2 a ) ~ 2 h ' h '' - a 2 ~ 2 h h ''' - O.

(6.15)

Let h-

b~ k + . . .

(6.16)

when ~--+0.

Substituting this expression in Equation (6.14), isolating terms of the smaller power 2 k - 1 in ~, cancelling ~2k-1, and collecting similar terms, we obtain the equation - 2 b 2 k ( k - 1)(ka - a + 1) - 0. It has a solution b ~ 0 only when k - 0, k - 1, and k - ( a - 1 ) / a - 1 - p l . Since pl E (0, 1), for all these solutions k <_ 1; and by the second boundary condition (6.13), it must be k > 1. Hence Equation (6.15) has no such solution of the form (6.16) with a finite k > 1, which satisfies conditions (6.13). Let us check as yet whether Equation (6.15) has a solution (6.16) with k - oc. For that purpose we make the logarithmic transformation g - log h. Then h' - hg',

h" - h(g

+ g"),

h'" - h(g

+ 3g'g" + g'").

6. The viscous fluid flow around a plate

307

Substituting this expression in Equation (6.15) and cancelling h 2, we obtain the equation (a - 1)(a - 2)g' - (a - 1)(a - 2)~g '2 - (a 2 + a)~(g '2 + g") + +

-

+ g") -

+ g ' g " + g'") -

0.

In the plane (ord ~,ord g), its support consists of three points: ( - 1 , 1 ) , ( - 1 , 2), ( - 1 , 3). When ~ -~ 0, we are interested in solutions g = b~ - k + . . . to the equation, i.e. pz _< 0, p2 _ 0. For Pl _< 0 < P2, the boundary subset of the support consists of one point ( - 1 , 3). To it there corresponds the truncated equation (a 2 - 2a)~2g '3 - a 2 ~ 2 g '3 - 0, i.e. - 2 a ~ 2 g '3 - 0. It has no non-trivial solutions. When pl - 1 the point (1, 1, 1) E U~ 1). The corresponding truncated equation is (6.4). Substituting in it expressions (6.14) with a = 1, multiplying by x 2 and collecting similar terms, we obtain the equation ~2 h' h" + h' h" + 2~hh" + ~2 hh'" + hh'" - O.

It has the first integral def

(~2 + 1)hh" - c - const~ Since when ~ --+ 0, its left hand side tends to zero by virtue of the boundary condition (6.13), then c = 0, i.e. hh" - O. Since h ~ 0, we obtain the equation h" - 0. Its solution h ~ 0 is a linear function of ~, and it does not satisfy the second condition (6.13). When pz E (1, 2) the point (pz, 1, 1) E U~~ The corresponding truncation of Equation (6.3) is Equation (6.5) with zero instead of the right hand side. Substituting in it expressions (6.14), cancelling a and collecting similar terms, we obtain the equation hh"' + h'h" - 0. It has the first integral h h " def

c - const In the same way as in the case pl - 1, it follows from here the absence of the solution (6.12) with the property (6.13). [:] R e m a r k 6.1. In the situation of Lemma 6.1, one can prove not only the absence of self-similar solutions but the absence of the pseudo-homogeneous as well. But such a proof is significantly more cumbersome. L e m m a 6.2 W h e n pz - 2, the problem (6.11) for the corresponding truncated equation has the sel/-similar solution (6.12), (6.13). P r o o f . When Pz - 2 the point (2, 1, 1) E U~ 1). The corresponding truncation of Equation (6.3) is Equation (6.5). Substituting in it expressions (6.14) with a - 1/2, we obtain the equation hh'" + h'h" + 2h (4) - 0. It has the first def

integral h h " + 2h'" - c - const. According to (6.13), when ~ ~ c~ its left

Chapter 6. Systems of arbitrary equations

308

hand side tends to zero. Consequently, c - 0, and the equation takes the form hh" § 2h'" = 0. It is the Blasius [1908] equation (see also Examples 3.2 and 7.2 of Chapter 3, and Example 7.1 of Chapter 7). The boundary conditions (6.13) determine its unique solution h = b(~), which is called the Blasius solution. It was studied analytically and numerically [Blasius 1908; Kochin et al. 1948; Schlichting 1965]. See also Theorem 7.2 of Chapter 7. In particular, it turned out that b"(0) de_f i~ and b ( ~ ) - ~ +/~ + o(~) when ~--4 oc,

(6.17)

where & ~ 0.33206, ~ ~ -1.72077. In the plane (q~,q2) = (ord x, ord y), the support of the function x/'~b(yvf~) is placed in the straight line 2q~ -t- q2 = 1 denoted as L1 in Fig. 6.9. The Blasius solution describes the asymptotics of the stream function in the boundary layer

lyv l < o ).D L1

(6.18) q2

q2 2

-4

-3

\1 \

-2

-1

01

ql

~(1)

1

ql

'-1

Q2 Q

Figure 6.9 (left): Straight lines L1 and L2 in the plane (ql, q2). Figure 6.10 (right)" The projection of the pentahedron r in the plane (ql, qz). From Lemmas 6.1 and 6.2, it follows T h e o r e m 6.1 If x 2 +y2 _+ oc, then the first approximations to the solution r to Equation (6.3) with boundary conditions (6.8) are r = x/~b ( y / v f~) inside the boundary layer (6.18) and ~b2 - y outside it. 3. T h e second a p p r o x i m a t i o n to t h e solution in infinity. To study the second approximation to the stream function r outside the boundary layer, we put r = y + ~o in (6.3), and we obtain the equation for ~o

6. The viscous fluid flow around a plate

309

where A is the Laplace operator, and boundary conditions O~o/Ox - O~o/Oy - 0 when x ~ -oo

or

[y[ ~ c~.

(6.20)

The support of Equation (6.19) is S - SU {Q8, Qg}, where Q8 - (-3, 0,1) and Q9 - ( - 1 , - 2 , 1 ) (Fig. 6.10). The convex hull F of the support S is the pentahedron; it has the edge ~1) D {Qs, Qg} with I~l~1) - {P:0 < pl - p2 > p3}. According to equalities (6.17), to the Blasius asymptotics r - v f x b ( y / v f - x ) w h e n y/vf-x ~ oo there corresponds ~ ~ j3v~, i.e. the point Q~ - (1/2, 0) in the plane q3 - 0. There exists the unique plane MR passing through points E3 and Q~ with P e ~1). For it P - (1, 1, 1/2) and Lp def {Q: ql + q2 - 1/2} def L2 (Fig. 6.10). The plane Hp, which is the supporting one to F, intersect F along the edge ~1). To the edge there corresponds the truncated equation 03~o/Ox 3 + 03~o/OxOy 2 - O.

We seek its self-similar solution as a function ~o- ~ def ~ h ( y / x ) with the support in the straight line L2. The boundary conditions h(0) - / ~ and (6.20) are satisfied only by the odd solution - cl sgn y V/x/x 2 + y2 + x

with

cz - / ~ / v ~ .

Thus, without any physical assumption we have proved T h e o r e m 6.2 If x 2 + y2 _~ oc, then the solution r to Equation (6.3) with boundary conditions (6.8) has the asymptotics ~bl - v/-xb(y/v/-x) when x -~ +oo, ly/v/-xl < c~ and ~b2 - y

+ / ~ s i g n y v / ( ~ / x 2 + y2 + x ) / 2 in other cases.

We note that to the lower face (with q3 - 1) of the pentagon I" there corresponds the Oseen linear approximation [Oseen 1910], frequently used in the flow problems. In [Imai 1957] there were obtained the second approximation to the solution in the boundary layer and the third approximation outside it. They contain terms with log x. In the book [Goldstein 1960, Chapter 8], the theory of further approximations was constructed, which are pseudo-homogeneous ones, and which include terms with powers of log x. All this is in agreement with the theory of Sections 1-3, 5 of this Chapter. 4. T h e first a p p r o x i m a t i o n to the s o l u t i o n in t h e origin. We consider the problem of the flow around the plate {x > 0, y - 0} (i.e. Problem (6.3) with u - 1, (6.9), (6.10)) near the leading edge of the plate, i.e. in the neighborhood of the point x - y 0. The boundary conditions (6.9), (6.10) here take the form r y) - - r (6.21)

Chapter 6. Systemsof arbitrary equations

310 Or

when x > 0 ,

y-0,

(6.22)

Or162

when x < 0 ,

y-0.

(6.23)

In particular, from (6.21) it follows that r - 0. Since here r is small, then when x, y are small the corresponding first approximation to the Helmholtz equation (6.3) is the Stokes equation (6.6), which is a biharmonic one. According to Theorem 5.1, we seek the first approximation to the stream function r as a self-similar solution r - xkh(y/x), k >_ 1 to the biharmonic equation (6.6). In the polar coordinates r, 9: x - r cos g, y - r sin 9 such a solution is written in the form r

k>_l.

(6.24)

Accordingly, the boundary conditions (6.21), (6.22) take the form g(lr - 9) - -g(Tr -t- 9), 0 < 9 _ lr,

(6.25)

g' (0) - g' (27r) - 0,

(6.26)

where the prime means the derivative with respect to 9. In general, the boundary condition (6.23) is not necessarily satisfied for the first approximation. But if it is satisfied for the function (6.24), then it has the form 9'(lr) ~ 0. (6.27) T h e o r e m 6.3 Equation (6.6) has Solution (6.24) satisfying the boundary conditions (6.25), (6.26) only with integer k > 3 and half-integer k > 1. This solution has g(9) - b[ksin(k - 2 ) 9 - ( k - 2)sinkg], g(O)-a[cos(k-2)O-coskO],

k e Z,

k-l+l/2,

k >_ 3,

leZ,

l>__l,

(6.28) (6.29)

where a, b de_.fconst r 0.

P r o o f . In the polar coordinates the biharmonic equation (6.6) has the form

04r

1 04r 2 03r 2 04r I .0r 4. .r 2 0r . 2002 + - ~ - 0 ~ ~ r Or 3

2 03r r 3 OrOO2

{

4 02r r 4 002

1 02r 1 0r I -0. r 2 Or 2 r 3 Or

Substituting in it the expression (6.24), cancelling r k-4 and collecting similar terms, we obtain the ordinary differential equation for g(9) g(4) + 2(k 2 _ 2k + 2)g" + k 2 ( k - 2)2g - 0.

(6.30)

6. The viscous fluid flow around a plate If we seek a solution in t h e form g characteristic equation

311 exp(A~), t h e n for A we o b t a i n t h e

A4 + 2(k 2 - 2k + 2)A 2 + k 2 ( k - 2) 2 = 0. Its r o o t s are

A~,2 - •

2),

A3,4 - •

For k >_ 1, a m o n g t h e s e roots t h e r e are multiple ones only w h e n k k - 2. W e consider t h e s e cases. k - 1. In this case t h e general solution to E q u a t i o n (6.30) is g - al cos 0 + bl sin 0 + a20 cos 0 + b20 sin 0,

(6.31) 1 and

(6.32)

w h e r e ai, bi - c o n s t . F r o m t h e condition (6.25) it follows t h a t g(O) - g(~') - g(21r) - O .

(6.33)

For t h e f u n c t i o n (6.32) t h a t m e a n s t h a t al-0,

al+a27r-0,

a1+a227r=0,

i.e. al - a2 = 0. T h e c o n d i t i o n (6.26) m e a n s t h a t bl-0,

bl+b22~r-0,

i.e. bl - b2 - 0. C o n s e q u e n t l y , in this case t h e r e is no n o n - z e r o solution to E q u a t i o n (6.30) satisfying conditions (6.25), (6.26). k - 1. In this case t h e general solution to E q u a t i o n (6.30) is g - al cos 20 + bl sin 20 + a2 + b20.

(6.34)

C o n d i t i o n s (6.33) m e a n t h a t az + a2 - 0,

al + a2 + b27r - 0,

az -t- a2 + b227r - 0,

i.e. al + a 2 - b2 - 0. C o n d i t i o n s (6.26) m e a n t h a t 2bl + b 2 = = 0. C o n s e q u e n t l y , az + a2 - bl - b2 - 0, a n d t h e f u n c t i o n (6.34) has t h e f o r m al (cos 2~ - 1) = - 2a 1 sin2 ~. T h i s f u n c t i o n does is also no r e q u i r e d If k > 1 a n d k g e n e r a l solution to

n o t satisfy the condition (6.25). T h u s , in this case t h e r e solution. ~t 2, t h e n all eigenvalues (6.31) are different. Hence t h e E q u a t i o n (6.30) is

g - az cos(k - 2)~ + bl sin(k - 2)~ + a2 cos kO + b2 sin kO.

(6.35)

Chapter 6. Systemsof arbitrary equations

312 Conditions (6.33) mean that al + a2 -- 0,

(al + a2) cos k~r + (bl + b2) sin klr - 0, (al + a2) cos 2k~r + (bl + b2) sin 2k~r = 0, i.e.

(6.36)

(bl + b2) sin k r - 0.

al ~- a2 -- 0, Conditions (6.26) mean that

bl (k - 2) + b2k - O,

(6.37)

and - [ a l (k - 2) + a2k] sin 2 k r + [bl (k - 2) + b2k] cos 2 k r - 0. Taking into account (6.36) and (6.37), from the last equality we obtain 2al sin 2klr - 0.

(6.38)

If al = 0, then a2 -- 0 by virtue of (6.36), besides bi + b2 ~ 0 by virtue of (6.37), i.e. sin k r = 0 by virtue of (6.36). Consequently, k is integer and (6.37) is satisfied. Such solution (6.35)is (6.28). If al ~ 0, then a2 = - a l by virtue of (6.36), besides the number 2k is integer according to (6.38). If k is not integer, then sin k r ~ 0 and bl + b2 - 0 according to (6.36). Together with (6.37) that means that bl = b2 = 0. Such solution (6.35)is (6.29). If al ~ 0 and k is integer, then Solution (6.35) with properties (6.36)(6.38) is g - a[cos(k - 2)0

cos k0] + b[ksin(k - 2 ) 8 - (k - 2)sin k0].

It satisfies the condition (6.25) only when a - 0, i.e. it is again Solution (6.28). D L e m m a 6.3 I/the solution r to Equation (6.3) has the first approximation (6.24), then it expands in the asymptotic series oo

r - Z

gm (r, O)r kin,

(6.39)

rn--1

where functions gin(r, O) have the zero order in r and gl

def - g(O).

P r o o f . The support of the series (6.39) in r consists of points km with m > 1. If we substitute the series in Equation (6.3), then the left hand side of the

6. The viscous fluid flow around a

plate

313

equation will contain terms of powers k m l - 1 + k i n 2 - 3 - k ( m l + m2) - 4, where ml and m2 are natural numbers, i.e. the number ml + m2 > 2 and integer. The right hand side of the equation (6.3) will contain terms of powers k i n - 4 , where m is natural. In this way, the supports of both parts of Equation (6.3) lie in the set of numbers k m - 4, where m is natural. Equating terms with the same power r, we obtain the infinite system of equations for functions gin, which is solved one-by-one over m. Here the powers of r do not leave the set. D The more concrete definition of the solution in comparison with Theorem 6.3 can be obtained by studying the second and further approximations to Solution (6.39) satisfying the boundary conditions. Thus, for example, the condition (6.27) is not satisfied for functions (6.28), and it is satisfied for functions (6.29). The function (6.29) in the open interval E (0, r) turns to zero I - 1 times; but the stream function must have even number of such zeros. The smallest value k permitted by Theorem 6.3 equals 3/2. For this case in [Carrier and Lin 1948] the first approximation to Solution (6.29) was suggested, and the second one, which was proved to be wrong. It was corrected in IVan-Dyke 1964] (see there Formula (3.24)). In [Carrier and Lin 1948] it was supposed that the asymptotic expansion near the leading edge (6.39) is directly matched with the asymptotic expansion in the boundary layer. In fact, from the second approximations near the leading edge and in the boundary layer it is clear that these asymptotic expansions near the plate cannot be matched directly. In [Van de Vooren and Dijkstra 1970, McLachlan 1991] the results of computations of the flow around the finite and semi-infinite plate respectively are cited. The preliminary results of this Chapter were published in [Bruno 1994, 1995a, b, c, 1996b, c, 1997a, b, 1998c, 1999] for Sections 1-4 and in [Bruno and Vasiliev 1995, 1996, 1997, 1998; Bruno 1999] for Sections 5 and 6. More complicated problems with the boundary layer were considered with methods of Power Geometry by Vasiliev [1998, 1999].