802 Kleine Mitteil ungen

a dopc dP/dw2 : 0, for the first timc in tho positive P-half of the P-(02 plane. I n the first branch this zero slope is reached from positive values (dP/dW2 > 0) and in the second branch from negative values (dP/dw2 < 0). [See Figure 1.1

The critical divergence occurs when the eigencurve meets the positive P-axis for the first time. The second branch can not meet the positive P-axis prior to the first branch unless they meet in themselves in the positive P-plane already, which means critical flutter prior t o critical diver- gcnce, which is not the case the theorem is concerned with.

a )

P1

b)

f C

Fig. 2

divergence flc/fter anddivergence

1‘1g 3 Big. 4

So the first branch has to meet the positive P-axis and as it has to meet from the positive &-half of the P-w2 plane, the slope dP/do2 0 a t critical divergence. I n the case of the equality sign, i t has t o be reached from dP/dw2 < 0. (Figure 2.) This is in contradiction to the slope requirements for critical flutter as obtained in the previous paragraph. Hence the critical flutter load can not be equal t o the critical divergence load if the eigencurve is smooth.

Let us consider the case when the eigencurve is ”two straight lines“ (degenerated case). They can meet (coalesce) only once and hence both the flutter loads coincide, i. e., I’E 1 = Prx2. (Figure 3). To have critical divengence along with critical fh t te r , the two straight lines have t o meet at the positive P-axis (Figure 4). In such a case the eigencurve can meet the P-axis only once and so both the divergence loads coincide. i. e., P D ~ = PDZ. Hence it is proven that PF = P D if, and only if,

Ppi = Pp.2 and Pu i = P D ~ . (7) Though the above proof was restricted t o positive loading

conditions only, it can similarly be proven for negative loading conditions also.

Con clii s i o n The theorem is valid for continuous systems reduced t o two- degree-of-freedom systems by modal approximations like GALERHIN’S method [3] or by any other approximations.

A c k n o w l e d g e m e n t s

The author is grateful t o Professor H. LEIPIIOLZ for his guidance during the course of this investigation. This research was supported by t,he National Researvh Council of Canada through Grant Xo. A 7297.

R e f e r e n c e s

1 HEHRXANN, G . , Stability of 13quilihrium of Elastic Systems Subjertrd to Nonconservative Forces, Applied M.~zchanics %view, 20, pp. IIKi - 108 I1 467) l_””.,.

2 ZIEQLER, H., Principlcs of Striictural Stability, niaisdeil l’til,lishing

3 LRIPHOLZ, J I . , Stability Theory, Awdcnri(’ I’rrss, R‘cw J-orli 1970, pp. Company, Massachusettes 1968, p. 109.

181 - “ 5 .

Eingereicht am 1 4 . 1 . 1973

Anschrijt : Dr. C. SUNDARARAJAN, Postdoctoral Fellow, Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada

ZAMM 53, 802 SO5 (1973)

A. QADIR / M. A. RASHID

Integration of Functions Satisfying a Second-Order Differential Equation

Starting from a second-order differentrial equation P”(z) + a(x) F’(z) + b(z) F ( z ) = 0 ( 1 ) we can express the indefinite integral of P(z) as1)

where the functions f(z) and g(z) do not explicitly depend upon the given function F ( z ) . Indeed, differentiating Eq. (2) and comparing the result with Eq. ( 1 ) we obtain

f (4 = 44 g(4 - g’(4

f’(4 = 1 + b ( 4 g(4 *

g”(z) - a(z) g’(z) + (b(z) - a’(z)) g(z) -1- 1 = 0 .

(3) (4)

Eliminating f(z) from Eqs. (3) and ( 4 ) we arrive a t a

(5 )

second-order inhomogeneous differential equation

We note that the coefficients in this differential equation do not involve the function P(z) and are obtainable directly from the coefficients appearing in the differential equation ( 1 ) for P ( Z ) . ~ )

After solving for g(z), we can obtain f (z ) from Eq. (3).3) Next we show that in Eq. (2) we require only a particular

integral of the differential equation ( 5 ) of g(z). If we write

g(4 = g P ( 4 + m(4 7 f (4 = f P ( 4 + ff&) 7

where gp(z) is a particular integral and gH(z) is a solution of the homogeneous equation obtained from ( 5 ) while f P ( x ) and!&) are

we get d

fa(+= a ( 4 B P ( 4 - s;c4 f fH(4 44 g&- g&)

& V H ( 4 P(4 + g r I ( 4 F’(z)l = 0

on using ( 1 ) and the homogeneous equation satisfied by gH(z). Henceforth gp(z) and fp(z) will be written as g(x) and f(z).

We do not attempt to solve Eq. ( 5 ) in general. However, when we consider the case of special functions, in most of these cases an explicit particular solution for g(z) can bc

l) R. A. WALDRON (1968) has done this problem for the BESSET, fiinctions. We are closely following his notation.

s, Eqs. (1) and ( 5 ) in the form given in this paper can be written as L ( s ) F ( x ) = 0 and L+(z)g(a) + 1 = 0 where L+ is the adjoint of the differential operator L(z).

”) Since the functions f(z) and g(a) do not explicitly depend upon F(z), the same ffx) and gfz) are to be used in Eq. (2) for every solution of tire differential Eq. (1).

Kleine Mitteilungen 803

obtained by using FROBENIUS' method. We discuss t,he special situation where Eq. (5) , after possibly transforming the independent variable, takes the form

(6) where u's and 11's are constants and k is some integer. I n this case, writing g(y) = Z an yn, we obtain

~ ~ ~ ( 7 1 , n (72, - 1) + vI n + v,) + an-i (u, (n - 1) (n - 2 ) + (7) + zl, (n - 1 ) + uo)l yn-2 + yk = 0 ,

i. e . arc I 2(% ( k + 1 ) ( k + 2 ) + w1 ( k + 2 ) + no) + + U k + 1 (UZ k ( k + 1) + u1 (k+ 1 ) + u,) t- 1 = 0 (8 a)

for n < k + 2 . (8 c ) Thus, in general, there are two solutions:

m --m

where

m i - 2 = - l / { % (k + 1 ) (k + 2 ) + Wl (k + 2 ) + vo}, b k t l = - 1 / { U , k ( k + 1 ) + U l ( k + 1) + UOJ

and other an's and bn's are obtained from the recursion relations (8b) and (8c).

The series for g,(y) (g2(y)), if non-terminating, converges for IYI ,< Iv2/u21 (Ivl > Iv2/u21). However, when IyI = lvz/u21, both series converge absolutely (diverge) if

u v 1 4 - 2 + ' > 0 ( 5 0 ) .

u2 v2

E x a m p l e s 1. Con f 1 u e n t h y p e r g e o m e t r i c f II n c t i o n s

Here c - x

a(2) = -~ , b(x) = - 5 , X X

therefore g(x) satisfies the differential equation

The two particular solutions are given by m -03

where

and n - 1 - a

(n - 1 ) (n - c) a n = - . an-i (n > 2 )

and (n - 1) (n - e)

(n - 1 - a ) 6%-1 = - -b,, (n < 2 ) .

The series for the second solution terminates a t n = 1 and so should be more useful. This gives

(9)

For the special case of the function IFl(a, c; z), we can even compute the constant obtained when thg lower limit is taken to be zero. Thus I

2 - c + l ,F1(a, c ; t ) dt = $,(a, c ; 2) d' 1 - a

The above result can be written for the LAGUERRE funct,ions in the form z

2. D e g e n e r a t e h y p e r g e o m e t r i c f u n c t i o ns Here

I - Y2

b(x) = >- -t 22- ; 1 4

therefore g(x) satisfies the differential equation

a(x) = 1 ,

The two particular solutions for g(x) are given by --m

where a2 = - 1 , b l = r - - A 1

(i- - Y) (; + P )

and

1 1

n - 1 - 1 = ~ ~~ ~ - _ _ ' b n . ( n < 2 )

When ,u f 2 k + l -~ , where k is a positive integer, only the 2 -

series for gl(x) is useful, the one for g2(x) being divergent for every x. This gives

r( P + +) I'(+ - P ) g (2) = - x r ( 2 -A)

x". (13)

In particular, for the solution qqP(x) where

r p n , L ( x ) = x ~ / ~ + P e - ~ , ~ l - a , 2 p + i : z )

(0 -+)

804 Kleine Mitteilungen

a e obtnin 2

P j / J ( t ) fi(>) y;/t(T) 4 gl(r) P i p ( z ) (14)

x31ierefl(T) and g,(r) are given in Eqs. (12) and (13) above.

Hou cvw, 15 hen / t = , where k IS a positive integer,

the second serics only can be used. which in this case ter- minates a t n = - k , giving

2 k + l

(15b) ivhilc

1 1 T ( n --I) T(-n -1 k ) (-k(l + k ) +I(n + 1)) zn, -~ T(n + z 4 k ) x 2 ( - 1’11

I1 0

(16b)

Above, Eqs. (15a) and (16a) have been given t o indicate their coiinedion with Eqs. (12) and (13). The expressions (15b) and (16b) have. 111 themselves, the built-in termination a t n - -- I. and n = - k - 1 rcspectively.

3 A s s o c i a t r d L E O E X I I R E f u n c t i o n s Here

thrrcfore g ( x ) satisfies the differential equation

+ 1 = 0 . Substituting 1 - x2 = y we arrive a t

+ I - 0 . Thc t\+o particular solutions for g(z) are given by

03 -cc gl(r) = Z , U ~ (1 - ?’)n, g2(x) == 2 6 , (1 - z 2 ) n ,

ri - _ n 1

u hcrr

and

Here the two series have a radius of convergence e q d t o 1 and the series are also absolutely convergent a t 1 - xz= & 1 (i.e. z2 = 0 or 2 ) . This happens for all values of ,u and Y except when the coefficient in any one of the recursion relations (17) and (18) explodes. I n such a case, the other

series terminates. Pinallyf(z) ran he computed from Eq. (3) to give

--a)

f2(z) = - 2 x 2 (n - 1 ) b, (1 T?)” ’ . (19) 91-1

Since a t z = f 1, both solutions for g ( r ) and f(z) vanish, for any solution P;(z) regular, say a t z = I , w e can w i t c

S P,Y,(f) dt 7

f(z) J’;M + s ( 4 l’;Sr), I

where the index 1 or 2 for f (z ) and ~ ( r ) has heen omitted, since any solution can be used.

4. HERMITE f u n c t i o i i s This example is mentioned to shou tlint tliv integcr h !n Eq. (6) is not necessarily zero. Herr a ( r ) = - 2 z , b ( i ) = 2 m , i.e., g(x) satisfies the differential equation f(z) + 2 x g‘(z) + 2 (m + 1) g(z) j 1 - 0 .

Writing y = z2, this takes the form

Thus the two particular solutions for g ( r ) arc W --m

g&) = zn , x” , &(X) z b 2 & r z ’ z , 91 :1 71 . n

where 1 1 al---, b - - - 2 O-Z(rn+ 1,

and

The second solution t.ermiiiates at n = 0, giving

For the standard HERMITE polymmitls one can imme-l ihlg compute the coefficient resulting from taking the lower limit as zero. This finally gives

The general solution to the HERMITR equation which is regular a t x = 0 is of the form*)

f l y&) = a. hP)(z) + a, h‘lp’(z) ,

4, HERMITE functions h?) (i = 0, 1 ) a l e given hy

where the normalization has been chosen such that the coeffirient of xi in hip’ is 1. The authors have noticed that these functions are nut mcntiortcd in the literature, though they are perfectly well defined. Nolicr: that alicn p is an integer, one of the h p ) hecomes a standard HEKXITG polynomial up to

where i = 0,l for p = even, odd, rcspectively.

Klei n e Mittci 1 u ngen 806

ivhtre J L ~ ( S ) has a constant term, whereas h,(z) has none. If we ncrmalize the coefficient of x in h,(z) to 1, we will obtain T

.I. .~, ,~,c..- "1 ~~~~

4 - 1 1)

In lfy,'. y siguiI'ies t'hltt we are taking lt geiicral solution. Note t,liat, Y t L ahove need riot be a integer,

A c k n o w 1 e d g m e n t s The authors would like to thank Prof. ABDUS SALAM, the lnteriiatiorial Atomic Energy Agency and UNESCO for hospitality a t the International Centre for Theoretical Physics, Trieste.

li c f'er e n c c s \ \ALIUWX; It . A., J . Illst. hlatlls. :i1Jplivs. 4, 315 (1968).

Eingereicht am 31. 1 . 1973

An<schrVt: A. QAUIR and M. A. RASHID, University a t Islamabad, Institute of Mathematics, Islamabad Post Box KO. 1090, Pakistan

ZAMM 58, 805 (1973)

J. Focrtr:

Ein Ziisammenhang rwisehen Konditionszahlen

%u ciner regularen Matrix A : IZn -+ Rn bzw. zu deren Gauss- Transformierten G = AT A betrachten wir die Spektralkon- dition

k = I r A S ; (1)

K I D(V?L/N)n, ( 2 )

und die in [l] arigegebene Koriditionszahl (vgl. auch [a] urid 131)

wobei

0 < d, 5 d2 5 . . . 5 A,, ,V = llAlj$uk, -= Spur G = A l + A, + . . . + A n , I), = Uet2 A = Det G = A1 A, . . . ,In . Diebe gestattet fdr 1, die folgende Abschhtzung (vgl. [l])

Eigenwerte von G ,

Wir wollen in dieser Note noch einen anderen Zusammerihang zwischen den Eigenwerten bzw. der Spektralkondition und der Kontlitionszahl K herleiten. Dazu bilden wir das Mittel

(4)

urid schatzen im Falle n > 2 mit der Ungleichung zwischen arithmetiscliem und geometrischem Mittel weiter a b 1 2 ( k . + k-1)

Dureh Einfiihrung der Funktion

H ( t ) = (Nz - ( n - 2) t= ) t = N2t - ( n - 2 ) t 11 2 - .. 2

, t 2 0 (6)

konnen wir in ( 5 ) ersetzen

(7)

Wir bildeii nun das Maximum von H ( t ) fur t 2 0. Oa. U ( t ) streng konkav ist, so ist fur die Maximalstelle to hinreichend

und damit ergibt sich fur den Maximaluert

Max I I ( t ) = IJ(to) = 2 (7)

Damit konnen wir in (7) weiter abschatzen und erhalten (nach (4) auch fur TL = 2 gultig)

oder nocli etwas umgefornit

Diese Abschatzung ist scharf in dem Sinne, dal3 fur d, : A, - _ . . . = I n - 1 = N Z / n das Gleichheitszeichen angenommen wird.

Zur Anwendung dieser Ungleichung betrachten wir das quadratische Minimumproblem

F ( z ) = x T G x - 2 gTx = Min!, x E E n , (12) und seine Losung durch das Gradientenverfahren. Fur dieses gilt nach KANTOROWITSCR [4] die Konvergenzabschitzung

An -A, 2 x

An + 11 E ' ( x ~ ) - Fmin 5 (F(z ' ) - F m i n ) (---)

Mit (11) erhalten wir

F ( x ~ ) - Fmin 5 (F(x ' ) - Pmin) (1 - K')' (14)

und haben damit den konvergenzerzeugenden Faktor durch die leicht numeriscli realisierbare Konditionszahl K ausge- driickt.

L i t e r a t ur

1 U'OCKE. J.. uber die Konditioii linearer Uleicllunass~,vstelne, \\ iss. X. - . l i M U ieipzig 11. 41-43 (1962).

chungssysteme, ZAMM 48, 568 (1963). 2 HEINRICH, H., Bemerkung zu einem IionditiotlamaU fur lincare Glci-

3 Z U R M ~ H L , R., Yatrizen, 4. Aufl., Berlin-Oottingen-Heidelberg 1904. 4 KANTOROIVITSCH, I,. W. und AKILOW, G . P., Yunktionalanalysis in nor-

mierten Rsumen, Berlin 1964.

Eingereicht am 5. 4. 1973

Anschrift: Prof. Dr. JOACH~M FOCKE, Sektion itlathe- matik, KMU Leipzig, 701 Leipzig, Karl-Marx-Platz, DDR