T he p ro b lem can be sta te d as finding th e accep tab le form of h (t) such th a t (2.1) h a s a n a n a ly tic a l so lu tio n

VNU. JOURNAL OF SCIENCE, Mathematics - Physics, T.XXI, Nq2, 2006

C O N D ITIO N S FOR THE APPROXIM ATED A N A L Y T IC A L SO L U TIO N O F A PARAMETRIC O SCIL LA TIO N PR O BLE M

D E S C R IB E D BY THE MATHIEU EQUATION N g u y e n D a n g B ic h

In s titu te o f B u ild in g Science a n d Technology N go D in h B ao N a m

College o f Sciences - V ietnam N a tio n a l U niversity

A b s tra c t: This paper presents the scientific detailed basis and the involving conditions for finding the approximated analytical solution of a param etric oscillation problem described by the Mathieu equation.

1. Introduction

T he m eth o d for fin d in g an ap p ro x im ated a n a ly tic a l so lu tio n of a p a ra m e tric o scillation pro b lem d escrib ed by th e M ath ieu e q u a tio n h a s b een p re s e n te d in [1].

H ow ever, th e sc ien tific b a sis a n d o th e r involving co n d itio n s have n o t b een d etailly defined ex cep t for th e n e ce ssa ry conditions. T his p a p e r in v e s tig a te s in d e ta ils th e scientific b a sis a n d th e rela tiv e re la tio n sh ip am ong p a ra m e te rs in th e m ethod for fin d in g th e a p p ro x im a te d solution p resen te d in [1],

2. The scientific basis for finding the approximated analytical solution of a parametric oscillation problem

C o n sid er a 2nd o rd e r d ifferen tial M ath ieu eq u atio n

in which: h(t) - a periodic function, ủ s ta n d s for th e 2nd d e riv a tiv e of h(t) w ith resp ec t to t.

T he p ro b lem can be sta te d as finding th e accep tab le form of h (t) such th a t (2.1) h a s a n a n a ly tic a l so lu tio n . And th e n we d e sire th is a n a ly tic a l so lu tio n to be an a p p ro x im a te d so lu tio n of th e following eq u atio n

(2.1)

(2.2)

w h a t re la te d c o n d itio n s m u st be found.

1.1. F o r m o f f u n c t i o n h (t)

C o n sid er th e s u p p le m e n ta ry eq u atio n

X = a x2, (2.3)

9

10 Nguyen D a n g Bichy Ngo Dinh Bao N a m

in which: a(t) - an y c o n tin u o u s non-zero function of t.

By re a rra n g in g p a ra m e te rs we have

From t h a t it yields:

ủ l à ax = ---

u 2 a

ủ ủ àx + ax = - + — - —

u u at

l a ) 2 a D im in ish — from (2.4), (2.5) we h av e

u

ax = a2x2 + ii d u dt

1 à 2 a .

/ 1 o '

^2 a R eplacing X d e te rm in e d by (2.3) in to (2.6) it yields

ủ + d ' l Õ'

- Í - - T

dt , 2 a ) u a ) u = 0.

C om pare (2.7), (2.1) th e form of h(t) can be considered as c0^{h ( t ) = d ' l ả ' ' 1 Ớ'}

d t ^ 2 CL , K2 a;

The solu tio n of (2.3) can be ex p ressed as ax = _I a

ịa d t + Cx

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

in which:

c x

S u b s titu tin g ax c a lc u la te d from (2.9) into (2.4) we have

Ù _ a l a

u te _ 2 a *

ịa d t + Cj

(2.10)

From t h a t i t yields

u a XA

t Cj + C2 ịa d t

0

(2.11)

in which: C2 - in te g ra l c o n sta n t.

It can be s ta te d t h a t if th e fu n ctio n h(t) h a s a form of (2.8), th en (1) h as an exact solu tio n in th e form of (2.11). From (2.11) a n d (2.8) it can be in ferred th a t if a(t) is a co n tin u o u s non-zero fu n ctio n , th e n u co n tin u o u sly depends on a(t).

C onditions for the A p p r o x im a te d A n a ly tic a l S olu tion of... 11 B ecause a(t) is an y co n tin u o u s function, th e n in in v e stig a tio n of (2.1) w hen h{t) is a periodic function, a(t) can be chosen in th e form of a periodic function as

a ( t ) = (A + Pcoscot)'

. c

Ị<y2 +CCẦ + aPcoscot^

(2.12)

Ả (Ú

in which: p , —— - p a ra m e te rs th a t need to be defined d u rin g th e solving procedure.

S u b s titu te (2.12) in to (2.8), (2.1 1) we have

h ^ _ 2yco2________2 y a + 3Ã CO2 + 3CCẢ + 2ỵ a 2

(/Í. + j3cosũ)tỶ A + fỉcosũ)t ũ)2 + aẪ + a/3 COS Cút ’ (2.13)

w here ỵco2 _ Ả2 ỵa _

p2 ~ ữ2 ’ p - U 2 T h u s, th e so lu tio n of (2.1) now is of th e form

ap_

co'1

u = cư2 + aẢ + a/3 cos cot A + p cos a t

(Ả + p coscot)

J * I

0 Ịứ>2 +aÁ + a/? cos cor'j

d r (2.14)

F o rm u la s (2.13) a n d (2.14) a re ex act so lu tio n s p re s e n te d in [1].

1.2. A p p r o x im a te d s o lu tio n

E q u atio n (2.2) w ith th e condition u * 0 can be re w ritte n as

— + Ũ)2 (k + pcoscot) = 0. (2.15)

S u b stitu te (2.14) into (2.15) an d denote th e left h a n d side of (2.15) by f{t) we have

/ X -1

/■(«)=^CD‘ (k + p c o sco t)- 2/ũ)2_________2aỵ + 3/1 6>2 + 3tt/l + 2 /q2 (/I + /3 COS cot)2 Ẳ + fỉcos(ot ú)2 + aẲ + a/3 coscot D enote

#(*) = Ẳ + pcoscot,

ta k in g in to acco u n t (2.13), th e f(t) fu n ctio n can be w ritte n as f { t ) = co2[ g ( t ) - h ( t ) ]

I f f { t ) = 0 V i, th e n (2.14) becom es a n ex act so lu tio n of (2.15).

If t h a t m ea n s h(t) « g(t) w ith every t, th e n (2.14) can be co n sid ered a n a p p ro x im a te d solution of (2.15). T h e e rro r of th is ap p ro x im ated so lu tio n d ep en d s on th e e rro r of th e a p p ro x im atio n of h(t) to g(t).

(2.16)

(2.17)

12 Nguyen D a n g Bich, Ngo D in h B a o N a m T herefore, th e scientific basis of the m ethod is: The solution of (2.1) continuously depends on th e function h(t), hence w hen h(t) is approxim ated by g(t) w ith every t th en th e exact solution of (2.1) becom es a n approxim ated solution of (2.15).

3. Relating conditions for parameters

3.1. C o n d itio n s in [1]

It h a s b een s ta te d in [1] t h a t for h ự ) can be ap p ro x im ate d by g(t) w ith every t, th e follow ing e q u a tio n s a n d in e q u a tio n s should be sa tisfie d

a. The e q u a tio n s

CO a p •

1 Ầ

k * p fi

A

p + p

(3.1)

- 1

A p

Ằ 2 2 1

p2

+ p a p p

\ / 2 3

co + _ + l À , a p p

CO2 Ả

(3.2)

6. The in eq u a tio n s

Ằ ^ 1 CO2 Ẳ

p a p + p

ý

8 +0)

a p 1 - 4 ^CO

2 '\

i l/?2

>1,

8 + —CO

aft 1 + 4 0)

>0,

(3.3)

(3.4)

3.2. S u p p l e m e n t a r y e q u a tio n s From (3.1), (3.2) i t y ield s

2p ( k - l ) — + k2 ^{+ p2 - k}

<J_ A _ ’ p

a P p ị k2 + p2 -& ) — + 2p k

(3.5)

(3.6)

C on dition s fo r th e A p p r o x im a te d A n a ly tic a l S olu tion of... 13

D enote

2 p ( k - l ) — + k 2 + p 2 - k

[ k 2 + p 2 - k ^ + 2 p k

(3.7)

B ased on (3.5) it can be proved th a t

2 p ( k - l ) ± + k * + p > - k

k * - p * - k x Ị ẵ 2 + p 2 - k ^ — + 2 p k b ~ p + k p

(3.8)

H ence

CO2 _ 2 p ( k 2 - p 2 ) p i

™a~ l2 J , L 7 ' (3.9)

T he co n d itio n ta k e n in to acco u n t

co2 Ả a p + p

a p k2- p 2 +k p

> 1 can be rep laced by \x\ >1 w h e n (3.6) a n d (3.8) are

co A. _ _ k2- p 2 - k Ẫ

" X ~ -2 _2

a / ? p k ~ p1 + k p (3.10)

S u b s titu te — c a lcu late d from (3.10) into (3.5) we h av e

[k2 - p2 + k j z2 - 2PX - [k2 - p2 - = 0 . (3.11) E q u a tio n s (3.5), (3.11) a re th e su p p le m e n ta ry e q u a tio n s for fin d in g the c o n d itio n s sa tis fy in g th e in e q u a tio n (3.3).

3.3. T h e c o n d itio n _>1

T he so lu tio n of (3.5) can be w ritte n as Ả P ± JA

w here

f i k 2 - p * - k '

I t is o b serv ed t h a t A > 0 w hen k2 - p2 < 0 or k2 - p2 > 1.

(3.12) (3.13) (3.14) B ased on (3.12), th e firs t condition of (3.3) lea d s to th e follow ing co n d itio n

p ± VÃ

k 2 - p 2 - k

From th e above, i t can be seen th a t w hen

> 1. (3.15)

14 Nguyen D a n g Bichy Ngo D in h B ao N a m

e q u a tio n (3.5) h a s one solution

k2 ^{- p}2 ^<0;

>1 , a n d w hen

k2 - p2 >1, k >1 , e q u a tio n (3.5) h a s two so lu tio n s

3.4. T h e c o n d itio n

> 1.

CO2 _{4_ __}Ẳ a p ¹p

> 1 o r \x\ > 1 T he solu tio n of (3.11) can be w ritte n as

p ± VÃ k2 - p2 +k T h e condition of ị%\ > 1 lead s to

± VÃ k2 - p2 + k

>1

From th e above it can be seen th a t w hen k2 - p2 <0 , e q u a tio n (3.11) h a s one solu tio n \x\ > 1; a n d w hen

k2 - p2> 1, k < - l , e q u a tio n (3.11) h a s two so lu tio n s Ix\ > 1 ■

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

3.5. N e c e ssa r y a n d s u f f ic i e n t c o n d itio n s f o r (3.3) to be s a tis fie d s i m u l t a n e o u s ly

F o r th e two conditions in (3.3) to be sa tisfie d sim u lta n eo u sly , it is e sse n tia l t h a t th e conditions p a irs of (3.16), (3.20) an d (3.17), (3.21) m u st be sa tisfie d . I t is o b serv ed t h a t th e re is only one condition for th o se re q u ire m e n ts to be m et, th a t is

k2 - p2 < 0 . (3.22)

W ith th is condition one solu tio n — of th e eq u atio n (3.5) a n d one so lu tio n X ° f e q u a tio n (3.1 1) have th e a b so lu te value g re a te r th a n 1.

3.6. N e c e s sa r y c o n d itio n f o r (3.4)

B ased on (3.9), th e condition (3.4) can be r e w ritte n in th e form

C onditions fo r th e A p p ro x im a te d A n a ly tic a l Solu tion of... 15

, k2 - p2 + k + p - - 4 kt - 4k i

p 0

l 2 _ 2 ? A . . Á

k - p +k + p — + 4k — ( k2- p 2+k )

and on (3.5), from (3.23) it yields 4

(k2 - p2 + k)j

H ence, th e n e ce ssa ry condition for (3.4) can be derived as (£ 2 - p 2)(fc2 - p 2 -l)-1 6 Jfe 2 >0

> 0 (3.23)

(3.24)

(3.25) It h a s been sta te d in [1] th a t for h(t) and g{t) to be approxim ated eachother w ith every t, th e n th e form ula expressing the subtraction (2.17) betw een g(t) and h(t):

.2 N

f ( y ) = y a - 3 j y 2 - 7 Ì - + 3

a p p ^{+ 2}

y +

p a p p* p a p (3.26)

c a n n o t be v a n is h e d in th e in te rv a l [-1, 1], w here d en o te y = C O S c o t .

The m en tio n ed a p p ro x im atio n re q u ire m e n t m u st be sa tisfie d th e n

/■ (l)/‘( - l) > 0 (3.27)

IS the n e c e ssa ry condition, an d f ( y ) n ot v an ish ed in th e in te rv a l [-1 1] is th e sufficient co ndition.

T he co n d itio n (3.27) can lead to th e condition (3.4), so t h a t th e n ecessary condition (3.25) IS found, a n d now it m u st be to find th e su fficien t condition

3.7. S u f f i c i e n t c o n d itio n f o r (3.4) From (3.26) it can be in ferred th a t

(3.28)

^ = 3 y2 - 2 — y - Í7 Ã_-- 2 ₁ ũ)2 Ả ₁2Ì

dy y p y

[ 3 /?2 a p p 3 J

a y 6 (3.29)

B ased on th e condition (3.3), from (3.29) it can be in fe rre d th a t th e sign of f " ( y ) re m a in s u n c h a n g e d in th e in te rv a l [-1,1], th a t lead s to th e m onotone of f ' ( y ) in the in te rv a l [-1,1].

In tro d u c e a n a d d itio n a l condition

r ( i ) r ( - i ) > 0 , (3.30)

16 Nguyen D a n g Bich, Ngo Dinh B ao N a m a sso ciatin g w ith th e m onotone condition of f ' ( y ) in th e in te r v a l [-1,1], it c an be in ferred th a t th e sig n f ' ( y ) re m a in s u n ch an g ed in th e in te r v a l [-1,1], th ere fo re

f ị y ) is m onotonic in th e in te r v a l [-1,1].

From th e condition (3.27) a n d th e m onotone condition of f [ y) in th e in te rv a l [-1,1], it can be s ta te d t h a t th e sign of f ( y ) re m a in s u n c h an g e d in th e in te rv a l [-1,1], or in o th e r w ords, f ( y ) does not v a n ish in th e in te rv a l [-1,1].

B ased on (3.28), th e condition (3.30) lead s to

Z i l K.Ả

3 p 2+ a p p

+ 2a

p

' ĩ - Ế - — Ả

3 p ĩ + a p p ^{- 2} >0 A ssociating w ith (3.9) it yields

Í 4 k2- p 2- k ) Ă2 1

> 2 i 1^3 k2- p 2+ k y p2 3 n

(3.31)

(3.32) T his is th e su fficien t condition for (3.4) to be satisfied .

4. Conclusion

In su m m ary , for h(t), g(t) to be ap p ro x im ated each o th e r w ith every t, th ere are th re e re la tin g co n d itio n s for p a ra m e te rs , th a t is th e n e ce ssa ry a n d sufficient conditions (3.22), (3.25) a n d (3.32).

T hese conditions provide to find an ap p ro x im ated a n a ly tic a l so lu tio n to a p a ra m e tric o scillatio n problem d escribed by th e M ath ew ’s eq u atio n .

Acknowledgements T he p a p e r is com pleted w ith th e fin an cial su p p o rt from th e N atio n al C ouncil for N a tu ra l Science.

References

1. Dao H uy Bich, N guyen D ang Bich. On th e la te ra l o scillatio n problem of b eam s subjected to a x ia l load. VNU . J o u rn a l o f Science. M ữ thcm atics-P hysics. T .x x ,

N 4(2004), p p .1-10.