Calculation of eigenvalues for a class of crosswise vibration equation of the beam

Applied Mathematics and Computation 165 (2005) 143–154 www.elsevier.com/locate/amc

Calculation of eigenvalues for a class of crosswise vibration equation of the beamq Gao Jia a

a,* ,

Ying-Qing Song b, Qiong Li

c

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China b Department of Applied Mathematics, Hunan City University, Yiyang 413000, China c Department of Mathematics, Southeast University, Nanjing 210096, China

Abstract This paper considers the calculation of eigenvalues for a class of crosswise vibration equation of the beam by GalerkinÕs method and obtains the estimation of the errors. The results are very useful for studies of the eigenvalue problems. Some numerical examples are given.
2004 Published by Elsevier Inc. Keywords: Differential equation; Eigenvalues; Eigenfunctions; GalerkinÕs method

1. Introduction and main results About the upper or lower bound of the eigenvalues for the ordinary differential operators and the uniformly elliptic operators, there are many good estimates (see, for example, [1–7]). However, we could not find any result for the eigenvalue approximate computation about the afore-stated problems. q

Foundation item: the National Natural Science Foundation of China (19771048). Corresponding author. E-mail address: [email protected] (G. Jia).

*

0096-3003/$ - see front matter
2004 Published by Elsevier Inc. doi:10.1016/j.amc.2004.04.078

144

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

Let (a, b)  R1. In this paper we consider the calculation of eigenvalues for the following problem: 8 k ðkÞ ðkÞ > < ðpðxÞy Þ ¼ ð1Þ ky; x 2 ða; bÞ; ð1Þ yðaÞ ¼ y 0 ðaÞ ¼    ¼ y ðk1Þ ðaÞ ¼ 0; > : 0 ðk1Þ yðbÞ ¼ y ðbÞ ¼    ¼ y ðbÞ ¼ 0; where p(x) 2 Ck([a, b]) with ð2Þ

l1 6 pðxÞ 6 l2 ;

and l1, l2 are two positive numbers, k is a positive integer. By the differential equation theories, we know that the eigenvalues always exist, and ones are real, positive and discrete. Let k1, k2, . . ., kn, . . . be the eigenvalues in (1), and yi(x) be the eigenfunction of (1) corresponding to ith eigenvalue ki, i = 1, 2, . . ., satisfying Z b y i ðxÞy j ðxÞdx ¼ dij ; i; j ¼ 1; 2; . . . : ð3Þ a

By (1) and (3), integrating by parts, we have Z b Z b Z k ðkÞ kp y 2p ðxÞdx ¼ ð1Þ ðpðxÞy ðkÞ Þ y dx ¼    ¼ p p a

a

b 2

pðxÞðy ðkÞ p Þ dx; a

i.e., kp ¼

Z

b 2 pðxÞðy ðkÞ p Þ dx:

ð4Þ

a

By virtue of (2), we get Z b kp kp 2 6 ðy ðkÞ : p Þ dx 6 l2 l1 a

ð5Þ

Lemma 1. Let a < b and f (k)(x) 2 L2([a, b]), f(a) = f 0 (a) =    = f Then, for 0 6 l 6 k  1, holds Z b Z b 2 2ðklÞ 2 ðf ðlÞ ðxÞÞ dx 6 ðb  aÞ ðf ðkÞ ðxÞÞ dx: a

(a) = 0.

ð6Þ

a

Proof. Since f (k1)(a) = 0, this implies that f ðk1Þ ðxÞ ¼ x 2 (a, b). By Cauchy inequality, we have

ðf ðk1Þ ðxÞÞ2 ¼

(k1)

Z

x

f ðkÞ ðtÞ dt a

2 6 ðb  aÞ

Z a

Rx a

f ðkÞ ðtÞ dt, where

x

ðf ðkÞ ðtÞÞ2 dt:

ð7Þ

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

145

Integrating in (7) yields Z b Z b Z x 2 2 ðf ðk1Þ ðxÞÞ dx 6 ðb  aÞ dx ðf ðkÞ ðtÞÞ dt a a a Z b 2 ðt  aÞðf ðkÞ ðtÞÞ dt ¼ðb  aÞ a Z b ðf ðkÞ ðxÞÞ2 dx: 6 ðb  aÞ2 a

For 0 6 l 6 k  1, we find that Z b Z b Z b ðf ðlÞ ðxÞÞ2 dx 6 ðb  aÞ2 ðf ðlþ1Þ ðxÞÞ2 dx 6 ðb  aÞ4 ðf ðlþ2Þ ðxÞÞ2 dx a a a Z b 2ðklÞ 2 ðkÞ ðf ðxÞÞ dx:  6    6 ðb  aÞ a 2 Using (5) and Lemma 1, we get that y p ; y 0p ; y 00p ; . . . ; y ðkÞ p 2 L ð½a; b Þ, p = 1, 2, . . . 2 Since L ([a, b]) is a separable linear space, there then exist on [a, b] the system of basic functions (see [6])

u1 ðxÞ; u2 ðxÞ; . . . ; un ðxÞ; . . . which are: Rb 1. Orthogonal and unitary, i.e., a ui uj dx ¼ dij , i, j = 1, 2, . . . 2. Complete, i.e., there exists no other non-zero function which is orthogonal to all the functions. 3. The system has been chosen in such a manner that ui(x), i = 1, 2, . . . satisfy the boundary conditions ðk1Þ

ui ðaÞ ¼ u0i ðaÞ ¼    ¼ ui

ðk1Þ

ui ðbÞ ¼ u0i ðbÞ ¼    ¼ ui

ðaÞ ¼ 0;

ðbÞ ¼ 0;

and the eigenfunction yp is a linear combination of ui(x), i.e., there exist scalars Cip, i = 1, 2, . . . in R1 such that 1 X C ip ui ; p ¼ 1; 2; . . . : ð8Þ yp ¼ i¼1

Pm Let we define y mp ¼ i¼1 C ip ui ; and y mp ¼ y p  y mp , p = 1, 2, . . . Now we consider the property of the eigenvalues about the following problem ðkÞ ðpðxÞy ðkÞ ¼ ð1Þk kmp y mp : mp Þ

ð9Þ

Since y p ¼ y mp þ y mp , by (1) and a simple computation, we could obtain ðpðxÞy ðkÞ mp Þ

ðkÞ

ðkÞ ðkÞ

þ ðpðxÞðy mp Þ Þ

k

¼ ð1Þ kp ðy mp þ y mp Þ

146

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

and (using (9)) ðkÞ ðkÞ

k

ð1Þ ðkp  kmp Þy mp ¼ ðpðxÞðy mp Þ Þ

k

 ð1Þ kp y mp :

ð10Þ

Theorem 1. Suppose that y p ð1 6 p < 1Þ is the eigenfunction of (1) corresponding to the eigenvalue kp. Suppose that y mp ð1 6 p < 1Þ is the eigenfunction of (9) corresponding to the eigenvalue kmp. Then  1=2 l2 kp ðkÞ j kp  kmp j 6 ky ðkÞ ð11Þ p  y mp kL2 ; 2 ky mp kL2 l1 Rb where kf kL2 ¼ ð a f 2 ðxÞdxÞ1=2 . Proof. Since u1, u2, . . . are orthogonal on [a, b], we know that Z b y mp y mp dx ¼ 0:

ð12Þ

a

By (10) and (12) and integrating by parts, we find  Z b  Z b ðkÞ ðkp  kmp Þ y 2mp dx ¼ ð1Þk pðxÞðy mp ÞðkÞ ð1Þk kp y mp y mp dx a Za b  Z b ðkÞ k ¼ ð1Þ pðxÞðy mp ÞðkÞ y mp dx  y mp y mp dx a a Z b ¼ pðxÞðy mp ÞðkÞ y ðkÞ ð13Þ mp dx: a

By (2), (5), and (13) and Cauchy inequality, we have

Z b

Z b

2 ðkÞ ðkÞ ðkÞ

j kp  kmp j y mp dx ¼ pðxÞðy p  y mp Þy mp dx

a

a

Z

1=2 Z b 1=2 2 ðkÞ ðkÞ 2 ðy ðkÞ Þ dx  ðy  y Þ dx mp p mp a a  1=2 kp ðkÞ 6 l2 ky ðkÞ p  y mp kL2 : l1 b

6 l2

Hence, we get j kp  kmp

 1=2 kp ðkÞ j 6 j y ðkÞ p  y mp jL2 : 2 j y mp jL2 l1 l2

Since Z lim

m!1

a

b 2

ðkÞ ðy ðkÞ p  y mp Þ dx ¼ 0;



G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

147

from (11), we obtain kp ¼ lim kmp ; p ¼ 1; 2; . . . ; m!1

and (11) is an upper bound of the error of the estimate for the eigenvalue kp by kmp. 2. The approximate calculation of the eigenvalues In this section, our aims are to establish the approximate method of the eigenvalues by GalerkinÕs idea (see [7]), and to give some examples for the approximateP calculation about the eigenvalues problem (1). m Let y m ¼ i¼1 C i ui , where Ci(1 6 i 6 m) are constants to be fixed, and ui(i = 1, 2, . . .) satisfy the afore-stated conditions. By (9) and Theorem 1, we can suppose that there is a number  km such that ðpðxÞy ðkÞ m Þ

ðkÞ

k km y m : ¼ ð1Þ 

Since ui ðaÞ ¼ ui ðbÞ ¼ 1, 2, . . ., we then have

u0i ðaÞ

ð14Þ ¼

u0i ðbÞ

¼  ¼

ðk1Þ ui ðaÞ

¼

ðk1Þ ui ðbÞ

¼ 0, i =

y m ðaÞ ¼ y m ðbÞ ¼ y 0m ðaÞ ¼ y 0m ðbÞ ¼    ¼ y mðk1Þ ðaÞ ¼ y mðk1Þ ðbÞ ¼ 0: By virtue of the orthogonality of ui and (3), multiplying two sides in (14) by ui and using integrating by parts (k-times), we have Z b m m X X ðkÞ ðkÞ Ci pðxÞui uj dx ¼  km dij C i ; j ¼ 1; 2; . . . ð15Þ a

i¼1

i¼1

Let we define Z b ðkÞ ðkÞ aij ¼ pðxÞui uj dx:

ð16Þ

a

We have aij = aji, (i, j = 1, 2, . . .) and (from (15)) m X ðaij   km dij ÞC i ¼ 0; j ¼ 1; 2; . . . ; m:

ð17Þ

i¼1

By the knowledge in linear algebra (see, for example, [8]) the homogeneous system about the unknowns Ci have non-zero solutions in (17) if and only if detðA   km IÞ ¼ 0, where I is an identity matrix, i.e.,

a11   km a12  a1m

a a22   km    a2m

21 ð18Þ

¼ 0:

     

a am2    amm   km m1

148

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

Since A = (aij)m·m is a real symmetric matrix and detðA  km IÞ is a polynomial of degree m in the variable  km , we will find (18) possesses the real characteristic values  kmp ðp ¼ 1; 2; . . . ; mÞ. Further, we could obtain the characteristic functions ymp(p = 1,2, . . ., m) corresponding to  kmp . The core of our method is to take  kmp as the (mp + 1)th approximation for the eigenvalue kp in (1), and to take ymp as the (mp + 1)th approximation for the eigenfunction yp corresponding to kp in (1). This idea is called the wellknown GalerkinÕs method. Here, we give two numerical examples. Example 1. Find approximate values of the eigenvalues to the boundary value problem  00 y ¼ ky; x 2 ð0; 1Þ; ð19Þ yð0Þ ¼ yð1Þ ¼ 0: pffiffiffi pffiffiffi pffiffiffiSolution. It is easy to see that u1 ¼ 2 sin px, u2 ¼ 2 sin 2px, . . ., um ¼ 2 sin mpx, . . ., satisfying the afore-stated conditions. Since p(x) = 1, direct computing and using (16) produces

a11 ¼

Z

1 2

ðu01 ðxÞÞ dx ¼ 2p2

Z

0

1

cos2 pxdx ¼ p2 ;

0

a12 ¼ a21 ¼

Z

1

u01 ðxÞu02 ðxÞdx ¼ 0;

0

a22 ¼

Z

1 2

ðu02 ðxÞÞ dx ¼ 4p2 :

0

Substituting aij(i, j = 1, 2) into (18), gives

2

p  k 0

¼ 0;

0 4p2  k and we have the approximate estimate for k1 and k2 k21 ¼ p2 ;

k22 ¼ 4p2 :

As a matter of fact, k21 = p2, k22 = 4p2 are the exact values for k1 and k2. Example 2. Find approximate values of the eigenvalues to the boundary value problem ( 00 ððx2 þ 1Þy 00 Þ ¼ ky; x 2 ð0; 1Þ; yð0Þ ¼ yð1Þ ¼ y 0 ð0Þ ¼ y 0 ð1Þ ¼ 0:

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

149

Solution. Since p(x) = x2 + 1, x 2 (0, 1). So 1 6 p(x) 6 2, and we can take l1 = 1, l2 = 2. If we choose u1 ¼ p4ffiffi6 sin px sin px, u2 ¼ 2 sin px sin 5px, u3 ¼ 2 sin px sin 9px; . . ., um ¼ 2 sin px sinð4m  3Þpx; . . ., by direct computation, we have Z 1 ui uj dx ¼ dij ; i; j ¼ 1; 2; . . . 0

and ui ð0Þ ¼ u0i ð0Þ ¼ 0, i = 1, 2, . . . Therefore ui satisfy the afore-stated conditions. It is easy to see that 8p2 u001 ðxÞ ¼ pffiffiffi cos 2px; 6 u002 ðxÞ ¼ 36p2 cos 6px  16p2 cos 4px; u003 ðxÞ ¼ 100p2 cos 10px  64p2 cos 8px; u004 ðxÞ ¼ 196p2 cos 14px  144p2 cos 12px; u005 ðxÞ ¼ 324p2 cos 18px  256p2 cos 16px; u006 ðxÞ ¼ 484p2 cos 22px  400p2 cos 20px: Integrating by parts, we find Z 1 2 a11 ¼ ðx2 þ 1Þðu001 ðxÞÞ dx ¼ 699:266614; 0

a12 ¼ a21 ¼

Z

1

ðx2 þ 1Þu001 ðxÞu002 ðxÞdx ¼ 52:640103;

0

a13 ¼ a31 ¼

Z

1

ðx2 þ 1Þu001 ðxÞu003 ðxÞdx ¼ 5:184302;

0

a14 ¼ a41 ¼

Z

1

ðx2 þ 1Þu001 ðxÞu004 ðxÞdx ¼ 1:545918;

0

a15 ¼ a51 ¼

Z

1

ðx2 þ 1Þu001 ðxÞu005 ðxÞdx ¼ 0:664694;

0

a16 ¼ a61 ¼

Z

1

ðx2 þ 1Þu001 ðxÞu006 ðxÞdx ¼ 0:346077;

0

a22 ¼

Z 0

1 2

ðx2 þ 1Þðu002 ðxÞÞ dx ¼ 97958:093711;

150

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

a23 ¼ a32 ¼

Z

1

ðx2 þ 1Þu002 ðxÞu003 ðxÞdx ¼ 3258:836701;

0

a24 ¼ a42 ¼

Z

1

ðx2 þ 1Þu002 ðxÞu004 ðxÞdx ¼ 277:824243;

0

a25 ¼ a52 ¼

Z

1

ðx2 þ 1Þu002 ðxÞu005 ðxÞdx ¼ 83:167230;

0

a26 ¼ a62 ¼

Z

1

ðx2 þ 1Þu002 ðxÞu006 ðxÞdx ¼ 36:812800;

0

a33 ¼

Z

1 2

ðx2 þ 1Þðu003 ðxÞÞ dx ¼ 883818:2415;

0

a34 ¼ a43 ¼

Z

1

ðx2 þ 1Þu003 ðxÞu004 ðxÞdx ¼ 21180:601261;

0

a35 ¼ a53 ¼

Z

1

ðx2 þ 1Þu003 ðxÞu005 ðxÞdx ¼ 1529:734403;

0

a36 ¼ a63 ¼

Z

1

ðx2 þ 1Þu003 ðxÞu006 ðxÞdx ¼ 425:335032;

0

a44 ¼

Z

1 2

ðx2 þ 1Þðu004 ðxÞÞ dx ¼ 3:7020314261  106 ;

0

a45 ¼ a54 ¼

Z

1

ðx2 þ 1Þu004 ðxÞu005 ðxÞdx ¼ 74669:334105;

0

a46 ¼ a64 ¼

Z

1

ðx2 þ 1Þu004 ðxÞu006 ðxÞdx ¼ 4847:471455;

0

a55 ¼

Z

1

ðx2 þ 1Þðu005 ðxÞÞ2 dx ¼ 1:0663613425  107 ;

0

a56 ¼ a65 ¼

Z

1

ðx2 þ 1Þu005 ðxÞu006 ðxÞdx ¼ 193832:020522;

0

a66 ¼

Z 0

1 2

ðx2 þ 1Þðu006 ðxÞÞ dx ¼ 2:4647410205  107 :

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

151

The procedure about the eigenvalues approximation can be listed below. First approximation From a11  k = 0, we get the first approximation about the first eigenvalue k1: k1  k11 ¼ 699:266614: Second approximation From det½ð699:266614  x; 52:640103Þ;

ð52:640103; 97958:093711  xÞ ¼ 0;

we get 6:84961  107  98657:4x þ x2 ¼ 0: Solving the equation yields k21 ¼ 699:2381;

k22 ¼ 97958:1233;

and we have k1  k21 ¼ 699:2381;

k2  k22 ¼ 97958:1233:

Third approximation From det½ð699:267  x; 52:6401; 5:1843Þ; ð52:6401; 97958:1  x; 3258:84Þ; ð5:1843; 3258:84; 883818  xÞ ¼ 6:05306  1013  8:7253  1010 x þ 982475:x2  x3 ¼ 0; we have k31 ¼ 699:2381;

k32 ¼ 97944:6233;

k33 ¼ 883832:2431;

and k1  k31 ¼ 699:2381;

k2  k32 ¼ 97944:6213;

k3  k33 ¼ 883832:2431:

152

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

Fourth approximation From det½ð699:266614  x; 52:6401; 5:1843; 1:5459Þ; ð52:6401; 97958:1  x; 3258:84; 277:82Þ; ð5:1843; 3258:84; 883818  x;  21180:6Þ; ð1:5459; 277:82; 21180:6; 3:70203  106  xÞ

¼ 2:24055  1020  3:23029  1017 x þ 3:72396  1012 x2  4:68451  106 x3 þ x4 ¼ 0; solving this equation we find k41 ¼ 699:2381;

k42 ¼ 97944:5715;

k43 ¼ 883672:3567;

6

k44 ¼ 3:70218919  10 ; and k1  k41 ¼ 699:2381;

k2  k42 ¼ 97944:5715;

k3  k43 ¼ 883672:3567;

k4  k44 ¼ 3:70218919  106 :

Fifth approximation From det½ð699:266614  x; 52:6401; 5:1843; 1:5459; 0:664694Þ;  ð52:6401; 97958:1  x; 3258:84; 277:82; 83:1672Þ; ð5:1843;  3258:84; 883818  x; 21180:6; 1529:73Þ; ð1:5459; 277:82;  21180:6; 3:70203  106  x; 74669Þ; ð0:664694; 83:1672;  1529:73; 74669; 1:06636  107  xÞ ¼ 2:3889  1027  3:44439  1024 x þ 4:00283  1019 x2  5:36721  1013 x3 þ 1:53481  107 x4  x5 ¼ 0; we get k51 ¼ 699:2381;

k52 ¼ 97944:57057; 6

k54 ¼ 3:70138869  10 ;

k53 ¼ 883672:9098;

k55 ¼ 1:066440097  107 ;

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

153

and k1  k51 ¼ 699:2381;

k2  k52 ¼ 97944:57057;

k3  k53 ¼ 883672:9098;

k4  k54 ¼ 3:70138869  106 ;

k5  k55 ¼ 1:066440097  107 : Sixth approximation From det½ð699:266614  x; 52:6401; 5:1843; 1:5459; 0:664694;  0:346077Þ; ð52:6401; 97958:1  x; 3258:84; 277:82; 83:1672;  36:8128Þ; ð5:1843; 3258:84; 883818  x; 21180:6; 1529:73;  425:335Þ; ð1:5459; 277:82; 21180:6; 3:70203  106  x; 74669;  4847:47Þ; ð0:664694; 83:1672; 1529:73; 74669; 1:06636  107  x; 193832Þ; ð0:346077; 36:8128; 425:335; 4847:47; 193832;  2:46474  107  xÞ ¼ 5:88717  1034  8:48856  1031 x þ 9:89898  1026 x2  1:36273  1021 x3 þ 4:31925  1014 x4  3:99955  107 x5 þ x6 ¼ 0; solving this equation, we get k61 ¼ 699:2381;

k62 ¼ 97944:5705; 6

k64 ¼ 3:70138838  10 ;

k63 ¼ 883671:8991;

k65 ¼ 1:066171634  107 ;

k66 ¼ 2:465008692  107 ; and k1  k61 ¼ 699:2381;

k2  k62 ¼ 97944:5705;

k3  k63 ¼ 883671:8991;

k4  k64 ¼ 3:70138838  106 ;

k5  k65 ¼ 1:066171634  107 ;

k6  k66 ¼ 2:465008692  107 :



From the afore-approximated results, we could find the fact is, by the second approximation, k1 correct to four significant figures; and by the fifth approximation, k2 correct to four significant figures. Therefore, the rate of convergence of the method is very high, and we want to point out that the method is a very good approximation for the eigenvalues about the problem (1). In addition, in order to get the more of the eigenvalues or the more higher degree of accuracy for some eigenvalues, we might take the seventh approximation, the eighth approximation and so on.

154

G. Jia et al. / Appl. Math. Comput. 165 (2005) 143–154

References [1] G. Jia, X.P. Yang, C.L. Qian, On the upper bound of second eigenvalues for uniformly elliptic operators of any orders, Acta Math. Appl. Sin. (English Ser.) 1 (2003) 107–116. [2] G. Jia, On the upper bound of second eigenvalues for differential equations with higher orders, J. Math. Study 3 (1999) 30–35. [3] G. Jia, Estimation of eigenvalues for crosswise vibration equations of the beam, J. Anhui Univ. 2 (1997) 29–33. [4] Z.C. Chen, C.L. Qian, On the upper bound of eigenvalues for elliptic equation with higher orders, J. Math. Anal. Appl. 3 (1994) 821–834. [5] G.N. Hile, R.Z. Yeh, Inequalities for eigenvalues of the biharmonic operator, Pacific J. Math. 112 (1984) 115–133. [6] K. Yosida, Functional Analysis, sixth ed., Springer-Verlag, Berlin, 1980. [7] H.P. Ma, W.W. Sun, A Legendre–Petrov–Galerkin and Chebyshev collocation method for third-order differential equations, SIAM J. Numer. Anal. 5 (2000) 1425–1438. [8] K. Hoffman, R. Kunze, Linear Algebra, Englewood Cliffs, New Jersey, 1971.