@Q 1
@a i
¼
Z b
a
v i Kð
X q
j¼1
a j v j Þ dx þ
Z b
a
ð
X q
j¼1
a j v j ÞKv i dx ¼ 2
Z b
a
v i Kð
X q
j¼1
a j v j Þdx
¼ 2
X q
j¼1
Z b
a
v i Kv j dx a j ¼ 2
X q
j¼1
k ij a j ; i ¼ 1; q
@Q 2
@a i
¼ Á Á Á ¼ 2
X q
j¼1
l ij a j ; i ¼ 1; q;
ð2:80Þ
where the coefficients k ij and l ij are defined by:
k ij
Z b
a
v i Kv j dx; l ij
Z b
a
v i Lv j dx; i; j ¼ 1; q:
ð2:81Þ
Substituting (2.79)–(2.80) into (2.78), a set of q linear and homogeneous
equations in the coefficients a i is obtained, the so-called Galerkin equations:
X q
j¼1
k ij À Kl ij
À
Á
a j ¼ 0; i ¼ 1; q:
ð2:82Þ
The Galerkin equations are easily recognized as an algebraic EVP:
Ka ¼ KLa;
ð2:83Þ
where K and L are q by q symmetrical matrices with components k ij and l ij ,
respectively, K is an eigenvalue and a is an eigenvector with components a i .
Solving the algebraic EVP (e.g., by locating zeroes of |K − KL|) one obtains a set
of eigenvalues K j , j = 1, q, which can be arranged in ascending order:
K 1 K 2 Á Á Á K q :
ð2:84Þ
These values are regarded as approximations to the correspondingly ordered
eigenvalues k 1 , k 2 , …, k q of the original differential EVP. For self-adjoint and
completely definite EVPs, all K j are upper bounds for the true eigenvalues:
K j ! k j ; j ¼ 1; q:
ð2:85Þ
78
2 Eigenvalue Problems of Vibrations and Stability