Example 2.10. The EVP of the fixed-free vibrating rod (Sect. 2.5.1) has differential equation c
2 u′′ = –x
2 u and boundary conditions u(0) = u′(l) = 0. The EVP
is completely definite for any c 6 ¼ 0, since:
Z b
a
uKu
ð
Þdx ¼
Z l
0
u c
2 u
00
À
Á
dx ¼ uc
2 u
0
Â
à l
0
À
Z l
0
c
2 u
0
ð Þ
2 dx
¼ À
Z l
0
c
2 u
0
ð Þ
2 dx \0 for all u 2 u TF ;
Z b
a
uLu
ð
Þdx ¼
Z l
0
u Àu
ð Þdx ¼ À
Z l
0
u
2 dx \0 for all u 2 u TF :
ð2:47Þ
The attribute of complete definiteness is associated with the operators of an
EVP. Other notions of definiteness are tied to the sign of the eigenvalues. Thus, an
EVP with eigenvalues k j , j = 1, ∞, is said to be positive definite if k j > 0 (all
j implied) and negative definite if k j < 0. It is positive semidefinite if k j ! 0 and
negative semidefinite if k j
0. A definite EVP is either positive or negative
definite, and a semidefinite EVP is either positive or negative semidefinite.
2.6.6 Orthogonality
Two real-valued functions u(x) and v(x), integrable on the interval [a;b], are said to
be orthogonal if:
Z b
a
uv dx ¼ 0
ð2:48Þ
Similarly, the functions are called G-orthogonal if, for a given linear operator G:
Z b
a
uGv dx ¼ 0
ð2:49Þ
where the letter ‘G’ should be replaced by the symbol of the operator in each case.
This kind of orthogonality is sometimes referred to as general orthogonality.
Example 2.11. The functions u = sin(x) and v = cos(x) are orthogonal on x2
[0;p], satisfying (2.48). Also, the functions are K-orthogonal for the operator K = –
d
2 /dx
2 on x2[0;p], satisfying (2.49) with G = K.
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2 Eigenvalue Problems of Vibrations and Stability