2.6.3 Function Classes: Eigen-, Test-,
and Admissible Functions
For every EVP on the form (2.41)–(2.42), we define this hierarchy of functions:
1. Eigenfunction: Any non-zero function u(x)2C
2m [a;b], satisfying the differential
equation as well as all boundary conditions of the EVP.
2. Test function: Any non-zero function u(x)2C
2m
[a;b] satisfying all boundary
conditions of the EVP. (Also called a comparison function.)
3. Admissible function: Any non-zero function u(x)2C
m [a;b] satisfying all essential
boundary conditions of the EVP.
We shall employ the shorthand notation u2u EF , u2u TF and u2u AF , respectively,
to indicate the class of a function u(x). Note that eigenfunctions are also test- and
admissible functions, and that test functions are also admissible functions.
Example 2.7. Consider the differential equation c
2 u′′ = –x
2 u of a vibrating
straight rod with fixed-free boundary conditions u(0) = u′(l) = 0 (cf. Sect. 2.5.1).
Then, u(x) = sin(x x/c) with x = pc/(2l) is an eigen-function of the EVP, whereas
u(x) = x(x − 2l) is a test function, and u(x) = x is an admissible function.
2.6.4 Adjointness
The EVP (2.41) is termed self-adjoint if, for any two test functions u(x) and v(x):
Z b
a
uKv À vKu
ð
Þ dx ¼ 0 and
Z b
a
uLv À vLu
ð
Þ dx ¼ 0; u; v 2 u TF : ð2:43Þ
Self-adjointness of differential operators parallels symmetry of matrices.
To test for self-adjointness one inserts the actual operators K and L and integrate
by parts, attempting to achieve integrands that vanish due to symmetry in u and v,
and limit-terms that vanish due to boundary conditions (which are satisfied by any
test function u, v).
Example 2.8. The EVP of the fixed-free vibrating rod (cf. Sect. 2.5.1) has differential equation c
2 u′′ = –x
2 u and boundary conditions u(0) = u′(l) = 0. Then,
with K = c
2 d
2 /dx
2 and L = −1:
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2 Eigenvalue Problems of Vibrations and Stability
and Admissible Functions
For every EVP on the form (2.41)–(2.42), we define this hierarchy of functions:
1. Eigenfunction: Any non-zero function u(x)2C
2m [a;b], satisfying the differential
equation as well as all boundary conditions of the EVP.
2. Test function: Any non-zero function u(x)2C
2m
[a;b] satisfying all boundary
conditions of the EVP. (Also called a comparison function.)
3. Admissible function: Any non-zero function u(x)2C
m [a;b] satisfying all essential
boundary conditions of the EVP.
We shall employ the shorthand notation u2u EF , u2u TF and u2u AF , respectively,
to indicate the class of a function u(x). Note that eigenfunctions are also test- and
admissible functions, and that test functions are also admissible functions.
Example 2.7. Consider the differential equation c
2 u′′ = –x
2 u of a vibrating
straight rod with fixed-free boundary conditions u(0) = u′(l) = 0 (cf. Sect. 2.5.1).
Then, u(x) = sin(x x/c) with x = pc/(2l) is an eigen-function of the EVP, whereas
u(x) = x(x − 2l) is a test function, and u(x) = x is an admissible function.
2.6.4 Adjointness
The EVP (2.41) is termed self-adjoint if, for any two test functions u(x) and v(x):
Z b
a
uKv À vKu
ð
Þ dx ¼ 0 and
Z b
a
uLv À vLu
ð
Þ dx ¼ 0; u; v 2 u TF : ð2:43Þ
Self-adjointness of differential operators parallels symmetry of matrices.
To test for self-adjointness one inserts the actual operators K and L and integrate
by parts, attempting to achieve integrands that vanish due to symmetry in u and v,
and limit-terms that vanish due to boundary conditions (which are satisfied by any
test function u, v).
Example 2.8. The EVP of the fixed-free vibrating rod (cf. Sect. 2.5.1) has differential equation c
2 u′′ = –x
2 u and boundary conditions u(0) = u′(l) = 0. Then,
with K = c
2 d
2 /dx
2 and L = −1:
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2 Eigenvalue Problems of Vibrations and Stability