2.8.1 Closed-Form Solutions
When the differential equation of the EVP (2.41) has constant coefficients, it is often
possible to express its general solution in closed form in terms of 2m arbitrary
constants a m . Requiring the 2m boundary conditions to be satisfied, a set of
2m linear and homogeneous equations in the constants a m results. For non-trivial
solutions to exist, the coefficient-determinant of the equations must vanish. This
gives a single algebraic equation, typically transcendent, for determining the
eigenvalues k. The corresponding eigenfunctions u are then calculated by substituting back each eigenvalue into the system of equations, solving for the constants
a m , and in turn substituting these constants into the general solution of the differential equation. For each eigen-function one of the constants remain undetermined,
free to be scaled for normalization. When applicable this technique is straightforward, as illustrated by the examples in Sects. 2.5.1 and 2.5.2.
2.8.2 The Method of Eigenfunction Iteration
This technique parallels the method of inverse iterations for algebraic EVPs:
Eigenfunction Iteration. For the differential EVP Ku = kLu, with boundary
conditions B l u = 0 that are independent of k:
1. Choose an arbitrary initial function u [0] (x) and let k [0] = 1
2. Iterate for k = 0, 1, …, until |k [k+1] − k [k] |/k [k] < e ( 1:
(a) Solve Ku [k+1] = kLu [k] with B l u [k+1] = 0 for the eigenfunction estimate u [k+1]
(b) Compute the associated eigenvalue estimate as the Rayleigh quotient
of u [k+1] :
k k þ 1
½
¼ R u k þ 1
½
Â
à ¼
Z b
a
u k þ 1
½
Lu k
½ dx
0Z b
a
u k þ 1
½
Lu k þ 1
½
dx
ð ! k 1 Þ
ð2:71Þ
If convergence has been achieved at step k = p, then let k 1 % k [p+1] and
u 1 (x) % u [p+1] (x).
Example 2.18. The EVP associated with vibrations of an elastic rod of length
l = 1 may be written −u′′ = ku with u(0) = u′(1) = 0, where k (x/c)
2 . The
scheme for eigenfunction iterations becomes:
2.8 Methods of Solution
75
When the differential equation of the EVP (2.41) has constant coefficients, it is often
possible to express its general solution in closed form in terms of 2m arbitrary
constants a m . Requiring the 2m boundary conditions to be satisfied, a set of
2m linear and homogeneous equations in the constants a m results. For non-trivial
solutions to exist, the coefficient-determinant of the equations must vanish. This
gives a single algebraic equation, typically transcendent, for determining the
eigenvalues k. The corresponding eigenfunctions u are then calculated by substituting back each eigenvalue into the system of equations, solving for the constants
a m , and in turn substituting these constants into the general solution of the differential equation. For each eigen-function one of the constants remain undetermined,
free to be scaled for normalization. When applicable this technique is straightforward, as illustrated by the examples in Sects. 2.5.1 and 2.5.2.
2.8.2 The Method of Eigenfunction Iteration
This technique parallels the method of inverse iterations for algebraic EVPs:
Eigenfunction Iteration. For the differential EVP Ku = kLu, with boundary
conditions B l u = 0 that are independent of k:
1. Choose an arbitrary initial function u [0] (x) and let k [0] = 1
2. Iterate for k = 0, 1, …, until |k [k+1] − k [k] |/k [k] < e ( 1:
(a) Solve Ku [k+1] = kLu [k] with B l u [k+1] = 0 for the eigenfunction estimate u [k+1]
(b) Compute the associated eigenvalue estimate as the Rayleigh quotient
of u [k+1] :
k k þ 1
½
¼ R u k þ 1
½
Â
à ¼
Z b
a
u k þ 1
½
Lu k
½ dx
0Z b
a
u k þ 1
½
Lu k þ 1
½
dx
ð ! k 1 Þ
ð2:71Þ
If convergence has been achieved at step k = p, then let k 1 % k [p+1] and
u 1 (x) % u [p+1] (x).
Example 2.18. The EVP associated with vibrations of an elastic rod of length
l = 1 may be written −u′′ = ku with u(0) = u′(1) = 0, where k (x/c)
2 . The
scheme for eigenfunction iterations becomes:
2.8 Methods of Solution
75