2.6.1 Multiplicity
If, for an eigenvalue k, there are p > 1 linearly independent eigenfunctions u(x),
then k is a p-fold eigenvalue – or we say that k has multiplicity p or is a repeated
eigenvalue. Most eigenvalues are simple, however; they occur only once, with a
single eigenfunction. Two or more simple eigenvalues are called distinct.
2.6.2 Boundary Conditions: Essential or Natural/
Suppressible
The 2m boundary conditions B l u = 0 in (2.41)–(2.42) generally contain derivatives
of the orders 0, 1, …, 2m − 1. Assume that derivatives of order m and higher have
been eliminated as far as possible – e.g., by forming linear combinations of the
original conditions. Then boundary conditions containing derivatives of order 0, 1,
…, m − 1 are called essential, whereas those containing derivatives of order m,
m + 1, …, 2m–1 are called natural (or suppressible).
Example 2.5. Consider an EVP with a second order differential equation and
boundary conditions u(0) = c 1 u′(0), u(l) = c 2 u′(0). In this case m = 1, and we
should first eliminate, as far as possible, derivatives of order one. We can do no
better than eliminate u′(0) from one of the conditions. Substituting u′(0) = u(l)/c 2
from the second into the first condition yields the new set: c 2 u(0) − c 1 u(l) = 0 and
u(l) − c 2 u′(0) = 0. The first condition is seen to be essential whereas the second is
natural.
Every boundary condition is either essential or natural/suppressible. However, it
is not unusual to encounter EVPs totally lacking one or the other class of
conditions.
Example 2.6. The free flexural vibrations of a uniform beam are governed by an
EVP with differential equation EIu′′′′ = x
2 qAu (cf. Sect. 2.5.2). Since m = 2, the
essential boundary conditions are those containing derivatives of order zero and
one; all others are natural. Thus, if the beam is clamped-free there are two essential
boundary conditions: u(0) = u′(0) = 0, and two natural: u′′(l) = u′′′(l) = 0. With
clamped-clamped supports: u(0) = u′(0) = u(l) = u′(l) = 0, all boundary conditions are essential, whereas with free-free supports: u′′(0) = u′′′(0) = u′′(l) = u′′′
(l) = 0 they are all natural. Note that the geometric boundary conditions are
essential, and that the static conditions are natural.
2.6 Concepts of Differential EVPs
63
If, for an eigenvalue k, there are p > 1 linearly independent eigenfunctions u(x),
then k is a p-fold eigenvalue – or we say that k has multiplicity p or is a repeated
eigenvalue. Most eigenvalues are simple, however; they occur only once, with a
single eigenfunction. Two or more simple eigenvalues are called distinct.
2.6.2 Boundary Conditions: Essential or Natural/
Suppressible
The 2m boundary conditions B l u = 0 in (2.41)–(2.42) generally contain derivatives
of the orders 0, 1, …, 2m − 1. Assume that derivatives of order m and higher have
been eliminated as far as possible – e.g., by forming linear combinations of the
original conditions. Then boundary conditions containing derivatives of order 0, 1,
…, m − 1 are called essential, whereas those containing derivatives of order m,
m + 1, …, 2m–1 are called natural (or suppressible).
Example 2.5. Consider an EVP with a second order differential equation and
boundary conditions u(0) = c 1 u′(0), u(l) = c 2 u′(0). In this case m = 1, and we
should first eliminate, as far as possible, derivatives of order one. We can do no
better than eliminate u′(0) from one of the conditions. Substituting u′(0) = u(l)/c 2
from the second into the first condition yields the new set: c 2 u(0) − c 1 u(l) = 0 and
u(l) − c 2 u′(0) = 0. The first condition is seen to be essential whereas the second is
natural.
Every boundary condition is either essential or natural/suppressible. However, it
is not unusual to encounter EVPs totally lacking one or the other class of
conditions.
Example 2.6. The free flexural vibrations of a uniform beam are governed by an
EVP with differential equation EIu′′′′ = x
2 qAu (cf. Sect. 2.5.2). Since m = 2, the
essential boundary conditions are those containing derivatives of order zero and
one; all others are natural. Thus, if the beam is clamped-free there are two essential
boundary conditions: u(0) = u′(0) = 0, and two natural: u′′(l) = u′′′(l) = 0. With
clamped-clamped supports: u(0) = u′(0) = u(l) = u′(l) = 0, all boundary conditions are essential, whereas with free-free supports: u′′(0) = u′′′(0) = u′′(l) = u′′′
(l) = 0 they are all natural. Note that the geometric boundary conditions are
essential, and that the static conditions are natural.
2.6 Concepts of Differential EVPs
63