Cain G., Meyer G.H. Separation of Variables for Partial Differential Equations: An Eigenfunction Approach
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where
is an eigenfunction of
with eigenvalue
E.
As in Chapter 3 we can multiply this equation by Φ and integrate over to conclude a priori that any
eigenvalue
E
is real and that Φ can be chosen to be real. The eigenvalue, however, does not need to be
positive. In fact, it generally is of interest to discover conditions under which the equation admits
negative eigenvalues.
We rewrite the eigenvalue equation in spherical coordinates and obtain
where
and As in Example 9.8 we write
where denotes an eigenfunction of the Laplacian on the surface of the unit sphere with
corresponding eigenvalue Then must be a continuously
differentiate solution of
It is subject to the constraints that
and
This equation is slightly more complicated than (9.2). We consider special cases now.
For
m
=0 we can write the equation for
α(r)
as in Example 6.7 in the form
If
then we would obtain a linear or trigonometric function for r > a which is not square integrable
over
Hence any eigenvalue necessarily be negative. A square integrable solution for
r
>
a
is
The corresponding bounded solution on [0, a) is
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Continuity of
α(r)
and
α′(r)
at
r=a
leads to the equations
where we have set This system can have a nontrivial solution only if the determinant of
the coefficient matrix vanishes. After canceling non-zero common factors we find that must be a
negative solution of
For
γ
is imaginary and there is no root of γ=0 is a root but leads to the trivial
solution. For
the sign pattern of the trigonometric polynomial shows that there is an
eigenvalue
when Hence there are at most finitely many eigenvalues for
m
=0 with corresponding radial solution
Because of the exponential decay, it is easy to verify as in Chapter 3 that the set
is orthogonal
in
Let us now consider the general case when m≠0. We shall assume first that and set
Then the radial equations
must be solved. Both of these equations are of the form of equation (9.2) with
We already know from Example 9.6 that a bounded solution on [0, a) is given by the spherical Bessel
function
αm(r)=c
1
jm(γr).
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For
r
>
a
the solution is a linear combination of the spherical Hankel functions
The Hankel functions
and decay and increase exponentially for Hence we
require
d
2=0 and observe that then
αm(r)
belongs to It remains to match the two solutions
at
r=a
. As before we need a nonzero solution of the system
or in matrix form
where ′ denotes the derivative of
jm(z)
and with respect to
z
. is a possible (scaled) eigenvalue if
the determinant of the coefficient matrix vanishes, which leads to the nonlinear equation
For
m
≥1 this equation can be simplified with the recursion formula
which is one of several available for spherical Bessel function.
We obtain after some algebra
(9.4)
It is known that (9.4) has no solution for imaginary γ, that γ=0 is a root but leads to the trivial solution,
that there are at most finitely many negative roots with their number depending on
V
0, and that for
positive
the Hankel function solution is not square integrable on Hence for each m≥0 there are
at most finitely many eigenvalues
Their values for a given
V
0 are found numerically from (9.4).
If the eigenvalues for a given m are indexed as and the corresponding solutions of the radial
equation are labeled
{αℓm(r)},
then for each such eigenvalue we have the radial eigenfunctions
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They are mutually orthogonal in the inner product
and can be used to find eigenfunction expansions for the time-dependent Schrödinger equation. For
example, let
denote the smallest root of The smallest eigenvalue is then
The corresponding solution of the radial equation is
and the eigenfunctions for the expansion of solutions of the time-dependent Schrödinger equation are
The question of how well an initial condition
u
0
(x)
for the Schrödinger equation (9.3) can be
approximated in terms of such eigenfunctions is best left to experts in quantum mechanics, but we do
know that if
is a set of
K
eigenvalues and orthogonal eigenfunctions for (9.3), then the
approximating problem
with
has the analytic solution
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Bibliography
[1] L.C.Andrews,
Special Functions for Engineers and Applied Mathematicians,
Macmillan, ISBN 0-020-
948650-5, 1985.
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Vorlesungen über Geschichte der Mathematik
IV, Teubner, 1908.
[3] R.Courant and D.Hilbert,
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[4] H.F.Davis,
Fourier Series and Orthogonal Functions,
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Partial Differential Equations,
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Elliptic Partial Differential Equations of Second Order,
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Equations,
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8247-0466-5, 2001.
[11] Abdul J.Jerri,
The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations,
Kluwer, ISBN 0792351096, 1998.
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[12] E.Kamke,
Differentialgleichungen: Lösungsmethoden und Lösungen I, Gewöhnliche
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Special Functions and Their Applications,
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Second Order Parabolic Differential Equations,
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Theory of Hydrodynamic Lubrication,
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Index
artificial discontinuity, 197
basis, 26, 55, 69
beat, 170
best approximation, 31
boundary
condition consistent, 206
periodic, 229
reflection, 12
Cauchy data, 11, 19
collocation, 236
compatibility condition, 5
consistency condition, 206
convection dominated diffusion, 137
convergence
in the mean, 69
point wise, 71
uniform, 71
dimension, 26
dispersion relation, 175
divergence theorem, 9
domain decomposition, 221
domain imbedding, 234
Duhamel’s principle, 156, 191
for the heat equation, 17
for the wave equation, 22
eigenfunction, 45
eigenvalue, 45
eigenvector, 45
energy estimate, 23, 151, 189
equation
associated Legendre, 246
backward heat, 17, 129
beam, 182
Bessel’s, 143, 244
Black-Scholes, 139
Cauchy-Euler, 228
diffusion, 11
heat, 11
homogeneous, 4
Laplace’s, 3
Poisson’s, 4, 5, 226
Reynolds, 218
telegraph, 171
wave, 18
equipotential lines, 217
Euclidean norm, 29
financial option, 139
finite element estimate, 249
formal splitting, 197, 199
Fourier coefficients, 68
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cosine series, 78
sine series, 78
Fourier series, 68
convergence, 75
differentiation, 82
integration, 80
partial sums, 84
function associated Legendre, 246
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Bessel, 142, 143, 176, 244, 270
delta, 240
Green’s, 239, 241
Heaviside, 35
piecewise continuous, 75
piecewise smooth, 75
smooth, 1
spherical Bessel, 270
spherical Hankel, 270
weight, 34
function spaces
C[a, b],
34
L
2
(D)
34
L
2
(D, w),
34
Gibbs phenomenon, 86, 89
Gram-Schmidt process, 42
homogeneous
boundary condition, 4
equation, 4
inequality
Bessel’s, 71
Gronwall’s, 24, 153, 189
Poincaré, 84, 190, 249
Schwarz’s, 28, 189, 249
triangle, 26
inner product, 27
interface condition, 59, 147
Laplacian, 3
least squares approximation, 33
Legendre polynomial, 246
Maple, 128
maximum principle, 7, 13
norm, 26
orthogonal, 27
orthogonal projection, 29
existence, 30
Parseval’s identity, 71
partial differential operator, 1
elliptic, 2
hyperbolic, 3
parabolic, 3
periodic extension, 68, 78
preconditioning, 199, 202, 205
problem
approximating, 99
biharmonic, 253
Dirichlet, 4
ill posed, 2
Neumann, 4