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- 1 .
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Computational Procedure for SturmLiouvflle Problem
MIKHAIL D. MIKHAILOVAND NIKOLAI L. VULCIIANOV
Appl ied Mathematics Centre, Saf in, Hulgi j r i Received September 24, I981
The approaches or DatzelT (C. K . Ami . D i ' l y , . Sc i . 17- (1959), 113, 285) and Wituick
and Williams [Quart . J . Mcch. App! . Mc. ih . XXIV (1971). 263; I n t . J . Mec. ' i . S r i .
If, (197-1). 209) have been incorporated to develop a procedure for the automatic
computation of the eigen-values and the eigenfunctions of one-dimensional linear Sturm-
Liouville boundary value eigenproblems, bolh singular and nonsingular. The continuous
coefficients of a regular Slurm-Liouville problem have been approximated by a finite number
of step functions. In each step the resulting boundary value problem has been integrated
exactly and the solutions have then been matched to construct, the continuously
differentiable solution of the original problem and the corresponding eigencondition. Step
sizes have been chosen automatically so that the local error has been held in a
predetermined interval. Representative test examples have been computed lo illustrate the
accuracy, reliability, and efficiency or the algorithm proposed.
I. INTRODUCTION
The solution of various problems in the field of mathematical physics and engineering
is closely related to the solution closely corresponding one-dimensional linear Sturm-
Liouville boundary value cigenproblem. As particular examples, one could imagine the
one-dimensional Sehrodinger equation with space-dcpcndent potential function and
numerous problems of physico-chemical kinetics [5].
A number of methods have been developed lo solve one-dimensional Sturm- Liouville
problems, among these finite difference methods [6,7 ], imbedding techniques |8], and
shooting type procedures |9, 10] based on integrators for systems of ordinary
differential equations. Regardless of the particular method utilized, one always has
difficulties with the computation of the higher-order eigenvalues and eigenfunctions due
to their rapid oscillation character. This results in the need for finer interval or
integration resolution, thus making the computer time involved unreasonable.
Recently Hargrave (11) and Bailey [12,13] proposed lo solve linear Sturm-Liouville
eigenproblems utilising a Pruffer Transformation of the original problem. This turned
out to be helpful because the approach allowed one lo compute the eigenvalues solving
only one ordinary differential equation numerically. Moreover, in the corresponding
initial value problem the order of the eigenvalue was present explicitly, thus making the
computational procedure safe f r o m e i g e n v a l u e s .
One othcr feature of the method discussed is that the general purpose integrators are
safer when applied to the phase function instead of being applied directly on the
eigenfunction [12]. After analyzing in detail the integration methods available to solve
numerically Sturm-Liouville eigenproblems, Bailey developed a general purpose solver
based on the Prtiffer transformation that was tested on a number of examples [J2, J3].
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In [1,2] Datzeff developed a method enabling one to obtain principally an approx-
imation of the whole infinite spectrum of the one-dimensional linear space-dependent
Schrodinger equation. A stepwise approximation of the external potential function was
applied, thus replacing the original problem with a sequence of finite sets of Sturm-
Liouville problems with constant coefficients defined on subintervals of the initial
interval of integration. He also proved that the spectra of the approximate problems
tend to the spectrum of the original problem as the length of the largest subinterval
tends to zero. AnX-^Qll.gpPrO-^.^^tion~af^tTc^n^t_na_j_ pro_bjem_can be integrated
_cxactix.jn_t_erms of circular and hyperboTIcJmciiOils. The solutions defined on
consequent subintervals can then be (Tnatche^at the corresponding interfaces to
ensure continuous differentiability of the solution of the approximate problem and to
derive the eigencondition of the problem. Such an approach guarantees roughly the
same accuracy of the ground and the higher-order eigenvalues and eigenfunctions.
These papers were published at the beginning of the computer era and thus no
numerical examples were presented to illustrate the computational advantages of this
method.[ The same approach was later developed by Canosa and Olivcira [I4j. They ) illustrated
its efficiency by many numerical examples. The method was tested again | by Canosa
(15] to show its applicability for some practically asymptotic problems. A / trial and
error procedure was used to locate the eigenvalues, based on the computation of
dct[/v(A)] for randomly chosen values of X and simply observing where the
determinant changed sign. Such a procedure, however, is generally not safe ^J"rom
missing eigenvalues in the course of the computation.
Wittrick and Williams |3, 4] developed an extremely efficient algorithm (in terms of
computer time) for ihe safe and automatic computation of the natural frequencies and
buckling loads of linear clastic skeletal structures. It permits one to estimate exactly
how many natural frequencies lie below any fixed value of the frequency parameter
without calculating them explicitly, thus being able to obtain upper and lower bounds
for any required eigcnfrcqncncy to any desired accuracy.
Purpose of the- present paper is lo combine ihe approaches of Datzeff and of
Wituick and Williams to develop a computational procedure for the solution of one-
dimensional linear Slurm-Liouville boundary value eigenproblems. The applicability
of the algorithm will be demonstrated by recalculating several problems
previously published.324
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COMPUTING STURM-I.IOUVIJ.U- I'UOHl.liMS 325
2. STATEMENTANO ANALYTICAL ASPECTS orTHE PROBLEM
present discussion concerns the one-dimensional linear Sturm-Liouville . . _~
_5ry value eigenproblem
(d/dx)(p{xMclx) T(x, A )) f- (Ar(x) - q(x)) Y'(.v, A ) - 0, a < x 0 is __________________________.________________
"17
, , sin(a),.(jrt -A)) , , % sin(o>T(JF A*.,)) A
where xk_, < x < x
k, o>J
kjn
yj= 1, 2,.... ir coj
:= 0, then Eq. (4a) takes the form V*(a%
A) = V*(**_ XX(xk
- x) / lk) -I- y
k{x
k, X ) ( (x - x
k_ ,)//
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where
(5d)
and
and &
= 1,
2,..., n.
Equati
ons
(5a)-
(5c)
form a
linear
syste
m of
(// + I)homog
eneou
s
equati
ons
for the
deter
minati
on of
&
0.1
2,...,
n. In
matrix
B
>
= Pk">Jsin(a,\/J, Ak
= Bk
cos (cokl
k) . * > 0, (6a
)
Bk
= PJh* Ak
col = (6b)
Bk
= inh(cu*/A), A k = B k cosh(a>*/t). u^ k = 1 , 2 , 3 -
1 An=A
n+ {a JP
n\
and
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notati
on it
has
the
form
|*(A)|{v,(A)}=0,
where
(7c) (7d) (7e)
{y
(A
)}
T
=
{
^(
A)
,
^(
A)
,..
.,
0 0 - 0 0 0
Ax -B 2 0 - 0 0 00 -Bj A
20 0 0
0 -B k* 0
0 0
0 ... 0 0 A
.(7b)
COMPUTING S rURMLIOUVILLE PRODLEMS 7
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v
^(
A)
}
(7
Q
is the
transp
ose of
{^(A)}
. In
order
that a
nonzer
o
solutio
n for |
y/(A)}
exists,
one
has
dct([AT(A)j = 0.
(8)
The
infinit
e
numbe
r of
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real
roots
of
transc
enden
tal
equati
on (8)
arc
the
eigenv
alues
of the
appro
ximate
proble
m of
Eqs.
(3a>-
(3f).
Their
compu
tationis the
core of
the
proced
ure to
be
descri
bed in
the
sectio
ns to
follow.
/
COMPUTING S rURMLIOUVILLE PRODLEMS 9
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M
IKHA1LOVANDVULCHANO
V
3.PROCEDUR
EFORTHECOMPUTATI
ONOF
THEEIGENVAL
UES
Witlri
ck
and
Willia
ms (3,
4]
devel
oped
an
efficie
nt
proce
dure
to
comp
ute
the
eigenf
reque
ncies
and
the
buckli
ng
loads
of
clastic
linear
skelet
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al
struct
ures.
Since
Eq.
(8) is
prese
nted
in the
same
form
as
that
utilize
d by
Wittri
ck
and
Willia
ms, it
is not
difficu
lt to
adapt
their
proce
dure
to the
proble
m
consid
ered
in the
prese
nt
paper.Willia
ms
and
Wittri
ck
have
shown
COMPUTING S rURMLIOUVILLE PRODLEMS 11
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that
the
numb
er of
positi
ve
eigen
values
A'(^)
lying
betwe
en
zero
and
some
prescr
ibed
positi
ve
value
o f t
X is
=
(9)7
where
N0( l )
is the
numb
er of
positi
ve
eigen
values
not
excee
ding
when
all
comp
onent
s of
the
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vector
corres
pondi
ng
to }
*'(/[)]
arc
zero
an
5(|
A'(2)])
denote
s the
"sign
count"
of
(A"(7.)|
.
To
find
Nn(X)
one
takes
into
accou
nt the
fact
that
since
all
comp
onent
s of
the
vector
are
zero,
the
syste
m of
equati
ons
COMPUTING S rURMLIOUVILLE PRODLEMS 13
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(3a)
degen
erates
into a
decou
pled
set
of
equati
ons
-
I-
co lv k(
xt
l ) =
0,xk- \
< x 0,namely,
d)kl
k= jn , j = . 1,2
(lib)
Thus,Ar
0(A)
isequalto thetotalnumbe
r ofeigenvaluesof thekind(Eq. (1lb))
Noa)= V int /Mi,
COMPUTING S rURMLIOUVILLE PRODLEMS 15
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' (12)
t I V 71 / /1
where
the
value
or the
functi
on
int(-)
is the
larges
t
intege
r not
excee
ding
the
value
of the
argum
e
n
t
of tlie
functi
on.'
< The
"sign
count" is
shown
(3| to
be
equal
10 the
numb
328
(10b)
(10c)
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er of
negati
ve
eleme
nts
along
the
main
diago
nal of
the
matri
x
|/v*(/)l
-
which
is the
triang
ulated
fo;m
of the
matri
x
(A'(2)|
, or
equiv
alent!
)', the
numb
er of
negati
ve
eleme
nts in
(he
sequence
COMPUTING S rURMLIOUVILLE PRODLEMS 17
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D\tr
Dn/ t
>n-n
wher
e Dk y
k =
1 , 2
d e n
o t e
s
the
Ath
order
p
r
i
n
c
i
p
a
l
m
i
n
or
o
f
t
h
e
m
a
t
r
i
x
[
K
(7.)
(13)
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\
,
D
0
=
l
t
>
,=
,
7
,
.
I
n
v
i
e
w
of
t
h
e
t
r
i
d
i
a
g
o
na
l
s
t
r
u
c
COMPUTING S rURMLIOUVILLE PRODLEMS 19
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t
u
r
e
o
f
t
h
i
s
m
at
r
i
x
,
o
n
e
h
a
s
"v
*
D* =Dk- ,(>4*+A
k_,)
~D^ Ik= 2,3,...,
n .J
T
h
e
c
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a
s
e
o
f
D
i
r
i
c
h
l
e
t
b
o
u
n
d
a
r
y
c
o
n
d
i
t
i
o
n
s
a
t
x
=
COMPUTING S rURMLIOUVILLE PRODLEMS 21
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x
0
o
r
a
t
x
=
x
n
o
r
a
t
b
ot
h
e
j
i
d
p
o
m
t
s
c
o
r
r
e
s
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p
o
n
d
s
t
o
/
J
0
=
0
o
r
/
a
n
d
/
?
=
0
,
t
h
us
r
e
s
u
l
COMPUTING S rURMLIOUVILLE PRODLEMS 23
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t
i
n
g
i
n
/
T
,
"
-
c
o
o
r
/
a
n
d
c
o.
I
n
a
n
y
o
f
th
e
s
e
e
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a
s
e
o
n
e
s
i
m
p
l
y
n
e
g
l
c
c
t
s
t
h
e
c
o
r
r
e
s
p
on
d
i
n
g
r
COMPUTING S rURMLIOUVILLE PRODLEMS 25
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o
v
T
a
n
d
c
o
l
u
m
n
I
i
s
j
h
c
y
do
n
o
t
i
n
f
l
u
q
ac
e
t
h
e
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e
l
i
m
i
n
a
t
i
o
n
p
r
o
c
e
s
s
d
e
s
c
r
i
b
e
d
b
y
E
q.
(
1
3
)
COMPUTING S rURMLIOUVILLE PRODLEMS 27
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a
n
d
i
m
p
I
y
/
^
&
r
X
)
=
0
o
r
/
a
nd
^
f
t
?
!
)
=
0
.A
s
b
o
t
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h
t
e
r
m
s
o
n
t
h
e
r
i
g
h
t
-
h
a
n
d
s
i
d
e
o
f
E
q
.
(
9
)
c
a
COMPUTING S rURMLIOUVILLE PRODLEMS 29
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n
b
e
c
o
m
p
u
t
e
d
,
o
n
e
h
a
s
t
he
f
o
l
l
o
w
i
n
g
al
g
o
r
i
t
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h
m
t
o
c
a
l
c
u
l
a
t
e
a
n
y
p
o
s
i
t
i
v
e
e
i
g
e
n
v
al
u
e
o
f
COMPUTING S rURMLIOUVILLE PRODLEMS 31
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t
h
e
a
p
p
r
o
x
i
m
a
t
e
p
r
o
b
l
e
mofEqs.
(3a)-(30:
S
t
e
p
I
.
P
r
e
s
c
r
i
b
e
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v
a
l
u
e
s
o
f
/
,
t
h
e
o
r
d
e
r
o
f
t
h
e
e
i
g
e
n
va
l
u
e
X
,
COMPUTING S rURMLIOUVILLE PRODLEMS 33
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d
e
s
i
r
e
d
a
n
d
e
f
,
t
h
e
(
a
b
so
l
u
t
e
)
a
c
c
u
r
ac
y
i
n
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X
,
;
s
e
t
l
o
w
e
r
b
o
u
n
d
f
o
r
X
,
,
A
,
=
0
,
u
p
p
e
r
,
COMPUTING S rURMLIOUVILLE PRODLEMS 35
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b
o
u
n
d
f
o
r
X
,
,
A
=
0
,
c
u
rr
e
n
t
v
a
l
u
e
o
f
t
h
e
A
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-
p
a
r
a
m
e
t
e
r
,
X
=
0
;
i
n
c
a
s
e
t
h
e
r
e
i
s
a
d
d
i
t
i
o
COMPUTING S rURMLIOUVILLE PRODLEMS 37
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n
a
l
i
n
f
o
r
m
a
t
i
o
n
c
o
n
c
e
r
n
i
ng
X
,
y
s
e
t
m
o
re
a
p
p
r
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o
p
r
i
a
t
e
v
a
l
u
e
s
f
o
r
t
h
e
p
a
r
a
m
e
t
e
r
s
l
is
t
e
d
t
o
COMPUTING S rURMLIOUVILLE PRODLEMS 39
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s
a
v
e
c
o
m
p
u
t
e
r
t
i
m
e
.
Step
2.
Defi
ne
the
incre
ment
S in
X, 5
* 0;
set 1
= I -f
A =
com
pute
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(^g^
)
Step
3. If
N{1)
> /,
go to
Step
5,
else
go to
Step
4.
Step
4. '
Para
mete
r I is
a
lowe
r
boun
d for
A,;
set
A,,=
2, go
COMPUTING S rURMLIOUVILLE PRODLEMS 41
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to
Step
2.
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S T E P 5. PARAMETER 1 ISANUPPERBOUNDFOR A,; SET AU
== I. . S T E P 6. COMPUTE AX = ABS(A - A,); IF
/, SET AU= 1 ANDGOTO STEP 6, ELSEGOTO STEP 9. S T E P 9. SET A, = I; GOTO STEP 6. S T E P 10. X, =
(AU-F A,)/2;Y330 MIKMAIL.OVANDVULCHANOV
into Eq. (5a), one obtains
V/.iO*,) = (0
+ 1 Pi))/B\- (Mb)
Equation (5b) can be rewritten as
VM + IO* / ) {{A k + A k 4 ,) - B k v/4_ ,(A(.))//;k4 ,, k I , 2................(I4c)
The last recurrence relation, Eq. (14c), enables one to compute all the components of the
vector {y(A,)| of Eq. (7f), which arc the values of the eigenfunction v,(a\ A,)
corresponding to the eigenvalue Afat the endpoints of the subintervals l
ktk= 1,2,..., n.
From Eqs. (5) it can be seen that in case the exact eigenvalues of the approximate
problem (Eqs. (3)) arc known and if the cocfficicnts A k and B k, k 1, 2,..., of Eqs. (6)
can be computed exactly, then Eq. (5c) should be satisfied identically. But to the extent
that the computation cannot be performed exactly and the eigenvalues can be computed
approximately in the sense of the algorithm described in Section 3, the left-hand side of
Eq. (5c) cannot become zero. Its value can be considered as an estimate for the global
error in the computation of the eigenfunction. For the special case of a Dirichlct
boundary, when Eq. (5c) is not considered, the accuracy estimate can be obtained by Eq.
(5b). written for k= n 1 and bearing in mind that that = 0. '
The norm r of the eigenfunction y/((.r, A,) can be defined as
= t V'L{x.X,)dx, (15)A r-I J.r,
and in view of Eqs. (4) and (6) one hasV
R v/J.Xv,;,.)^- = + vl){BlUPk-Ak)J ~~
+ (16) where y/k
Vi t
{xk, A,) and the constant (v '
k,
Ak, and D
kare calculated for A = some function \ j / , (x , X , ) that oscillates at least as
rapidly as the cigenfunction wanted, where X, is some upper bound for the eigenvalue
Af, on the basis of two different divisions of the interval [ a , b j , the second of therri j
COMPUTING S rURMLIOUVILLE PRODLEMS 43
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times finer than the first, and the corresponding values computed are equal in the sense
of some predetermined local error level, then the rougher of these two divisions of \a , b\
could be used to compute the desired eigenvalue and the corresponding cigenfunction
with approximately the same local accuracy.
Having in mind the procedure described in the previous section for the computation of
the eigenfunction, the following principal algorithm could be used to find the division of
[a, bJ:
Step 1. Prescribe the following quantities: upper and lower bounds zm a x
and mln
,
respectively, for the local error of the function \j/({x, A,) for an arbitrary subin- terval,
where / is the order of the eigenvalue A y and the corresponding eigenfunction y / , (x , X,)
to be computed; A, is an upper bound of /., and y/,(x, Af) oscillates at least as rapidly as
A,); / is the initial guess for the first subintcrval /, x, .v0; is the minimal admissible
length of an arbitrary subinterval of the division of [a, b) to be computed (the maximum
admissible length for the subintcrval is 6) itself): fix an integer j by means of which a
/times finer division of [,/>) could be generated.
Step 2. Set Jt = 1 ,([)= i0, .v = .
Step 3. Set l j = /// yl S
Step 4. Compute + A Xj} directly from Eqs. (14).
Step 5. Compute utilizing successively the values + /,,Af),-
V?, ( .x + 21J y
X v , ( x + i f -0b>Xt)> a n d E c l s -
S tep 6. If v\J \x + /, Af) and \ j / , (x + /, X,) are equal in the sense of the local error
prescribed, go to Step 9; else go to Step 7.Step 7. If / < /mln, go to Step 12; else go to Step 8.
Step 8. Decrease (set /=/,) or increase (set I = j ! ) the guess Tor the subintcrval
considered and go to Step 3.Step 9. Accuracy requirements arc satisfied; set l
h= /, x = x + / , k = k + I.
Step 10. Ifx + / = xy
go to Step 3; else go to Step 11.
Step 11. This is the correct exit of the algorithm; the last subintcrval has to be adjusted
so that / =xx in the sense of some error tolerance. At this place one has the number
of the subintervals for the eigenvalue considered n = k+\ and the division of |o, b\ into
n subintervals, thus being able to utilize the algorithms for the cdirtfnitation of A, and
A,) described earlier.
Step 12. This is the error exit, indicating in general that the accuracy requirements arc
not appropriately chosen.
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MJKLLA1LOVANDVULCHANOV
The estimate A,+
, for the next eigenvalue A( +
, can be obtained utilizing the division by
means of which A, and v/,(.v, X, ) have been computed and A, can be obtained starting
from some initial (e.g., uniform) division of [,/>).
A simpler approach can be used by finding an estimate A, for the eigenvalue Xm
, where
m is the order of the highest eigenvalue that has to be computed.
A check on the global error of the y function computed can be performed by making use
of Eq. (5c) or Eq. (5b) for A. = n 1 as was described in the previous section; thus theinequality should be satisfied
abs(- _,, Af) -f- A i j i ,{x , A,)) 0,O
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abs(
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having in mind that for thc case under consideration p{x)= I. This formulation
corresponds to [12] in thc sense lhat when a limit circle singularity is present, the
solution advances towards the singular endpoint following a trajectory prescribed by
the particular boundary condition imposed at thc singutar endpoint. Moreover, as Eqs.
(23) arc valid for co 2k
< 0, it is clcar that if q0
< 0 (depending on r(.r). which is
supposed bounded and nonnegativc), one may have a finite number of eigenvalues on
thc negative semiaxis, while otherwise thc whole spectrum will be continuous. J=!00
5.9S01 -0.00 304.453
80.S314
997.89.7
-263.279
COMPUTING S rURMLIOUVILLE PRODLEMS 49
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Moscow, 1960. |in Russian).
21.M. D.MIKHAILOV. "Transient Temperature Fields in Shells." Energy. Moscow. 1967. Jin Russian)
22.C. O. 1 (ORGANANDS. NUMAT-NASSER. Z. A / Igcu- . Moth . Phys . 30 (1979). 77.
23.'G.L7\ MUIIMIOLLANDANDM. H. COBBLE. I n t . J . Heat Mass T rans fer 15 (1872), 147.r. j. Nunc.e and w. n. Gill, Int. J . I leal , . Mass Transfer8 (1965).873.4. COMPUTATION OF THE EIGENFUNCTIONS
Once the eigenvalue X, has been located, one might want to know some values
of the cigenfunction V,(-Y,A,) at (possibly) previously prescribed positions. The
computation of the cigenfunction can be carried out making use of Eqs. (5) and (6). '
The choice of ft, and pjd fdx) satisfies
identically
Eq. (3e). Therefore, introducing .
5. CHOOSING STEP SIZES
The purpose of the procedure to be described is to construct the division of the interval
{
(20a)
and substitute Eq. (20a) into Eq. (la) to obtain
336
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