Paper Vulcanov

8/6/2019 Paper Vulcanov

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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].


3/50

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_ ,)//


6/50

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


7/50

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


9/50

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.

/



10/50

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


11/50

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



12/50

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


13/50

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



14/50

(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,



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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)


17/50

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



18/50

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)


19/50

\

,

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



20/50

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


21/50

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

=



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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


23/50

p

o

n

d

s

t

o

/

J

0

=

0

o

r

/

a

n

d

/

?

=

0

,

t

h

us

r

e

s

u

l



24/50

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


25/50

a

s

e

o

n

e

s

i

m

p

l

y

n

e

g

l

c

c

t

s

t

h

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c

o

r

r

e

s

p

on

d

i

n

g

r



26/50

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


27/50

e

l

i

m

i

n

a

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i

o

n

p

r

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c

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s

s

d

e

s

c

r

i

b

e

d

b

y

E

q.

(

1

3

)



28/50

a

n

d

i

m

p

I

y

/

^

&

r

X

)

=

0

o

r

/

a

nd

^

f

t

?

!

)

=

0

.A

s

b

o

t


29/50

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



30/50

n

b

e

c

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m

p

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t

e

d

,

o

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e

h

a

s

t

he

f

o

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o

w

i

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al

g

o

r

i

t


31/50

h

m

t

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a

l

c

u

l

a

t

e

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n

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p

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t

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i

g

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al

u

e

o

f



32/50

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


33/50

v

a

l

u

e

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o

f

/

,

t

h

e

o

r

d

e

r

o

f

t

h

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g

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n

va

l

u

e

X

,



34/50

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


35/50

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

,



36/50

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


37/50

-

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



38/50

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


39/50

o

p

r

i

a

t

e

v

a

l

u

e

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r

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p

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t

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t

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40/50

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


41/50

(^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



42/50

to

Step

2.


43/50

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



44/50

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 subinterval,

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.


45/50

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


47/50

abs(


48/50

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



50/50

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

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