Internat. J. Math. & Math. Sci.
567
Vol. 9 No. 3 (1986) 567-575
NUMERICAL METHODS FOR APPROXIMATING
EIGENVALUES OF BOUNDARY VALUE PROBLEMS
KIAZ A. USMANI
Department of Applied Mathematics
University of Manitoba
Winnipeg, Manitoba R3T 2N2
Canada
and
MOHAMMAD ISA
College of Engineering
King Abdul Aziz University
P.O. Box 9027
Jeddah, Saudi Arabia
(Received November 21, 1985)
ABSTRACT.
This paper describes some new finite difference methods
for the approxi-
mation of eigenvalues of a two point boundary value problem associated with a fourth
order
linear
differential
equation
of the type
(py’)
(q y’)
+ (r
s)y
O.
The smallest positive eigenvalue of some typical eigensystems is computed to demonstrate the practical usefulness of the numerical techniques developed.
KEY WORDS AND PHRASES.
Band matrices, Deflation, Finite-dlfference methods, General-
ized matrix eigenvalue problem, Inverse power iteration, The Smallest eigenvalue of a
matrix eigenvalue problem, Two-point boundary value problems.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
65L15.
In this paper we will consider the fourth order linear differential
equation
[p(x)y’(x)]
Ly
a
x
b,
[q(x)y’(x)]’ + [r(x)
%s(x)]y(x)
O.
associated with one of the following pairs of homogeneous boundary condi-
tlons
y(a)
y"(a)
y(b)
y’(b)
0
y(a)
y’(a)
y(b)
y’(b)
0
(1.2)
Boundary value problems of the type (I.I)-(I.2) and/or (I.I)-(l.3) together with some
of
their modifications
electrical engineering,
functions
p(x), r(x)
occur
see
and
frequently in applied mathematics, modern physics and
[5, 7, 8, 12].
s(x)
In (I.I), we assne that the real-valued
are continuous on
[a,b]
and satisfy the conditions
568
R.A. USMANI and M. ISA
p(x) C C2 [a,b], q(x) CC’[a,b] and p(x), q(x), s(x)
Recently,
>
O, r(x)
[a,b
0, x
techniques of order 2 and 4 have been developed for computing
numerical
of
values
approximate
>
for the boundary value problem (1.1-(1.3) with
%
p(x)
l,
In the fourth order method, the problem is dlscretlzed to yield
a generalized seven-band symmetric matrix elgenvalue problem of the form
q(x)
E
O,
Ah BY
AY
A
where
[1,2].
see
(1.5)
be converted
excessive
amount
with elgenvector
s(x).
matrix depending on the function
can
%
is an approximate value of
Y
B
and
is a diagonal
Consequently, the eigenvalue problem (1.5)
to
the usual standard problem of the type
of
computational
effort.
There are at
AY
MY
without any
several areas of
present
research activity surrounding the development and analysis of numerical methods for
A
approximating
see
satisfying the generalized matrix elgenproblems of the type (1.5),
[4, 9, I0, 11, 13].
Usmanl has developed [14] some new finite difference methods of order 2 and 4 for
of the
elgenvalues
computing
0, q(x)
(for p(x)
y’(a)
y(a)
differential
(I.I)
system
with
p(x)
E
I, q(x)
0
see [15]) associated with the boundary conditions
I,
E
y"(b)
y’"(b)
O.
(1.6)
The purpose of this work is to present some new finite difference methods for
computing approximate values of
(1.1)-(1.3).
A
where
%
for the boundary value problems (1.1)-(1.2) and
These methods lead to generalized elgenvalue problems of the form (1.5)
is a flve-band or seven-band matrix and
We
matrix.
preface
the
numerical
methods
by
B
is a diagonal positive definite
some
analytical
properties
of
the
elgenvalues and elgenfunctions of the boundary value problems under discussion.
2.
PROPERTIES OF EIGENVALUES AND EIGENFUNCTIONS.
stand for any of the two boundary value problems (1.1)-(1.2) and (I.I)-
Let
(1.3).
THEOREM 2.1.
If
Y2(X)
%1
and
%2
are two
distinct
eigenvalues
of
the
problem
I[
aand
Yl(X),
are the corresponding eigenfunctions, then
b
s(x) yl(x) y2(x) dx
0.
a
PROOF.
The proof is a direct consequence of Green’s identity (see [3], p. 86)
and the boundary conditions (1.2) and (1.3).
LEMMA 2.2.
If y(x) is an elgenfunction belonging to the elgenvalue
boundary value problem
PROOF.
,
then y(x)
%
of the
is an eigenfunction belonging to the eigenvalue
The proof is trivial and follows from
LY
0.
EIGENVALUES OF BOUNDARY VALUE PROBLEMS
,
THEOREM 2.3.
the boundary value
of
The eigenvalues
problem
569
together with ([.4),
are
In fact
real.
b
[p(y..)2 + q(y,)2 +
]dx
r
a
b
>
(2.2)
0.
s y2dx
a
,
+ i,
C R, be an eigenvalue of the problem I[ with
u(x) + iv(x) where u(x) and v(x) are real-valued functions.
Let
PROOF.
y(x)
eigenfunction
u(x)
y(x)
respect to the eigenfunction
I with
Now, from Theorem 2.1, it follows
is also an eigenvalue of the problem
From Lemma 2.2, it follows that
iv(x).
that
b
b
f
0
s(x)y(x)(x)
f
dx
a
>
s(x)
because
0
s(x)
a
y(x)
and
#
0.
ly(x) 12dx >
(2.3)
0
The contradiction in (2.3) suggests that
%
can-
not be complex, hence it is real as required.
In order to prove (2.2), we multiply (I.I) by y(x) and integrate the resulting
equation twice from
a
x
x
to
We consequently arrive at (2.2) on using the
b.
conditions (I.I)-(1.4).
A SECOND ORDER METHOD FOR COMPUTING % FOR (I.I)-(1.2).
3.
For
We
a
shall
positive
b
> 4, let h n+
Y(Xi)’ Pi P(Xi)
integer
also designate
n
Yi
a
and
x
i
a + ih, i
0(I) n+l.
We discretize the boundary
etc.
value problem (1.1)-(1.2) by the following set of difference equations
(a)
(b)
+
P2
h4slY
+
[4Pl
Pi_lYi_ 2
qi+
I/2
[2Pl
+
2P2
+
h2q3/2]Y2
/
P2Y3
0(h4),
[2Pi_l
(3.1a)
+
h4ri]yi
2Pi+l
+
h2
[2Pi+ 2Pi+l
qi+
1/2]Yi-I
+
[Pi-I
+
4Pi
+
Pi+l
+
h2(qi-I/2 +
h2qi+I/2]Yi+l + Pi+lYi+2 kh4siyl + O(h6)’
(3.1b)
2(1)n-l,
i
(C)
+
h2(ql/2+ q3/2 + h4rl]Yl
+
2Pn + h2qn-I/2]Yn-I
4
]Yn X h SnYn + 0(h4).
Pn-lYn-2- [2Pn-I
4
+ h r
+
+
[Pn-I
+
4Pn
+
h2(qn-I/2 + qn+I/2
(3.1c)
The difference equation (3.1b) os pbtaomed bu wrotomg (I.l) at
form
h2(py") i
h2(qy ’) i + h2(r i
As i )Yi
0,
x
xl,
in the
R.A. USMANI and M. ISA
570
+ h2 (r i-
(PY")i-I- 2(py")i+ (PY")i+I h[(qY’)i+I/f (qy)i- I/2
ksi)Yi
--O(h4 ),
!/2(Yi
Yi-! )]
or
Pi_l[Yi_2
2Yi_
+
Yi
+
2Yi-!
+
Yi+2
Pi+l [Yi
+ h4(r
2y
i
+
Yi+l
h2[qi+ I/2(Yi+l
qi-
Yi
0(h6),
sl)Y i
i
2Pi[Yi_
(3.2)
which can be arranged in the desired form (3.1b).
The difference equations (3.1a) and
(3.1c) are introduced so that the resulting coefficient matrix in (3.1) is a five-band
in the form
x
In order to obtain (3.1a), we write (I.I) at x
symmetric matrix.
h2(qy ’) +
h2 (PY")
+
2PlY
p0Y0-
h2
As
(rl
)Yl
0,
-ql/2Yl/2)
P2Y2- h(q3/2y3/2
+
h2(r
Isl )Yl
or
-2Pl(y0
q
+
2y
Y0 )]
I/2 (Yl
+
Y2
P2(Yl
h4(rl
+
2y
2
%sl)Yl
+
Y3
h2[q I/2(Y 2
yl
0(h4)"
(3.3)
The preceding equation is easily arranged in the form (3.1a).
The difference equation
(3.1c) is developed in an analogous manner by writing (I.I) at
linear equations (3.1) can be written in matrix form
4R)y
4Sy
(A + h
h
where A is a symmetric
x
n
The system of
(3.4)
+ t(h)
flve-band matrix.
[y!
The matrices
y2...yn]T,t(h)
are
diagonal matrices and y
and
i
2(1)n-l.
t.1
Thus, our method for computing approximations
0(h6),
x
R
diag
T
t2...tn].
[t
A
for
l
(rl),
Here
S =diag (s i)
tl,
t
0(h 4)
n
satisfying (I.I)-(1.2)
can be expressed as a generalized five-band symmetric matrix eigenvalue problem
(A +
h4R)y-- Ah4Sy.
(3.5)
It can be proved that A is a positive definite matrix and hence for any stepslze h
the approximations
r
>
0.
A
for
by (3.5) are real and positive for all p, q, s
That our method provides
established following Grigorieff
0(h
[6].
2)
convergent approximations
A
for
>
k
>
0,
0 and
can be
We omit the long and tedious details of conver-
gence proof for brevity.
Normally, only one or a few of the extreme elgenvalues of (1.1)-(1.2) are needed
in applications.
In what follows, we will compute only the smallest eigenvalue of the
571
EIGENVALUES OF BOUNDARY VALUE PROBLEMS
We consider the eigenvalue problems
system to illustrate our method based on (3.1).
[(I +
x2)y"]
[(I +
with boundary conditions
(1.2)
[eXy ’]’
[eXy"]
x2)y’]
at
smallest eigenbalue
2
k(l +
x)4]y
a
cos x]y
I.
0, b
a
(3.7)
0
We computed approximations to the
of these eigensystems by our method (3.1)
A
(3.6)
0
(I + x)
I, and
O, b
+ [sin x
with boundary conditions (1.2) at
I___
+
or equivalently
It is evident from
The corresponding relative errors are shown in Table I.
conthe entries of the accompanying table that our numberical method provides 0(h
that
assumed
we
errors,
relative
the
vergent approximations. In computing
(3.5).
with
2)
h
2
-8,
kl A1
cannot be obtained by analytical methods
k
because exact value of
for these eigensystems.
TABLE I
Observed relative errors for h
h
Problem
(3.6)
2
2
2
2
2
2
(3.7)
2
2
2
2
2
2
* We
-3
-4
-5
-6
-7
-8
-3
-4
-5
-6
-7
-8
2
A
-m
m
3(1)7
Relative Error
1
24.634,681
2.448-2*
25.085,489
6.068-3
25.199,984
1.497-3
25.228,721
3.563-4
25.235,913
7.125-5
25.237,711
0.0
19.548,553
2.551-2
19.921,847
6.294-3
20.016,196
1.551-3
20.039,847
3.691-4
20.045,764
7.380-5
20.047,244
0.0
write 2.448-2 for 2.44fl x i0
-2
FOR (I.I)-(1.3).
of the development of our numerical method but we redetails
We omit the lengthy
satisfying
mark that a second order method for computing approximations A to
4.
A METHOD FOR COMPUTING
(I.I)-(1.3)
is based on
(A’ +
h4R)y
Ah
4SY
(4.
572
R.A. USMANI and M. ISA
A’
where
differs from
A’
The first row of
A
introduced in (3.5) in the first and the last rows only.
is
[[2P0 + 4Pl + P2 + h2(q I/2 + q3/2 )}
and the last row of
[0
0
Pn-I
A’
[2Pl
h2q3/2} P2
0...0]
A’
is a five-band symmetric matrix.
for computing approximations
A
to
satisfying
is based on
(A" +
A",
+
-2Pn-I + 2Pn + h2 qn-I/2 {Pn-I + 4Pn + 2Pn+l + h2(qn-I/ qn+I/2)}]"
Another second order method
where
2P2
is
Again, as in the previous section, the matrix
(I.I)-(1.3)
+
h4R)y-- Ah4Sy
(4.2)
as before, differs from
The first and the last rows of
[{4P0 + 4Pl
+
P2
+
h2
(q
A"
A
in the first and the last rows only.
are
I/2+ q3/2 )}
/2P0 + 2Pl
+ 2P +
2
h2q3/2
P2
0...0]
and
[0...0
Pn-1
{2Pn-1+ 2Pn +I/2Pn+I +
h2qn-I/2
[Pn+l + 4Pn + 4Pn-
The numerical results for computing
respectively.
better than those based on (4.1), but the matrix A"
A
+
h2 (qn_I/ q n+I/2)}]
based on (4.2) are slightly
is no longer a symmetric matrix.
We illustrate our methods based on (4.1) and (4.2) by computing A
satisfying
(3.7) and the boundary conditions (1.3) with a 0 and b
I.
The smallest elgenvalue A
is computed employing inverse power iteration method and
the numerical results are summarized in Table II
Table II
Observed relative errors for Problem (3.7)-(1.3) with a
Method
h
2
2
2
2
2
2
(4.2)
2
2
2
2
2
2
-3
-4
-5
-6
-7
-8
-3
-4
-5
-6
-7
-8
A
I
841.550
0, b
i
Relative error
1.374-1
925.773
3.393-2
949.245
8.364-3
955.283
1.990-3
956.803
3.980-4
957.184
0.0
922.198
3.805-2
949.282
8.428-3
955.404
1.967-3
956.847
4.553-4
957.197
8.983-5
957.283
0.0
EIGENVALUES OF BOUNDARY VALUE PROBLEMS
5.
573
HIGHER ORDER METHODS FOR SPECIAL CASE OF (I.I).
In this section we consider the linear differential equation
y
(4)
% s(x))y
q(x)y" + (r(x)
0
associated with the boundary conditions (1.2)
h
size
and the sequence
{x i}
or (1.3).
For
n
>
7,
let the step-
be defined as in section 3.
The boundary value problem (5.1)-(1.2) is discretized by the following
Case I.
difference equations
12h2q! + 6h4rl)Y (38 + 6h4ql)Y 2 +
(38 + 8h2q2)Yl
(56 + 15h2q2 + 6h4r2)Y
2
(44 +
(a)
(b)
/
+ (12 + h2 q2)y
(c)
’/t-3
(39
(d)
6Ah4SlYl
Y4
(39 +
8h2q2)Y 3
Y2
(39 + 8h2q3)i_ + (56 + 15h2qi + 6h4ri)t
h2qi)i_
2
2
+ 8h2qi)Yi+ + (12 + --h2qi)Yi+2
Yi+3 6Ah4siYi, i 3(I)n-2,
2
+ (12 +
6h4r n-I )Yn-Yn-3
4
6Ah s2
3
+ (12 +
Yn-4
(e)
Y5
4
2
12Y
+
--h22 qn_l)Yn_3(38 +
(38 +
12Yn-2-
(39 +
8h2qn-I )Yn-2 +
(5.2)
5h2qn-I +
(56 +
8h2 qn-I)Y n 6Ah4s n-I Yn-
6h2qn)Yn-I +
(44 +
6Ah4SnYn"
12h2qn + 6h4rn )Yn
The system of equations (5.2) can be written in the form
h2QB
(A +
+
6h4R)y
6Ah4Sy
(5.3)
where
j3
A
I,
+
and
b
bll
6j2
(b
B
nn
(Jmn)
J
12, b
is a tridiagonal matrix with
Jmm
o
is a five-band matrix so that
1,2
bn, n-I
8,
bij
I/2’
6
and
i-
j
2.
The boundary value problem (5.1)-(1.3) is discretized by the difference
Case 2.
equations (see [I] also)
(a)
(b)
2h2q0 + 12h2ql + 6h4rl)Y (42 + 6h2ql)Y2 +
I13
+ 8hZqz)Y +
(42 +_
15h2q2 + 6 h4r2 )Y2
h2qo
---+
2
(76 +
2
+ (12 + h__
2
(c)
q2)Y4
Y5
12Y
3
(39 +
Y4
6Ah4SlYl
8h2q2)Y 3
6Ah4s2Y2
The same equation as (5.1c),
i
3(I) n-2,
(5.4)
R.A. USMANI and M. ISA
574
(d)
-Yn-4
2
+ (12 + h__
2
qn_l)gn_3
(39 +
8h2q n-I )Yn-2
13
--"
+
+
15h2qn-I +
8h2qn_l + qn+l)Yn =6Ah4s n-I YnYn-3 + 12Yn-2 (42 + 6h2qn )Yn-I + (76 + 12h2qn + 2h2qn+l + 6h4rn)Y n
6h4rn-l)Yn-I
(e)
2
(42 +
_4
6Ahs Y n
n
and (5.1)-(!.3)
with
+ x
q(x)
for the boundary value problems (5.1)-(1.2)
A
We computed approximations to
2,
r(x)
(l+x)
A
In both cases we assumed that
with
h
s(x)
2’
2
-8
(I + x)
4,
0, b
a
1.
It is abundantly clear from
the entries of the Table III that both our methods based on (5.2) and (5.4) are fourth
order methods.
TABLE III
Observed relative errors
Problem & method
A
h
Problem (5.1)-(1.2)
based on method
(5 2)
2
2
2
2
2
2
Problem (5.1)-(1.3)
based on method
(5 4)
2
2
2
2
2
2
ACKNOWLEDGEMENT:
Canada.
The
author
(Oh.4)?.converen.t .nu..er.ical techniqu.es)
This
also
-3
-4
-5
-6
-7
-8
-3
-4
-5
-6
-7
-8
19.807,299
7.203-4
19.820,723
4.256-5
19.821,518
2.469-6
19.821,564
1.324-7
19.821,567
5.515-8
19.821.567
0.0
93.932,101
3.772-3
94.265,002
2.268-4
94.285,062
1.398-5
94.286,299
8.632-7
94.286,376
4.800-8
94.286,380
0.0
work was
wishes
Relative error
I
to
part
by
a grant
supported
in
thank Mr.
Manzoor Hussain
implementation of the methods developed in this paper.
for
from NSERC of
the
numerical
EIGENVALUES OF BOUNDARY VALUE PROBLEMS
575
REFERENCES
I.
CHAWLA, M.M.
2.
A
New
Fourth Order Finite Difference Method for Computing EigenOrder Two Point Boundary Value Problems,IMA J. of Numerical
Analysis, 3 (1983), 291-293.
CHAWLA, M.M. and KATTI, C.P. A New Symmetric Five Diagonal Finite Difference
Method for Computing Eigenvalues of Fourth Order Two Point Boundary Value Problem,
J. Computational and Applied Maths., 8 (1982), 135-136.
values
of
Fourth
3.
CODDINGTON, E.A. and LEVINSON, N. Theory
McGraw-Hill Book Company, New York, 1955.
4.
EDWARDS, Brunce. Computation of Eigenvalues Using Two Point Boundary Value
Problem Codes, Internat. J. Numer. Methods Engrg., 18 (1982), 1736-1738.
FOX, L. The Numerical Solution of Two Point Boundary Value Problems in Ordinary
Differential Equations, Oxford University Press, Oxford, 1957.
5.
of
Ordinary
Differential
Equations,
6.
GRIGORIEFF, R.D. Diskrete Approximation
Mathematik, 24 (1975), 415-433.
7.
HILDEBRAND, F.B. Methods of Applied Mathematics, Prentice-Hall, Engelwood Cliffs,
N.J., 1952.
HILDEBRAND, F.B. Advanced Calculus for Applications, Prentic-Hall, Engelwood,
Cliff, N.J., 1964.
KELLER, H.B. On the Accuracy of Finite Difference Approximations to the Eigenvalues of Differential and Integral Operators, Numer. th., 7 (1965), 412-419.
KELLER, H.B. Numerical Methods for Two Point Boundary Value Problems, Blaisdell,
Waltham, Massachusetts, 1968.
MCCORMICK, STEPHEN F. A mesh refinement method for AX
ABX, Math. Comp. 36
(1981), 485-498.
REISS, E. J., CALLEGARI, A.J. and AHLUWALIA, D.S. Ordinary Differential Equations
8.
9o
I0.
II.
12.
yon
Eigenwertproblemen II, Numerische
with Applications, Holt, Rinehart and Winston, New York, 1976.
13.
TURYN, Lawrence.
Perturbation of Two Point Boundary Value Problems with EigenParameter in the Boundary Conditions, Proc. Roy. Soc. Edinburgh Section A,
94 (1983), 213-219.
value
14.
USMANI, Riaz A.
Finite Difference Methods for Computing Eigenvalues of Fourth
Order Boundary Value Problems, Int. J. Math. and Mathematical Sciences, 9 (1986),
137-143.
15.
USMANI, Riaz A. Some New Finite Difference Methods for Computing Eigenvalues of
Two Point Boundary Value Problems, Computers and Maths. with Applications,
(1985), 903-909.