Internat. J. Math. & Math. Sci.
([986) 137-143
137
Vol. 9 No.
FINITE DIFFERENCE METHODS FOR COMPUTING EIGENVALUES
OF FOURTH ORDER BOUNDARY VALUE PROBLEMS
RIAZ A. USMANI
Department of Applied Mathematics
University of Manitoba
Winnipeg, Manitoba, Canada
R3T 2N2
(Received September 7, 1984)
ABSTRACT.
This brief report describes some new finite difference methods of
order 2 and 4 for computing eigenvalues of a two point boundary value problem
associated with a fourth order linear differential equation
(p(x)
q(x))y
-
y(4)
0.
These methods are derived from the formula
h4yl(4)
(64
7
66
68
"’’’)Yi
Numerical restilts are included to demonstrate practical usefulness of our methods.
KEY WORDS AND PHRASES. Central difference formula, Finite difference methods, Generalized
symmetric eigenvalue problem, Positive definite matrix, Two-point boundary value problem.
1980 AMS SUBJECT CLASSIFICATION CODE: 65L15.
i.
INTRODUCT ION
We shall consider the fourth order linear differential equation
y(4)
[p(x)
q(x)]y
0
(1.1)
associated with one of the following pairs of homogeneous boundary conditions:
(a)
y(a)
y’(a)
y(b)
y’(b)
0
(b)
y(a)
y"(a)
y(b)
y"(b)
0
(c)
y(a)
y’(a)
y"(b)
y"’(b)
(1.2)
0
Such boundary value problems occur frequently in applied mathematics, engineering and
modern physics, see
[1,2,3].
In (1.1), the functions
p(x)
q(x)
E
C[a,b]
and they
satisfy the conditions
p(x)
_>
0
q(x)
0
x
[a,b]
(1.3)
Recently, Chawla and KaCi [4] have developed a finite difference method of order
2 for computing approximate values of
for a boundary value problem (l.l)-(l.2a).
For the same problem, a fourth order method was developed by Chala [5] which
leads o a generalized seven-band symmetric matrix eigenvalue problem.
The purpose of this note is to present some new finite difference methods for
computing approximate values of
and (l.l)-(l.2c).
for the boundary value problems (l.l)-(l.2b)
These methods lead to generalized five-band and seven-band
symmetric matrix eigenvalue problem and provide
0(h
2)
and
0(h
4)
-convergent
38
R.A. USMANI
approximations for the eigenvalues.
2.
A SECOND ORDER METIIOD FOR COMPUTING X FOR (i. I) and (l.2b)
For a positive integer N
(b-a)/(N+ iI and x.1
4, let h
We shall designate
O(1)N
P(xi) and qi
Yi Y(xi)
Pi
a
ih
The
q(xi)"
boundary ralue problem (l.l) and (l.2b) is discretized by the difference equations
(a)
-2Y0 SYl -4Y2 Y3
(b)
64yi h4y 4)
t.
-h2y; h4[- y (4)
y
4)
+
tl
2(1)N-
i
(2 1)
t
Note that the truncation errors
h6y (6)
59
3-
(x i_ 2,xi +2
i
h6y (6) (N)
59
3--6-d
are
(a,x3)
(l
h6y (6) (i)
ti
I(1)N
i
t.
N
CN
2(1)N-
(2.2)
(XN- 2 ’b)
The formula 2.1(b) is obtained from the well-known central difference formula
h4yi(4) [64
7
6
8
41
7560
2-
68
(2 3)
]Yi
2,3,4
(i
The discretizations 2.1(a) and 2.1(c) are introduced so that the resulting coefficient
matrix in (2.1) is a five-band symmetric matrix.
The system of linear equations
(2.1) can be written in matrix form
j2
where
Xh4Qy
h4p)y
(j2
(2.4)
t
is a symmetric five-band matrix and
J
(Jrs)
is a tridiagonal matrix
such that
r
2,
s
(2.5)
Jrs
The matrices
P
0,
otherwise.
and
Q
are diagonal matrices
P
diag[Pl P2"
Y
[Yl Y2
Q
PN
--diag[ql q2
qN
and
YN
T
t
[t
t
2
tN]
T
A for X of (1.1)- (1. 2b) can be
expressed as a generaiized five-band symmetric matrix eigenvalue problem
Thus, our method for computing approximations
(j2
h4p)
In fact, the matrix
h
0
Ah4Q
J
2
is a positive definite matrix and hence for any step-size
the approximations
_>
(2.6)
A for
X
by (2.6) are real and positive for all
2)
0 and q(x)
p(x)
0
That our method provides 0(h
convervent approximations
A for t can be established following Grigorieff [6]. We omit the detaiIs of
FINITE DIFFERENCE METHODS FOR COMPUTING EIGENVALUES
139
convergence proof for brevity.
3.
A FOURTI! ORDER IETHOD
The boundary value problem (1.1)-(1.2b) is now discretized by the following
difference equations
(a)
(b)
12Y3 Y4 -8h2y
10Y0 38Yi 56Y2 39Y3 12Y4 Y5
-17y 0
44YI
38Y2
{c)
{6
6)y
4
(4)
4
[Y Yo
6h4y (4)
h
6y
3-F6
h4[’lv4)
h2y
h6v((,)
"o
7
8 (8)
h
40
13
4)] -6h
6y6)
+
3{1)N= 2
i
Yi
6y
As in [7], the derivation of (3.1c) is immediate from (2.3), on truncating the
infinite series on the right of equality sign after the two terms. The additional
difference equations (3.1a, b, d, e) are chosen so that the resulting matrix associated
with the system of linear equations in (3.1) is a seven-band symmetric matrix.
turns out that our method for computing approximations
It
^
for
of (l.l)-(l.2b) can
be expressed as a generalised seven-band symmetric matrix eigenvalue problem
[(6j2 j3)
The matrix
6h4p]
+
j3
6j2
6Ah4Q
(3.2)
is a positive definite matrix and hence for any step-size
A for
by (3.2) are real and positive for all
As before, it follows from the results of Grigorieff (1975)
that our present method provides 0(h
-convergent approximations A for
h
p(x)
the approximations
0
0
and
q(x)
0.
4)
4.
METHODS FOR COMPUTING
For this section, let
value problem
(a)
(b)
FOR (l.1)-(l.2c)
a + ih
(b-a)/N and x
0(1)N.
(l.l)-(l.2c) is discretized by the following scheme
4y 0
7y
h
4Y2 Y3 2hy
64yi h4yi(4)
h4y (6) (i
4
h y
4)
i
3
h
(-
y
x
(xi- 2
...)
2)
2(13N- 2
(C)
YN- 3
(4.1)
4YN- 2 5YN(-j
(d)
YN- 2
2YN-
The boundary
YN
2y N
s (s)
h
YN
II
_h2y
h
4 (43
YN-
"’’)
h2y h3y
This system can be written in matrix form
7
n
4 (4)
YN
(- h 5y(N 53
140
R.A. USMANI
(A
h4Qy
h4p)y
t
A
and our method for computing approximation
for
X
of (I.I)-(1.2c) can be
written as a generalized five-band symmetric matrix eigenvalue problem
(A
h4Q
h4p)
(4.3)
where
7
-4
-4
6
-4
-4
6
-4
-4
6
0
(4.4)
-4
6
-4
-4
5
-4
-2
-2
It can be established (see Appendix A) that the matrix
matrix and the approximations A for X by (4.3) are
A
is a positive definite
0(h
2)
-convergent.
A third order method is obtained on discretizing (l.l)-(l.2c) by the difference
equations
(a)
(b)
27
--Y0- 42YI
113
--Y2- 39Y3
h4 (a" y4)+
(c)
(664
(d)
-YN-5
66)y i
(4)
+
7
hSyS)
-YN-4
-YN-3
(ga-
8 (8)
34yiq_
12YN- 3 34YN- 2 36YN-
13YN
-
26 ,4 (4)
YN-
s
+
--’)
3(1)N- 3
5 hSyN(S)
591_ h4yN(4)- 2
n
h6y6)
(- h Yi
Yi
(-
2h3y,
-4h3y
(f)
4
6_h
6y 4))
* ...)
12Y4- Y5 -3hy
12YN-4 39YN-3 54YN-2
+
(e)
--
-45Y0 76YI -42Y2 12Y3 Y4 24hy
h4(-y 4) 6y 4)) 4 hSy5) (- h6y6)
hSy
(4.5)
8YN -h2y
...)
4h2y
...)
8YN-2 13YN-1 6YN 5h2y 4y’
-h17 4.yN(4) 2372 h6y(6)(SN) N (XN- 3 ’b)
This third order method gives rise to a generalized seven-band symmetric matrix
eigenvalue problem
(B
where
h4p) Ah4Q
(4.6)
141
FINITE DIFFERENCE METHODS FOR COMPUTING EIGENVALUES
7-6
i
12
1
56
-39
12
1
1
12
-39
56
-39
12
i
12
-39
54
-34
8
1
12
-34
36
-13
1
8
-13
6
1-41-42
m
5.
12
-39
2
-42
113
2
-39
(4.7)
1
NU [ERICAL ILLUSTRATIONS
To illustrate our methods for order 2 and 4 for the approximation
(l.2b),
1
of (l.l)-
consider the eigenvalue problem
"e
y
(4)
1
(i
(5.1)
y"(0)
y(0)
0
y
+x)4
y(b)
l(b)
The smallest eigenvalues
y"(b)
for
0
1,2
b
-
416.324,564,86...
%(i)
are
I(2)
646.269,207,... respectively. We computed approximations ^(i) and
-m
^(2) by our methods (2.1) and (3.1) applied to the problem (5.1) for h
2
m
3(1)6. The corresponding errors 1
are shown in Table I.
It s easily
verified that our methods based on finite difference approximations (2.1) and (3.1)
and
do provide 0(h
2)
and
0(h) 4-
convergent approximations for the smaiiest eigenvalue
of (5.1).
TABLE I
b
Error
h
1/8
1/16
1/32
1/64
l(b) based on the method
1
(2.1)
(3.1)
2.68-2*
1.09-4
6.76-3
8.34-6
1.69-3
5.46-7
4.23-4
3.29-8
1/8
1/16
2.61-2
2.50-3
7.13-3
i. 70-4
1/32
1/64
1.82-3
1.08-5
4.58-4
6.39-7
*We write
2.68-2 for 2.68
i0
-2
We now illustrate our methods for the approximation of
approximating the value of the smallest eigenvalue
y
+
(I +
(1 + x)
y(0)
y’(0)
v"(1)
4
"(i)
I
1
I
of (i.i)-(1.2c) by
satisfying
0
y
(5.2)
0
142
R.A. USMANI
where
135.320,349,281,57
-m
We list the errors
1
for h
2
It is readily verified that the relative e rors
based on the finite
difference scheme (4.1) are
convergent and likewise the relative errors
m
1
3,4 5,6.
based on the scheme (4.5) are
O(h2)O(h3)-
convergent.
-
TABLE I I
Error
I
(4.1)
1/8
1/16
1/32
i/64
1/128
based on methods
(4.5)
1.40-2
2.80-3
3.41-3
8.41-4
3.50-4
4.31-5
4.78-6
2.08-4
5.13-5
APPENDIX A
j2
it follows that the matrices
positive definite matrix,
in equations
introduced in (2.4) is a
J
It is well known that the tridiagonal matrix
(6j2+j3)
and
introduced
(2.6) and (3.2) respectively are also positive definite matrices. In
A given by (4.4) is a positive
order to establish that the real symmetric matrix
suffices
definite matrix, it
to prove that the
(N- i)
principal minors
6
-4
1
6
-4
i
-4
5
-2
-4
6
-4
i
i
-2
i
1
-4
6
-4
i,
1
are each equal to i
and
AI
1
-4
6
1
-4
1
-4
5
-2
1
-2
1
2
ACKNOWLEDGEMENT
This work was supported in part by a grant from the Natural Sciences and
Engineering Research Council of Canada.
We also acknowledge the assistance of
Mr. Naiyer A. Usmani for making numerical calculations presented in Tables I and If.
REFERENCES
i.
FOX, L.
2.
HILDEBRAND, F.B.
3.
REISS, E.L.
4.
The numerical solution of two-point boundary value problems in
differential equation, Oxford University Press, Oxford, 1957.
ordinary
Advance Calculus for Applications, Prentice-Hall Inc.,
Inglewood Cliffs, N.J., 1964.
CALLEGARI, A.J. and AHLUWALIA, D.S. Ordinary Differential
Equations with Applicatins, Holt, Rinhart and Winston, New York, 1976.
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. Comput. and Appl. Math.,
(1982), 135-136.
FINITE DIFFERENCE METHODS FOR COMPUTING EIGENVALUES
143
A new fourth-order finite difference method for computing eigenvalues of fourth-order two-point boundary value problem, IMA Journal of
Numerical Analysis, 3 (1983), 291-293.
5.
CHAWLA, M.M.
6.
GRIGORIEFF, R.D. Diskrete Approximation von Eigenwertproblemen II, Number.
Math., 24 (1975), 415-433.
7.
USMANI, R.A. On a class of numerical integration methods for a problem of
plate deflection theory, Internat. J. Comput. Math., ii (1982), 305-318.
8.
USMANI, R.A. Finite difference methods for a certain two point boundary
value problem, Indian J. Pure Appl. Math., 14 (1983), 398-411.