cubic b-spline interpolation method for singular integral equations ...

CUBIC B-SPLINE INTERPOLATION
METHOD FOR SINGULAR INTEGRAL
EQUATIONS WITH LOGARITHMIC
SIGULARITIES
Moin Uddin 1 , Sheo Kumar 2 and A. L. Sangal 3
1,3 Department of Computer Science & Engineering
2 Department of Applied Mathematics
Dr. B. R. Ambedkar National Institute of Technology
Jalandhar-144011, (Punjab) India.
Abstract- A method for numerical solution
singular integral equation of the first kind with
logarithmic singularities in their kernels along
the integration interval using cubic B-Spline
interpolations has been developed.
Keywords: Singular Integral Equations;
Logarithmic Sigularities; Cubic B-Spline
1. Introduction
We consider the following singular integral
equation of the first kind with a logarithmic
singularity in its kernel.
1
∫
−1
w(
t)
[ ln t − x + k(
t,
x)
]
g(
t)
dt = f ( x),
−1
≤ x ≤1,
(1)
where the kernel k( t,
x)
and the right hand side
function f (x)
are regular functions on
−1, 1 and w (t)
is the weight function
[ ]
2 − 1
2
w ( t)
= ( 1−
t )
(2)
Singular integral equation of the first kind with
kernels presenting logarithmic singularities
appear frequently. Some problems where these
equations are encountered are mentioned by
Morland (1970) and Christiansen (1971).
Morland (1970) found closed-form solutions for
these equations in the special case of difference
kernels, whereas Christiansen (1971) proposed
two methods for their numerical solution.
Ordinary Cauchy-type integrals and integrals
with a logarithmic kernel result from each other
by a simple differentiation or integration. For
example, by differentiating (1) with respect to
x we obtain
1
∫
−1
1
w (t)
[ − + K ′ ( t,
x)
] g ( t)
dt
t − x
= ′ ( x),
−1
≤ x ≤ 1.
f (3)
The solution in the form of (3) of (1) has been
proposed in Ioakimidis (1981). Gakhov (1966)
considered in some detail complex potentials
expressed not only as Cauchy-type integrals, but
also as integrals with logarithmic kernels, which
give rise to analogous singular integral
equations. The appearance of singular integral
equations with logarithmic singularities in their
kernels in problems of potential theory and
elasticity is also reported in Delves & Walsh
(1974, p.255).
For the numerical solution of singular integral
equations with logarithmic singularities in their
kernels, several methods are available,
particularly for such equations of the second
kind. These methods are reported in the books
by Delves and Walsh (1974), Atkinson (1976)
and Baker (1977) and are based on the removal
of the diagonal term in the system of linear
algebraic equations approximating the singular
integral equation, the use of methods based on
approximate integration and particularly on
product integration rules, the use of special
quadrature formulae or the use of expansion
methods. Similar methods and particularly the
modified quadrature method and the methods
based on product integration rules are also
applicable to singular integral equations of the
first kind with logarithmic singularities Baker
(1977).

The method described here uses Langrange
interpolation on “practical” abscissas
xk = cos( kπ
/ n),
k = 0(
1)
n.
The related results are given below:
Let Ln ( f ; x)
denote the Lagrange polynomial
of degree n interpolating f (x)
at
= kπ
n k = We may write
x k
cos( / ), 0(
1)
n.
n
n
∑ k
k = 0
L ( f ; x)
= l ( x)
f ( x ), (4)
where the Langrange fundamental polynomials
lk (x)
are given by
n
lk ( x)
( 2bk
/ n)
b jT
j ( xk
) T j ( x),
= ∑
j=
0
k
k = 0(
1)
n,
1 b =
where 0 = n 2
(5)
b and
bk = 1, k = 1(
1)
n −1.
(x)
T j are
Chebyshev polynomials of the first kind. That
lk ( x j ) = δ kj follows from the discrete
orthogonality of the Chebyshev polynomials at
the practical abscissas. For details, please see
Chawla & Kumar (1979).
For latest works on B-Spline, please see, Liu
(1997), Prautzsch (2004) and Reif (2000).
In past methods for solution of singular integral
equations with logarithmic singularities have
been also discussed in Alaylioglu & Lubinsky
(1984), Atkinson & Sloan (1991), Atkinson
(1988), Chakrabarti & Manam (2003), Saranen
& Sloan (1992), Saranen (1991), Sloan & Burn
(1992), Kress & Sloan (1993) and Manam
(2003).
2. The Numerical Method
we consider the singular integral equation of the
first kind with a logarithmic singularity in its
kernel.
1
∫
−1
w(
t)
[ ln t − x + k(
t,
x)
]
g(
t)
dt = f ( x),
−1
≤ x ≤ 1,
where the kernel k( t,
x)
and the right hand side
function f (x)
are regular functions on
−1, 1 and w(t) is the weight function
[ ]
2 − 1
2
w ( t)
= ( 1−
t )
(7)
In past a method for the integral equation (6)
have been discussed in Ioakimidis (1981). In
this chapter, we have described a cubic B-spline
approximation method for the numerical solution
of (6), using uniqueness condition of the form
1
∫
−1
w ( t)
g(
t)
dt = C
(8)
where C is a known constant . Here, we have
evaluated the integrals
1
( 4)
( 4)
L( Bn,
j ; x)
w(
t)
Bn,
j ( t)
ln t − x dt,
= ∫
−1
m = 1(
1)
4,
j = 1(
1)
n
( 4)
using the representation of Bn , j ( t)
given
below:
(6)
(9)

B
( 4)
n,
j
3
⎧ ( t − t j )
⎪
,
t j ≤ t ≤ t j+
1,
⎪(
t j+
3 − t j )( t j+
1 − t j )( t j+
2 − t j )
⎪
⎪ ( t − t j ) ⎡ ( t − t j )( t j+
2 − t)
( t j+
3 − t)(
t − t j+
1)
⎤
⎪
⎢
+
⎥
( t j+
3 − t j ) ⎢(
2 )( 2 1)
( 3 1)(
2 1)
⎪ ⎣ t j+
− t j t j+
− t j+
t j+
− t j+
t j+
− t j+
⎥⎦
⎪
2
(t j+
4 − t)(
t − t j+
1)
⎪ +
, t j+
1 ≤ t ≤ t j+
2 ,
⎪ ( t j+
4 − t j+
1)(
t j+
2 − t j+
1)(
t j+
3 − t j+
1)
⎪
⎪ ( t j+
4 − t)
⎡ ( t − t j+
1)(
t j+
3 − t)
( t j+
4 − t)(
t − t j+
2 ) ⎤
( t)
= ⎨ ⎢
+
⎥
⎪(
t j+
4 − t j+
1)
⎢⎣
( t j+
3 − t j+
1)(
t j+
3 − t j+
2 ) ( t j+
4 − t j+
2 )( t j+
3 − t j+
2 ) ⎥⎦
⎪
2
⎪
( t − t j )( t j+
3 − t)
+
, t j+
2 ≤ t ≤ t j+
3,
⎪ ( t j+
3 − t j )( t j+
3 − t j+
1)(
t j+
3 − t j+
2 )
⎪
3
⎪ ( t j+
4 − t)
⎪
,
t j+
3 ≤ t ≤ t j+
4 ,
⎪(
t j+
4 − t j+
1)(
t j+
4 − t j+
2 )( t j+
4 − t j+
3 )
⎪
0,
otherwise.
⎪
⎪
⎩
(10)
≤ t ≤ t ≤ t ≤ ⋅⋅
⋅ ≤ t ≤ 1.
Thus
( 4)
and replacing Bn , j ( t)
by Lagrange
interpolation at the “practical” abscissas
xk = cos( kπ
/ n),
k = 0(
1)
n described in
Chawla & Kumar (1979) and described here by
equations (4) and (5). For derivation of (10),
please see Prautzsch, Boehm & Paluszny [2002,
pp. 61] & Phillips [2003, pp. 224]. The
following results also have been used in the
evaluation of singular integrals in (9):
1
∫ ( 0
−1
2 − 1
2 1−
t ) ln t − xT
( t)
dt = −π
ln 2 , (11)
− 1 0 1 2
n+
4
( 4)
{ Bn, j ( t)
}
{ } 4 n+
( 4)
t j and further B ( )
j=
0
n,
j t
over interval ( j , t j+
4 )
is a cubic B-spline basis with knots
is a cubic B-spline
t . Now, we have
approximated unknown function g (t)
as:
n
g ∑ j n,
j
j=
0
( 4)
( t)
≈ α B ( t)
. (14)
We get n equations from (6) using (14)
n 1
∑ ∫
−
[ ]
α j w(
t)
ln t − xi
( 4)
+ k(
t,
xi
) Bn,
j ( t)
dt
1
2 − 1
π
2
∫ ( 1−
t ) ln t − xT
j ( t)
dt = − T j ( x),
j
−1
j=
0
j ≥ 1,
(12)
by
1
choosing
= f ( xi
),
n collocation
(15)
points
T j + 1(
x)
= 2xT
j ( x)
− T j−1
( x);
T0
= 1,
T1
( x)
= x.
where T j (x)
are Chebyshev polynomials of the
first kind.
(13) −1 < x 0 < x1
< x2
< ⋅⋅
⋅ < xn−1
< 1.
Equation ( n + 1)
th given below is obtained
from equations (8) replacing g(t) by (14).
Here, we have described a product integration
method using cubic B-spline approximations for
unknown function g(t). Here, we have taken
n + 5 knots
n 1
( 4)
∑α j ∫ w( t)
Bn,
j ( t)
dt = C . (16)
j=
0 −1

The values of α j , j = 0(
1)
n is obtained from
the solution of above n + 1equations
using
matrix inversion by partition method.
3. Numerical Examples
To illustrate the above method computationally,
we consider the following two examples of
singular integral equations with logarithmic
singularities:
Example 1
Consider the singular integral equation
1
2 − 1 ⎡
1 ⎤
2
∫ ( 1−
t ) ⎢ln
t − x +
( )
( 10)
⎥ g t dt
1 ⎣ t + x +
−
⎦
= f ( x),
−1
≤ x ≤ 1 (17)
where the exact solution is given by
t
g ( t)
= e
and the values of f (x)
have been computed.
numerically at the various points of
discretisations under the unique condition:
1
∫
−1
2 − 1
2 ( 1−
t ) g(
t)
dt = 2.
493524565689
.
We solved the singular integral equation (17) by
the present method for n = 20 and 40 for equally
spaced mesh and choosing the collocation points
sk = (tk + tk+1)/2, k = 0(1)n; we have solved linear
algebraic system of equations using partition
method for matrix inversion. The corresponding
errors ek = | exact value (xk) − approximate
value (xk) |
for few points are listed in Table 1.
Example 2
Consider the singular integral equation
1
∫
−1
( 1−
t
2
)
−
1
2
t+
x
[ ln t − x + e ]
−1
≤ x ≤ 1,
g(
t)
dt = f ( x),
where the exact solution is given by
2 1
2
g( t)
= ( 1−
t )
and the values of f (x)
have been computed
numerically at the various points of
discretisations under the unique condition:
1
2 − 1
2 ( 1−
t ) g(
t)
dt = 2.
00 .
∫
−1
( 18)
We solved the singular integral equation (18) by
the present method for n = 20 and 40, for equally
spaced mesh and choosing the collocation points
sk = (tk + tk+1)/2, k = 0(1)n;
We have solved linear algebraic system of
equations using partition method for matrix
inversion. The corresponding errors ek = |
exact value (xk) − approximate value (xk) |
for few points are listed in Table 2.
Table 1
Integral equation (17)
n = 20 n = 40
x h = 1/
10 h = 1/
20
-1.00 9.16 E−05 1.52 E−06
-.800 3.46 E−05 6.45 E−07
-.600 3.29 E−05 2.99 E−06
-.400 5.08 E−06 5.12 E−07.
-.200 7.58 E−06 8.92 E−07
0.00 6.13 E−05 5.56 E−07
0.20 7.39 E−05 1.79 E−06
0.40 8.15 E−06 8.54 E−07
0.60 9.85 E−05 8.77 E−07
0.80 1.36 E−05 5.37 E−06
1.00 3.48 E−06 6.59 E−07
Table 2
Integral equation (18)
n = 20 n = 40
x h = 1/
10 h = 1/
20
-1.00 1.16 E−03 4.78 E−04
-.800 2.93 E−03 9.28 E−05
-.600 4.72 E−03 8.54E−04
-.400 6.09 E−04 1.19 E−05.
-.200 7.27 E−04 2.12 E−05
0.00 8.15 E−04 4.85 E−05
0.20 9.46 E−04 6.81 E−04
0.40 1.24 E−04 6.75 E−05
0.60 2.54 E−04 4.93 E−06
0.80 3.19 E−04 3..46 E−06
1.00 8.45 E−04 8.49E−06

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