Applied Mathematics E-Notes, 15(2015), 225-232 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Adapted Quadratic Approximation For Weakly
Singular Integrals
Mostefa Nadiry, Belkacem Lakehaliz
Received 25 December 2014
Abstract
In this work, we present a new approximation to the weakly singular integral,
for this goal, we use a modi…cation of the quadratic spline function and replace it
in the integral in order to eliminate the weak singularity. This approximation is
destined to solve numerically the weakly singular integral equation on a smooth
oriented curve or on an interval.
1
Introduction
Many physical and engineering problems, scattering theory, seismology, heat conduction and ‡uid ‡ow lead to weakly singular integral equations based on the Abel’s integral [2]. Various numerical approximations for Abel’s integrals are treated, based on
Legendre wavelet approximations [8], Bernstein polynomials [1] and Wavelet Galerkin
method [3].
The idea is to replace the Abel kernel by its approximations in the weakly singular
integral equation
'(t0 ) + b0 (t0 )
Z
'(t)
dt +
(t t0 )
Z
k(t; t0 )'(t)dt = f (t0 );
(1)
where designates an oriented smooth open curve, the points t and t0 are on and
0
< 1; b0 (t); k(t; t0 ) and f (t) are a given functions on :
The goal of this work is to present a new technical method based on the quadratic
spline functions, in order to give a good and e¢ ciency approximation to the weakly
singular integral
Z
'(t)
F (t0 ) =
dt; t; t0 2 ;
(2)
(t t0 )
where '(t) is a given function on :
Mathematics Subject Classi…cations: 45D05, 45E05, 45L05, 45L10 and 65R20
of Mathematics University of Msila 28000 Algeria
z Department of Mathematics University of Msila 28000 Algeria
y Department
225
226
2
Adapted Quadratic Approximation
Quadrature
We denote by t the parametric complex function t(s) of the curve
t(s) = x(s) + iy(s); a
s
de…ned by
b;
where x(s) and y(s) are continuous functions on the …nite interval of de…nition [a; b]
and have continuous …rst derivatives x0 (s) and y 0 (s) never simultaneously null. Divide
the interval [a; b] into N equal subintervals I1 ; I2 ; :::; IN by the points
s =a+
l
; l=b
N
a for
Further, we divide each of segments [s ; s
[s M ; s +1 ] where
s
M
=s +
+1 ]
= 0; 1; 2; ::::; N:
in two equals segments [s ; s
+1 ]
and
l
h
; h= :
2
N
In other words, we have for each subinterval [s ; s
[s ; s
M]
= fs < s
M
+1 ]
<s
the following subdivision
+1 g:
We introduce the notation
t = t(s ); t
M
= t(s
M );
t
+1
= t(s
+1 );
= 0; 1; 2; :::; N
1:
Assume that, for the indices ; = 0; 1; 2; ::::; N 1; the points t and t0 belong respec_
_
_
tively to the arcs t t +1 and t t +1 where t t +1 designates the arc with ends t and
t +1 [4,5,6].
For an arbitrary number = 0; 1; 2; :::; N 1; we de…ne the piecewise quadratic interpolation polynomial S2 ('; t; ) dependent on '; t and which represents the quadratic
approximation of the function density '(t) on the subinterval [t ; t +1 ] of the curve
: We interpolate the function density '(t) with respect to the values '(t ); '(t M )
and '(t +1 ) at the points t ; t M and t +1 respectively with a quadratic polynomial,
given by the following formula.
For t
t t +1 ;
(t t M )(t
(t M t )(t
(t t )(t
+
(t +1 t )(t
S2 ('; t; ) =
t
+1 )
'(t )
t )
t M)
'(t
t M)
+1
+1
(t
(t
M
t )(t
t )(t
t
+1
+1 )
t
M)
+1 );
this piecewise quadratic interpolating polynomial exists and is unique.
We de…ne for an arbitrary numbers
and ; such that 0
;
following continuous function
('; t; t0 ); dependents on '; t and t0
('; t; t0 ) =
U ('; t; )
0
'(t
V ('; t0 ; ; ) for t 6= t0 ;
for t = t0 :
M)
(3)
N
1; the
(4)
M. Nadir and B. Lakehali
227
The function U ('; t; ) represents a modi…ed quadratic interpolation of the function
density '(t) on the subinterval [t ; t +1 ] of the curve : Indeed, for t
t t +1 we
put
U ('; t; )
=
(t t0 )
(t t M )(t t +1 )
'(t )
(t M t )(t +1 t )
(t
t0 )
(t t )(t t +1 )
(t t0 )
'(t M )
(t M t )(t +1 t M )
(t M t0 )
(t t )(t t M )
(t t0 )
+
;
'(t +1 )
(t +1 t )(t +1 t M )
(t +1 t0 )
and the function V ('; t0 ; ; ) is given by
V ('; t0 ; ; )
(t t M )(t t +1 ) (t t0 )
(t M t )(t +1 t ) (t
t0 )
(t t )(t t +1 )
(t t0 )
S2 ('; t0 ; )
(t M t )(t +1 t M ) (t M t0 )
(t t )(t t M )
(t t0 )
+S2 ('; t0 ; )
:
(t +1 t )(t +1 t M ) (t +1 t0 )
= S2 ('; t0 ; )
Denote by
('; t; t0 ) the cubic approximation of the density '(t) at the point t 2
[t ; t +1 ]; t0 2 [t ; t +1 ] and 0
;
N 1 by
('; t; t0 ) = '(t0 ) +
('; t; t0 ):
(5)
Our idea is to replace the density '(t) by expansion (5) in the weakly singular integral
(2)
Z
'(t)
dt;
F (t0 ) =
(t t0 )
and obtain the following approximation noting by Fn (t0 ) given as
Z
Z
Z
'(t0 )
('; t; t0 )
('; t; t0 )
dt =
dt +
dt:
Fn (t0 ) =
(t t0 )
(t t0 )
(t t0 )
3
(6)
Main Results
We have
THEOREM 1. Let be an oriented smooth open curve and let ' be a function
density de…ned on : Then the following estimation
jF (t0 )
Fn (t0 )j
C
(2N )
1
holds, where the constant C depends only on the curve :
228
t
Adapted Quadratic Approximation
PROOF. Taking the points t 2 [t ; t
and t k t0 t +1 ;
+1 ]
+1
'(t)
('; t; t0 )
and t0 2 [t ; t
+1 ];
we write, for t
'(t0 )
('; t; t0 )
(t t0 )
'(t) '(t0 )
(t t M )(t t +1 )
(t
=
'(t )
(t t0 )
(t M t )(t +1 t )
(t
(t t0 )
(t t )(t t +1 )
'(t M )
(t M t )(t +1 t M )
(t M t0 )
(t t )(t t M )
(t t0 )
+
'(t +1 )
(t +1 t )(t +1 t M )
(t +1 t0 )
S2 ('; t0 ; )(t t M )(t t +1 ) (t t0 )
(t +1 t )(t M t )
(t
t0 )
S2 ('; t0 ; )(t t )(t t +1 ) (t t0 )
+
(t +1 t M )(t M t ) (t M t0 )
S2 ('; t0 ; )(t t )(t t M ) (t t0 )
:
(t +1) t M )(t +1 t ) (t +1 t0 )
=
t
'(t)
t0 )
t0 )
(7)
Taking into account the expression (7) we get
Z
'(t)
(t
N
X1 Z
('; t; t0 )
dt =
t0 )
=0 t
'(t)
t
+1
(t
('; t; t0 )
dt:
t0 )
(8)
Note that, the equalities (t
t0 ) = 0; (t M t0 ) = 0 and (t +1 t0 ) = 0 are
possible only when =
1; + 1 and : For these cases, it is easy to see that the
integral (8) exists when t tends to t0 or t M tends to t0 or t +1 tends to t0 as a weakly
singular integral. For the other case
= ; we can easily seeing that, the function
('; t; t0 ) contains (t
t0 ); (t M t0 ) and (t +1 t0 ) as factors, so for all cases
the function
('; t; t0 ) makes sense.
Indeed, for the points t; t0 2 [t ; t +1 ] such that t
t; t0 t +1 ; we write
('; t; t0 ) = U ('; t; )
V ('; t0 ; ; ):
Hence
=
('; t; t0 )
(t t M )(t t +1 ) (t t0 )
['(t ) S2 ('; t0 ; )]
(t M t )(t +1 t ) (t
t0 )
(t t0 )
(t t )(t t +1 )
['(t M ) S2 ('; t0 ; )]
(t M t )(t +1 t M ) (t M t0 )
(t t )(t t M )
(t t0 )
+
['(t +1 ) S2 ('; t0 ; )] :
(t +1 t )(t +1 t M ) (t +1 t0 )
In other words, we write
('; t; t0 ) = (t
t0 ) Q('; t; t0 );
(9)
M. Nadir and B. Lakehali
229
where the expression Q('; t; t0 ) is given by
Q('; t; t0 )
(t t M )(t t +1 )
1
['(t ) S2 ('; t0 ; )]
(t M t )(t +1 t ) (t
t0 )
(t t )(t t +1 )
1
['(t M ) S2 ('; t0 ; )]
(t M t )(t +1 t M ) (t M t0 )
(t t )(t t M )
1
+
['(t +1 ) S2 ('; t0 ; )] :
(t +1 t )(t +1 t M ) (t +1 t0 )
=
_
+1
Passing now to the estimation of the expression (8), for t0 2 t t
and we have
N
X1 Z
t t
=0
+1
dt
f('(t)
(t t0 )
and
6=
1;
'(t0 ))
(t t M )(t t +1 )
(t t0 )
'(t )
(t M t )(t +1 t )
(t
t0 )
(t t0 )
(t t )(t t +1 )
'(t M )
(t M t )(t +1 t M )
(t M t0 )
(t t )(t t M )
(t t0 )
+
'(t +1 )
(t +1 t )(t +1 t M )
(t +1 t0 )
S2 ('; t0 ; )(t t M )(t t +1 ) (t t0 )
(t +1 t )(t M t )
(t
t0 )
S2 ('; t0 ; )(t t )(t t +1 ) (t t0 )
+
(t +1 t M )(t M t ) (t M t0 )
S2 ('; t0 ; )(t t )(t t M ) (t t0 )
=O
(t +1 t M )(t +1 t ) (t +1 t0 )
f
Indeed, it is clear that
max
_
t0 2t t
+1
N
X1 Z t
=0
+1
t
('(t) '(t0 ))
dt = O
(t t0 )
and also we estimate the expression
N
X1 Z t
=0
t
(t
+1
(t t M )(t
(t M t )(t
t
+1
+1 )
t )
'(t )
(t
(t
t )(t t +1 )
(t t0 )
'(t M )
t )(t +1 t M )
(t M t0 )
(t t )(t t M )
(t t0 )
+
'(t +1 )
(t +1 t )(t +1 t M )
(t +1 t0 )
S2 ('; t0 ; )(t t M )(t t +1 ) (t t0 )
(t +1 t )(t M t )
(t
t0 )
(t
M
t0 )
t0 )
1
(2N )1
1
(2N )1
:
+1
230
Adapted Quadratic Approximation
+
S2 ('; t0 ;
(t +1
S2 ('; t0 ;
(t +1
)(t t )(t t +1 ) (t t0 )
t M )(t M t ) (t M t0 )
)(t t )(t t M ) (t t0 )
t M )(t +1 t ) (t +1 t0 )
1
dt = O
t0 )
(t
1
(2N )1
:
Naturally, the estimation given above is obtained by using expressions
(t
(t
t0 )
t0 )
= O(1);
(t t0 )
(t M t0 )
Further, for the cases where =
of ; we get
Z
'(t) '(t0 )
dt
(t t0 )
t t +1
1;
A
= O(1);
(t
(t
+1
t0 )
t0 )
= O(1):
+ 1 and ; using the condition of smoothness
Z
s
+1
(t
t0 )1
ds = O
s
1
(2N )2
;
where A represents the bound of the derivative '0 (t0 ) of the density function, say
j'0 (t0 )j A:
4
Numerical Experiments
Using our approximation, we apply the algorithm to weakly singular integrals and we
present results concerning the accuracy of the calculations. In this numerical experiment each table F represents the exact value of the weakly singular integral and Fn
corresponds to the approximate calculation produced by our approximation at points
values interpolation.
EXAMPLE 1. Consider the Abel integral
Z t0
'(t)
p
I = F (t0 ) =
dt;
t0 t
0
where the function F (t0 ) is calculated chosen so that the function '(t) is given
'(t) = t;
F (t0 ) =
4 3
t2 :
3
The approximate Abel integral Fn (t0 ) of F (t0 ) is obtained by the adapted quadratic
approximation
TABLE 1. We present the exact and the approximate values of the Abel integral
in the example 1 in some arbitrary points, the error for N = 10 is calculated.
Values of t
0:000000
0:200000
0:400000
0:600000
0:800000
1:000000
Exact integral F
0:000000e + 000
1:192570e 001
3:373096e 001
6:196773e 001
9:540557e 001
1:333333e + 000
Approx integral Fn
0:000000e + 000
1:192570e 001
3:373096e 001
6:196773e 001
9:540557e 001
1:333333e + 000
Error
0:000000e + 000
1:387779e 017
5:551115e 017
0:000000e + 000
1:110223e 016
2:220446e 016
M. Nadir and B. Lakehali
231
EXAMPLE 2. Consider the Abel integral
Z t0
'(t)
p
I = F (t0 ) =
dt;
t0 t
0
where the function F (t0 ) is calculated chosen so that the function '(t) is given
'(t) = t2 ;
F (t0 ) =
16 5
t2 :
15
The approximate Abel integral Fn (t0 ) of F (t0 ) is obtained by the adapted quadratic
approximation
TABLE 2. We present the exact and the approximate values of the Abel integral
in the example 2 in some arbitrary points, the error for N = 10 is calculated.
Values of t
0:000000
0:200000
0:400000
0:600000
0:800000
1:000000
Exact integral F
0:000000e + 000
1:908111e 002
1:079391e 001
2:974451e 001
6:105956e 001
1:066667e + 000
Approx integral Fn
0:000000e + 000
1:908111e 002
1:079391e 001
2:974451e 001
6:105956e 001
1:066667e + 000
Error
0:000000e + 000
0:000000e + 000
0:000000e + 000
0:000000e + 000
1:110223e 016
2:220446e 016
EXAMPLE 3. Consider the Abel integral
Z t0
'(t)
p
I = F (t0 ) =
dt;
t0 t
0
where the function F (t0 ) is calculated chosen so that the function '(t) is given
'(t) = p
1
;
1+t
F (t0 ) =
2
arcsin
1 t0
1 + t0
:
The approximate Abel integral Fn (t0 ) of F (t0 ) is obtained by the adapted quadratic
approximation
TABLE 3. We present the exact and the approximate values of the Abel integral
in the example 3 in some arbitrary points, the error for N = 10 is calculated.
Values of t
0:000000
0:200000
0:400000
0:600000
0:800000
1:000000
Exact integral F
0:000000e + 000
8:410687e 001
1:127885e + 000
1:318116e + 000
1:459455e + 000
1:570796e + 000
Approx integral Fn
0:000000e + 000
8:411104e 001
1:128199e + 000
1:318987e + 000
1:461079e + 000
1:573241e + 000
Error
0:000000e + 000
4:173975e 005
3:137339e 004
8:713448e 004
1:623303e 003
2:444518e 003
232
Adapted Quadratic Approximation
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