Adapted Quadratic Approximation For Weakly Singular Integrals

Adapted Quadratic Approximation For Weakly Singular Integrals

Mostefa Nadir

, Belkacem Lakehali

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 conduc- tion and ‡uid ‡ow lead to weakly singular integral equations based on the Abel’s in- tegral [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) (t t0) dt+

Z

k(t; t0)'(t)dt=f(t0); (1) where designates an oriented smooth open curve, the points t andt0 are on and 0 <1; b₀(t); k(t; t₀)andf(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

F(t₀) =

Z '(t)

(t t₀) dt; t; t₀2 ; (2)

where '(t)is a given function on :

Mathematics Sub ject Classi…cations: 45D05, 45E05, 45L05, 45L10 and 65R20

yDepartment of Mathematics University of Msila 28000 Algeria

zDepartment of Mathematics University of Msila 28000 Algeria

225

2 Quadrature

We denote bytthe parametric complex function t(s)of the curve de…ned by t(s) =x(s) +iy(s); a s b;

where x(s)and y(s)are continuous functions on the …nite interval of de…nition [a; b]

and have continuous …rst derivativesx⁰(s)andy⁰(s)never simultaneously null. Divide the interval[a; b]into N equal subintervalsI1; I2; :::; IN by the points

s =a+ l

N; l=b a for = 0;1;2; ::::; N:

Further, we divide each of segments [s ; s +1] in two equals segments [s ; s M] and [s M; s +1]where

s M =s +h

2; h= l N:

In other words, we have for each subinterval [s ; s ₊₁]the following subdivision [s ; s +1] =fs < s M < s +1g:

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 pointstandt₀belong respec- tively to the arcst t^_₊₁ andt t^_₊₁ wheret t^_₊₁ designates the arc with endst and t +1[4,5,6].

For an arbitrary number = 0;1;2; :::; N 1;we de…ne the piecewise quadratic inter- polation polynomialS₂(';t; )dependent on'; tand 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 pointst ; t _M andt ₊₁ respectively with a quadratic polynomial, given by the following formula.

Fort t t ₊₁;

S₂(';t; ) = (t t _M)(t t ₊₁)

(t M t )(t +1 t )'(t ) (t t )(t t ₊₁)

(t M t )(t +1 t M)'(t _M) + (t t )(t t M)

(t +1 t )(t +1 t M)'(t +1); (3)

this piecewise quadratic interpolating polynomial exists and is unique.

We de…ne for an arbitrary numbers and ; such that 0 ; N 1; the following continuous function (';t; t0);dependents on'; tandt0

(';t; t₀) = U(';t; ) V(';t₀; ; ) for t6=t₀;

0 for t=t₀: (4)

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 t _M)(t t ₊₁)

(t M t )(t +1 t )'(t ) (t t₀) (t t0) (t t )(t t +1)

(t M t )(t +1 t M)'(t M) (t t0) (t M t0) + (t t )(t t M)

(t ₊₁ t )(t ₊₁ t _M)'(t +1) (t t0) (t ₊₁ t₀) ; and the function V(';t₀; ; )is given by

V(';t₀; ; ) = S₂(';t₀; ) (t t _M)(t t ₊₁) (t M t )(t +1 t )

(t t₀) (t t0) S2(';t0; ) (t t )(t t +1)

(t M t )(t +1 t M)

(t t0) (t M t0) +S2(';t0; ) (t t )(t t M)

(t ₊₁ t )(t ₊₁ t _M)

(t t0) (t ₊₁ t₀) :

Denote by (';t; t₀)the cubic approximation of the density '(t) at the point t 2 [t ; t ₊₁]; t₀2[t ; t ₊₁]and0 ; N 1by

(';t; t0) ='(t0) + (';t; t0): (5) Our idea is to replace the density'(t)by expansion (5) in the weakly singular integral (2)

F(t₀) =

Z '(t) (t t₀) dt;

and obtain the following approximation noting by Fn(t0)given as Fn(t0) =

Z (';t; t0) (t t₀) dt=

Z '(t0) (t t₀) dt+

Z (';t; t0)

(t t₀) dt: (6)

3 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) F_n(t₀)j C (2N)¹ holds, where the constant C depends only on the curve :

PROOF. Taking the pointst2[t ; t ₊₁]and t₀2[t ; t ₊₁]; we write, fort t t +1 andt k t0 t +1;

'(t) (';t; t0) = '(t) '(t0) (';t; t0) (t t0)

= '(t) '(t₀) (t t0)

(t t _M)(t t ₊₁)

(t M t )(t +1 t )'(t ) (t t₀) (t t0) (t t )(t t +1)

(t M t )(t +1 t M)'(t M) (t t0) (t M t0) + (t t )(t t M)

(t ₊₁ t )(t ₊₁ t _M)'(t +1) (t t0) (t ₊₁ t₀) S₂(';t₀; )(t t _M)(t t ₊₁)

(t +1 t )(t M t )

(t t₀) (t t0) +S2(';t0; )(t t )(t t +1)

(t +1 t M)(t M t )

(t t0) (t M t0) S₂(';t₀; )(t t )(t t _M)

(t +1) t M)(t +1 t )

(t t₀)

(t +1 t0) : (7) Taking into account the expression (7) we get

Z '(t) (';t; t0) (t t0) dt=

NX1

=0

Z

t t +1

'(t) (';t; t0)

(t t0) dt: (8)

Note that, the equalities (t t0) = 0; (t M t0) = 0 and (t +1 t0) = 0 are possible only when = 1; + 1and : For these cases, it is easy to see that the integral (8) exists whent tends tot₀ort _M tends tot₀ort ₊₁tends tot₀as 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 pointst; t₀2[t ; t ₊₁]such thatt t; t₀ t ₊₁; we write (';t; t₀) =U(';t; ) V(';t₀; ; ):

Hence

(';t; t0)

= (t t M)(t t +1) (t _M t )(t ₊₁ t )

(t t0)

(t t₀) ['(t ) S2(';t0; )]

(t t )(t t ₊₁) (t M t )(t +1 t M)

(t t₀)

(t M t0) ['(t _M) S₂(';t₀; )]

+ (t t )(t t _M) (t +1 t )(t +1 t M)

(t t₀)

(t +1 t0) ['(t +1) S2(';t0; )]: (9) In other words, we write

(';t; t0) = (t t0) Q(';t; t0);

where the expressionQ(';t; t₀)is given by Q(';t; t₀) = (t t _M)(t t ₊₁)

(t _M t )(t ₊₁ t ) 1

(t t₀) ['(t ) S₂(';t₀; )]

(t t )(t t ₊₁) (t M t )(t +1 t M)

1

(t M t0) ['(t M) S2(';t0; )]

+ (t t )(t t M) (t +1 t )(t +1 t M)

1

(t +1 t0) ['(t +1) S2(';t0; )]: Passing now to the estimation of the expression (8), fort₀2t t^_₊₁and 6= 1; +1 and we have

NX1

=0

Z

t t +1

dt

(t t0) f('(t) '(t0)) f (t t M)(t t +1)

(t _M t )(t ₊₁ t )'(t ) (t t0) (t t₀) (t t )(t t +1)

(t _M t )(t ₊₁ t _M)'(t _M) (t t0) (t _M t₀) + (t t )(t t _M)

(t +1 t )(t +1 t M)'(t ₊₁) (t t₀) (t +1 t0) S2(';t0; )(t t M)(t t +1)

(t +1 t )(t M t )

(t t0) (t t0) +S2(';t0; )(t t )(t t +1)

(t ₊₁ t _M)(t _M t )

(t t0) (t _M t₀) S₂(';t₀; )(t t )(t t _M)

(t +1 t M)(t +1 t )

(t t₀)

(t +1 t0) =O 1 (2N)¹ : Indeed, it is clear that

max

t02t t^_+1

NX1

=0

Z t +1

t

('(t) '(t₀))

(t t0) dt =O 1 (2N)¹ and also we estimate the expression

NX1

=0

Z t +1

t

(t t M)(t t +1)

(t M t )(t +1 t )'(t ) (t t0) (t t0) (t t )(t t +1)

(t _M t )(t ₊₁ t _M)'(t M) (t t0) (t _M t₀) + (t t )(t t _M)

(t +1 t )(t +1 t M)'(t ₊₁) (t t₀) (t +1 t0) S₂(';t₀; )(t t _M)(t t ₊₁)

(t +1 t )(t M t )

(t t₀) (t t0)

+S₂(';t₀; )(t t )(t t ₊₁) (t +1 t M)(t M t )

(t t₀) (t M t0) S2(';t0; )(t t )(t t M)

(t ₊₁ t _M)(t ₊₁ t )

(t t0) (t ₊₁ t₀)

1

(t t₀) dt =O 1 (2N)¹ : Naturally, the estimation given above is obtained by using expressions

(t t₀)

(t t0) =O(1); (t t₀)

(t M t0) =O(1); (t t₀)

(t +1 t0) =O(1):

Further, for the cases where = 1; + 1and ;using the condition of smoothness of ;we get

Z

t t+1

'(t) '(t0) (t t₀) dt A

Z s +1

s

(t t₀)¹ ds=O 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 experi- ment 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 I=F(t₀) =

Z t0

0

p'(t) t0 tdt;

where the function F(t0)is calculated chosen so that the function '(t)is given '(t) =t; F(t0) =4

3t³²:

The approximate Abel integral Fn(t0) ofF(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 forN = 10is calculated.

Values of t Exact integral F Approx integralF_n Error

0:000000 0:000000e+ 000 0:000000e+ 000 0:000000e+ 000 0:200000 1:192570e 001 1:192570e 001 1:387779e 017 0:400000 3:373096e 001 3:373096e 001 5:551115e 017 0:600000 6:196773e 001 6:196773e 001 0:000000e+ 000 0:800000 9:540557e 001 9:540557e 001 1:110223e 016 1:000000 1:333333e+ 000 1:333333e+ 000 2:220446e 016

EXAMPLE 2. Consider the Abel integral I=F(t0) =

Z t0

0

p'(t) t₀ tdt;

where the function F(t0)is calculated chosen so that the function '(t)is given '(t) =t²; F(t0) = 16

15t⁵²:

The approximate Abel integral Fn(t0) ofF(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 forN = 10is calculated.

Values of t Exact integral F Approx integralFn Error

0:000000 0:000000e+ 000 0:000000e+ 000 0:000000e+ 000 0:200000 1:908111e 002 1:908111e 002 0:000000e+ 000 0:400000 1:079391e 001 1:079391e 001 0:000000e+ 000 0:600000 2:974451e 001 2:974451e 001 0:000000e+ 000 0:800000 6:105956e 001 6:105956e 001 1:110223e 016 1:000000 1:066667e+ 000 1:066667e+ 000 2:220446e 016

EXAMPLE 3. Consider the Abel integral I=F(t0) =

Z t0

0

p'(t) t0 tdt;

where the function F(t0)is calculated chosen so that the function '(t)is given '(t) = 1

p1 +t; F(t0) =

2 arcsin 1 t0

1 +t₀ :

The approximate Abel integral Fn(t0) ofF(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 forN = 10is calculated.

Values of t Exact integral F Approx integralFn Error

0:000000 0:000000e+ 000 0:000000e+ 000 0:000000e+ 000 0:200000 8:410687e 001 8:411104e 001 4:173975e 005 0:400000 1:127885e+ 000 1:128199e+ 000 3:137339e 004 0:600000 1:318116e+ 000 1:318987e+ 000 8:713448e 004 0:800000 1:459455e+ 000 1:461079e+ 000 1:623303e 003 1:000000 1:570796e+ 000 1:573241e+ 000 2:444518e 003

References

[1] M. Alipour and D. Rostamy, Bernstein polynomials for solving Abel’s integral equa- tion, J. Math. Computer Sci., 3(2011), 403–412.

[2] R. Goren‡o and S. Vessella, Abel Integral Equations, Analysis and applications.

Lecture Notes in Mathematics, 1461. Springer-Verlag, Berlin, 1991.

[3] K. Maleknejad, M. Nosrati and E. Naja…, Wavelet Galerkin method for solving singular integral equations, J. Comput. Appl. Math., 31,(2012), 373–390.

[4] M. Nadir, Adapted quadratic approximation for singular integrals, J. Math. In- equal., 4(2010), 423–430.

[5] M. Nadir, Adapted linear approximation for singular integrals, Math. Sci., 6(2012), Art. 36, 5 pp.

[6] M. Nadir, Adapted quadratic approximation for logarithmic kernel integrals, Fasc.

Math., 49(2012), 75–85.

[7] A. Saadatmandia and M. Dehghanb, A collocation method for solving Abel’s inte- gral equations of …rst and second kinds, Z. Naturforsch., 63a(2008), 752–756.

[8] A. Shahsavaran, Numerical approach to solve second kind Volterra integral equa- tion of Abel type using block-pulse functions and Taylor expansion by collocation method, Appl. Math. Sci., 5(2011), 685–696.