# An Automatic Quadrature Schemes and Error Estimates for Semibounded Weighted Hadamard Type Hypersingular Integrals

Authors: Sirajo Lawan Bichi; Z. K. Eshkuvatov; N. M. A. Nik Long
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The approximate solutions for the semibounded Hadamard type hypersingular integrals (HSIs) for smooth density function are investigated. The automatic quadrature schemes (AQSs) are constructed by approximating the density function using the third and fourth kinds of Chebyshev polynomials. Error estimates for the semibounded solutions are obtained in the class ofh(t)∈CN,α[-1,1]. Numerical results for the obtained quadrature schemes revealed that the proposed methods are highly accurate when the density function h(t) is any polynomial or rational functions. The results are in line with the theoretical findings.


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## Body

## 1. Introduction

In an attempt to solve the Cauchy’s problem for hyperbolic partial differential equations, Hadamard [1] introduced the concept of hypersingular integrals. He defined the hypersingular integrals to be the finite part of a divergent integral in which singularities are arranged at the endpoints of the interval for one-dimensional integrals and on the domain boundary for multidimensional integrals. These forms of integrals are later identified as Hadamards finite part integrals or simply Hadamards integrals. Numerous applications of the approximate solutions of Hadamard integrals have been found in mechanics, electrodynamics, aerodynamics, and acoustics with various numerical approaches [2–5]. Some authors applied various regularization methods [2, 4, 5] for the transformation of the hypersingular integrals into singular or weakly singular integrals, while others applied the direct numerical computation of the finite part integrals by using various quadrature or cubature formulae [4, 6]. Hypersingular integral represents a natural extension of singular integrals in the Cauchy principal value:
(1)H(h,x)=⨎-11w(t)h(t)t-x2dt=ddxC(h,x),
where
(2)C(h,x)=⨍-11w(t)h(t)t-xdt.Hui and Shia [7] developed Gaussian quadrature formula for hypersingular integrals with second-order singularities which have been extensively used for the solution of elasticity problems. Chen [8] used the hypersingular integral equation approach to study the plane elastic problem for crack problem. Nik Long and Eshkuvatov [9] formulated the multiple curved crack problems into hypersingular integral equations via the complex variable function method. Obayis et al. [10] developed an automatic quadrature scheme (AQS) based on Clenshaw-Curtis Chebyshev quadrature methods for the evaluation of HSIs of the form
(3)Qi(f,x)=⨎-11wi(t)f(t)t-x2dt,x∈(-1,1),i={0,1,2},
where w0(t)=1, w1(t)=1-t2, and w2(t)=1/1-t2 are the weight functions and f(t) is a smooth or continuous. Error estimation for the developed AQS for i=0 was obtained in the class of functions CN+2,α[-1,1].In this note, we have developed the automatic quadrature scheme based on Chebyshev Gauss quadrature formula for the semibounded solutions of the hypersingular integrals of the form(4)Hi(h,x)=wi(x)π⨎-11h(t)wi(t)t-x2dt,x∈(-1,1),
where w1(t)=1-t/1+t and w2(t)=1+t/1-t are the weights and h(t) is a smooth function. The proposed automatic quadrature schemes constructed cover the different weights functions that have not been considered by Obayis et al. [10]. Unlike the work of Obayis et al. [10] which takes advantage of Clenshaw-Curtis Chebyshev quadrature methods, the construction of our AQSs is based on Chebyshev Gauss quadrature methods. The method proposed by Hui and Shia [7] works only for even number N of knots points whereas AQS for our problems works well for all N and converges very fast to the exact solution.Present in Section2 are the mathematical concepts of the third and fourth kinds of Chebyshev polynomials and singular integrals. In Section 3, the construction of automatic quadrature schemes for semibounded solutions of hypersingular integrals is given. Rate of convergence of the suggested method is discussed in Section 4. Numerical examples and results are provided in Section 5.


## 2. Mathematical Concepts of Third and Fourth Kinds of Chebyshev Polynomials and Singular Integrals

The Chebyshev polynomialsVm(t) and Wm(t) are defined as follows.Definition 1 (Mason and Handscomb [11]).
The chebyshev polynomialVm(t) of the third kind on [-1,1] is a polynomial of degree m in t defined by
(5)Vm(t)=cos⁡(m+1/2)θcos⁡1/2θ,t=cos⁡θ,0≤θ≤π.Definition 2 (Mason and Handscomb [11]).
The chebyshev polynomialWm(t) of the fourth kind on [-1,1] is a polynomial of degree m in t defined by
(6)Wm(t)=sin(m+1/2)θsin1/2θ,t=cos⁡θ,0≤θ≤π.Both polynomials(7)Pim(t)=Vm(t),i=1,Wm(t),i=2,
satisfy the recurrence relation
(8)Pimt=2tPim-1t-Pim-2t,hhhm=2,3,…,i=1,2
with the same starting value
(9)Pi0(t)=V0(t)=1,i=1,W0(t)=1,i=2,Pi1(t)=V1(t)=2t-1,i=1,W1(t)=2t+1,i=2.The third and fourth kinds of Chebyshev polynomials are orthogonal on[-1,1] with the weights w1(t) and w2(t), respectively. They also have continuous orthogonality given in [11], respectively, as
(10)llVm,Vj=∫-11‍1+t1-tVmtVjtdt=0,m≠jπ,m=j,Wm,Wj=∫-11‍1-t1+tWmtWjtdti2=0,m≠j;(m,j≤N),π,m=j.The third kind and fourth kind of Chebyshev polynomials have discrete orthogonality, respectively, as(11)Vm,Vj=∑k=0N‍1+tkVmtkVjtk=0,m≠j,(m,j≤N),N+1,0≤j=m≤N,Wm,Wj=∑k=0N‍1-tkWmtkWjtk=0,m≠j,(m,j≤N),N+1,j=m,(0≤m≤N).The four kinds of Chebyshev polynomials are connected via the relations(12)(1+t)Vk(t)=Tk+1(t)+Tk(t),(13)Wk(t)=Uk(t)+Uk-1(t),(14)(1-t)Wk(t)=Tk+1(t)-Tk(t),(15)Vk(t)=Uk(t)-Uk-1(t),
and the differentiation formulae
(16)ddxTk(t)=kUk-1(t),ddxUk(t)=(k+1)Tk+1(t)-tUkt2-1.The Hilbert type integral transform is as follows.Lemma 3 (Mason and Handscomb [11]).
Consider the following(17)⨍0πcos⁡mθcos⁡θ-cos⁡αdθ=πsinmαsinα,
for any α∈[0,π], m=1,2,3,….Lemma 4 (Mason and Handscomb [11]).
Consider the following(18)⨍0πsinmθsinθcos⁡θ-cos⁡αdθ=-πcos⁡mα,
for any α∈[0,π], m=1,2,3,….The estimations for some integrals are established.Lemma 5 (Israilov [12]).
Let0<σ≤1 and 1-x≥δN, 0<δN<1/2, and for all i={1,2}(19)γix=wix∫-1x-δN‍dtwitt-x,γi*(x)=wi(x)∫x+δN1‍dtwi(t)(t-x),Ti(x,δN,σ)=wi(x)∫x-δNx+δN‍t-xσ-1dtwi(t),
where w1(x)=1-x/1+x and w2(x)=1+x/1-x.
The following estimations are true:(i)
|γi(x)|≤ln⁡2/δN+π/2,(ii)
γ1(-x)=-γ2*(x), γ2(-x)=-γ1*(x),(iii)
Ti(x,δN,σ)≤ϕi(σ)δNσ, max⁡{ϕ1(σ),ϕ2(σ)}≤10×2-σ+(1+10×2-1-σ)σ-1.


## 3. Automatic Quadrature Schemes for Semibounded Hypersingular Integrals

To construct AQSs for the semibounded HSIs (4), the density function h(t) is approximated with the truncated series of third and forth kinds of Chebyshev polynomials:
(20)h(t)≈SiN(t)=∑m=0N‍b1mVm(t),i=1,∑m=0N‍b2mWm(t),i=2.Substituting (20) into (4), we have
(21)Hih,x≈HiN(h,x)=wi(t)π⨎-11SiN(t)wi(x)t-x2dt=1-x1+x∑m=0N‍b1mddx1π⨍-11(1+t)Vm(t)1-t2(t-x)dt,i=1,1+x1-x∑m=0N‍b2mddx1π⨍-11(1-t)Wm(t)1-t2(t-x)dt,i=2,=1-x1+xq1(x),i=1,1+x1-xq2(x),i=2,
where
(22)qi(x)=ddx∑m=0N‍b1mWm(x),i=1,ddx∑m=0N‍b2mVm(x),i=2.By applying the orthogonality relations (10), we obtain the coefficients bim, i={1,2} as
(23)bim=1π∫-11‍1+t1-th(t)Vm(t)dt,i=1,1π∫-11‍1-t1+th(t)Wm(t)dt,i=2.For the discrete coefficientsbim, we choose the collocation points, t=tk, at the zeros of TN+1; namely,
(24)t=tk=cos⁡(2k+1)π2N+2,k=0,1,2,…,N,
and, using the orthogonality conditions (11) and relations (12) and (14), respectively, as well as applying composite trapezium rule, we arrive at
(25)bim≈1N+3/2∑k=0N‍1+tkVmtkhtk,i=1,1N+3/2∑k=0N‍(tk-1)Wm(tk)h(tk),i=2.


## 4. Error Estimation for the Semibounded Hypersingular Integrals

Let us introduce the classes of functions.(i)
Hα([-1,1],L) is a class of function satisfying Hölder condition on the interval with the index α and constant L.(ii)
CN,α[-1,1]={h(t):h(N)(t)∈Hα([-1,1],LN)}.(iii)
C[-1,1] is a class of continuous functions on the interval [-1,1] with the norm
(26)h=h(x)C[-1,1]=max⁡x∈[-1,1]h(x).(iv)
Cr[-1,1]={h(t):hr(t)∈C([-1,1])}.LeteiN(t)=h(t)-SiN(t) and let the maximum norm of the error be defined as
(27)eiN=max⁡t∈[-1,1]eiN(t).It is known in [11] that for any x∈(-1,1)(28)e1N(x)=h(N)(ηx)WN(x)2NN!,ηx∈[-1,1],e2Nx=hNηxVNx2NN!,ηx∈-1,1,(29)e1N′x=12NN!WNxddxhNηx+hNηxWN′x,ηx∈-1,1,e2N′x=12NN!VNxddxhNηx+hNηxVN′x,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhηx∈-1,1.As a consequence of (29), we arrive at the following estimations:
(30)e1N=e2N≤(2N+1)M12N·N!,hhhhhhiihhhhhhhihhhhhh(31)e1N′=e2N′≤M12·2N-3(N-3)!1+L1NN+7N-42(N-1)(N-2),
where
(32)M=max⁡M1,M2,M1=max⁡ηx∈-1,1hNηx,M2=max⁡ηx∈[-1,1]ddxh(N)(ηx)<∞,(33)L1N=1+3N+7N-42(N-1)(N-2)1+3N.Lemma 6.
The Lipschitz constantsLjiN, j={3,4}, i={1,2} of VN(x) and WN(x) are
(34)VNx-VNy≤L31Nx-y,VN′x-VN′y≤L32Nx-y,WNx-WNy≤L41Nx-y,WN′x-WN′y≤L42Nx-y,
where
(35)L31N=L41N=N3N+12N+1,L32N=L42N=N15N+1N+22N+1N-1.Proof.
LetL31N be the Lipschitz constant of the third kind Chebyshev polynomial VN(t); that is,
(36)VNt-VNx≤L31Nt-x,
where
(37)L31N=max⁡ζt∈[-1,1]VN′(ζt),
which gives sequence of numbers:
(38)0,2,10,28,60,110,182,280,408,570….
These sequences can be generated by(39)L31(N)=L31(N-1)+2N2,L31(0)=0,
which leads to
(40)L31N=N3N+12N+1.
In a similar way we can get(41)L41N=N3N+12N+1.
LetL32(N) be the Lipschitz constant of the derivative of third kind Chebyshev polynomial VN′(t); that is,
(42)VN′t-VN′x≤L32Nt-x,
where
(43)L32N=max⁡ζt∈[-1,1]VN′′(ζt).
This Lipschitz constants generate the following sequence of numbers:(44)0,0,8,56,216,616,1456,3024,5712,10032,…,
and its recurrence relation is
(45)L32(N+2)=L32(N+1)+MN,
where
(46)L32(0)=0,M(N)=8D(N+1),D(N+1)=D(N)+R(N+1),D(1)=1,R(N+1)=R(N)+N+12R(1)=1.
These recurrence relations lead to(47)L32(N)=N5(N+1)(N+2)(2N+1)(N-1).
In a similar way we obtain(48)L42(N)=N5(N+1)(N+2)(2N+1)(N-1).Main results are given in the following Theorem7.Theorem 7.
Leth(t)∈CN,α[-1,1] for 0<α≤1 and let the series of Chebyshev polynomials SiN(t) be defined by (20); then the AQS (21) has an error bound
(49)EihC≤0.11M2N-4N-4!1+2.38NL3N+0.57N-4N-1N-2,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi=1,2,
where M is defined by (32) and
(50)L3N=1+2.53ln⁡N0.03(23N3-69N2+100N-48)(N-1)(N-2)(N-3)(N-4)L1N+0.07L2N+0.427N-4N-1N-21+1.11ln⁡N+0.427N-4N-1N-21+1.11ln⁡N+0.03(23N3-69N2+100N-48)(N-1)(N-2)(N-3)(N-4),L1N=1+3N+7N-42(N-1)(N-2)+3(7N-4)2N(N-1)(N-2),L2N=1+3N+23N3-69N2+100N-48(N-1)(N-2)(N-3)(N-4)1+3N.Proof of Theorem7.
In view of error norms (27), we can compute the error bound of AQS (21) for i={1,2} as follows:
(51)Eih,x=Hih,x-HiNh,x=wixπ⨎-11eiNtdtwi(t)t-x2=wixπ⨎-11eiNt-eiNx-eiN′xt-xdtwi(t)t-x2+wi(x)π⨎-11[eiN(x)+eiN′(x)(t-x)]dtwi(t)t-x2.
Taylor series expansion ofeiN(t) around x is given by
(52)eiN(t)=eiN(x)+eiN′(ζt)(t-x),
where
(53)ζt∈(x,t),ift∈(x,1),(t,x),ift∈(-1,x).
Due to (51)-(52), we have
(54)Ei(h,x)≤Ei1(h,x)+Ei2(h,x)+Ei3(h,x),
where
(55)Ei1(h,x)=wi(x)π⨍-11[eiN′(ζt)-eiN′(x)]dtwi(t)(t-x),Ei2(h,x)=wi(x)πeiN(x)⨎-11dtwi(t)t-x2,Ei3(h,x)=wi(x)πeiN′(x)⨍-11dtwi(t)(t-x).
It is easy to check that(56)⨎-111+t1-tdtt-x2=⨎-111-t1+tdtt-x2=0,⨍-111+t1-tdtt-x=-⨍-111-t1+tdtt-x=π.
For the estimation ofEi1(h,x), we divide the interval (-1,1) into three subintervals (-1,x-δN), (x+δN,1), and [x-δN,x+δN]:
(57)Ei1(h,x)≤1πFi1(x)+Fi2(x)+Fi3(x),
where
(58)Fi1(x)=wi(x)∫-1x-δN‍[eiN′(ζt)-eiN′(x)]dtwi(t)(t-x),Fi2(x)=wi(x)∫x+δN1‍[eiN′(ζt)-eiN′(x)]dtwi(t)(t-x),Fi3(x)=wi(x)∫x-δNx+δN‍[eiN′(ζt)-eiN′(x)]dtwi(t)(t-x).
ForFi1(x), according to Lemma 5, we obtain
(59)Fi1x≤2eiN′wi(x)∫-1x-δN‍dtwi(t)(t-x)≤2eiN′ln⁡2δN+π2.
In the similar way, forFi2(x), we have
(60)Fi2x≤2eiN′wi(x)∫x+δN1‍dtwi(t)(t-x)≤2eiN′ln⁡2δN+π2.
Utilizing (30) and (31) into (59) and (60), we arrive at
(61)Fi1x=Fi2x≤1/6M2N-3N-3!1+L1NN+7N-4N-1N-2×ln⁡2δN+π2.
For the error ofFi3(x),
(62)Fi3(x)≤wi(x)∫x-σNx+σN‍eiN′(ζt)-eiN′(x)wi(t)ζt-xdt.
It follows from (29) and Lemma 6 that
(63)eiN′ζt-eiN′x≤12NN!×PiNζt-PiNxddxhNζt+hNζtPiN′ζt-PiN′xddxfNζt≤12NN!×LjiNζt-xddxhNζt+LjiNhNζtζt-x≤M480·2N-5N-5!N2N+12N+1N-1N-2N-3N-4×1+1N+3N2ζt-x≤M240·2N-5N-5!×1+23N3-69N2+100N-482N-1N-2N-3N-4×1+1N+3N2ζt-x≤M240·2N-5N-5!×1+L2NN+23N3-69N2+100N-482N-1N-2N-3N-4×ζt-x,
where
(64)L2N=1+3N+23N3-69N2+100N-48(N-1)(N-2)(N-3)(N-4)1+3N.
Substituting (63) into (62) yields
(65)Fi3x≤M240·2N-5(N-5)!×1+L2NN+23N3-69N2+100N-482N-1N-2N-3N-4×wi(x)∫x-δNx+δN‍dtwi(t).
Using Lemma5 gives
(66)Fi3x≤3.37M240·2N-5N-5!×1+L2NN+23N3-69N2+100N-482(N-1)(N-2)(N-3)(N-4)δN.
By choosingδN=2/N2 for N≥5, we obtain
(67)Fi3x≤0.11M2N-3N-3!×1+L2NN+23N3-69N2+100N-482N-1N-2N-3N-4.
By substituting (61) and (67) into (57), we arrive at
(68)Ei1h,x≤0.64Mln⁡Nπ·2N-3N-3!×0.17(23N3-69N2+100N-48)(N-1)(N-2)(N-3)(N-4)1+1.27ln⁡N+2.11ln⁡NL1N+0.08L2N+2.547N-4N-1N-21+1.11ln⁡N+0.17(23N3-69N2+100N-48)(N-1)(N-2)(N-3)(N-4).(69)Ei1h,x≤2.46Mln⁡N2N-3N-3!×10.59(23N3-69N2+100N-48)2(N-1)(N-2)(N-3)(N-4)1+0.10ln⁡N+L1NN+0.10L1NNln⁡N+23.167N-4N-1N-2+0.05(7N-4)(N-1)(N-2)+10.59(23N3-69N2+100N-48)2(N-1)(N-2)(N-3)(N-4).
Due to (31) and (56),
(70)Ei2(h,x)=0,hhhhhhhhhhhhhhhhhhhhhhhEi3h,x≤21-x20.08M2N-3N-3!×1+L1NN+7N-42(N-1)(N-2).
Since, from the hypothesis in Lemma5,
(71)1-x2≥1-1-δN2,11-x2≤11-1-δN2≤23δN=N3,
we have
(72)Ei3h,x≤0.09M2N-3(N-3)!×N1+L1NN+7N-42(N-1)(N-2).
Substituting (69)–(72) into (54) yields
(73)Eih,x≤0.68Mln⁡Nπ·2N-3N-3!×+0.1723N3-69N2+100N-48N-1N-2N-3N-41+1.27ln⁡N+2.11ln⁡NL1N+0.08L2N+2.547N-4N-1N-21+1.11ln⁡N+0.1723N3-69N2+100N-48N-1N-2N-3N-4+0.42Nln⁡N+0.42ln⁡NL1N+0.42N(7N-4)2ln⁡NN-1N-2+0.1723N3-69N2+100N-48N-1N-2N-3N-4.
ChoosingN≥5 the statement of Theorem 7 follows.


## 5. Numerical Results and Discussion

In this session, we evaluate the hypersingular integrals (4) for different choices of h(t), and compare with the exact solution.Example 1.
Consider the HSIs of the form(74)H1(h,x)=1π1-x1+x⨎-111+t1-th(t)t-x2dt,
where h(t)=2t2-3t+5. The exact solution is H1(h,x)=21-x/1+x(-1+4x).The numerical results in Table1 show that the AQS gives a very good result for the quadratic function h(t)=2t2-3t+5 for the case i=1 and N=2. This is further illustrated in Figure 1(a) where the graphs of the exact and AQS solutions are superimposed for the same choices of singular point values. Details of the accuracy of the proposed AQS is observed from the absolute error in Figure 1(b).Table 1
Results of HSIs forw1(t).
N
x
Exact solution
Approximate solution
Absolute error
2
−0.99999
−2236.0444988237596342
−2236.0444988237596345
3.13
×
10
-
16
−0.91111
−21.535241487484869307
−21.535241487484869310
3.08
×
10
-
18
−0.71111
−9.3563357403104129848
−9.3563357403104129863
1.42
×
10
-
18
−0.31111
−3.0963683860810919235
−3.0963683860810919240
5.86
×
10
-
19
0.00000
−1.0000000000000000000
−1.0000000000000000003
3.00
×
10
-
19
0.31111
0.17718528456311131724
0.17718528456311131709
1.27
×
10
-
19
0.71111
0.75786494952828070264
0.75786494952828070264
6.39
×
10
-
21
0.91111
0.57031832778553277005
0.57031832778553277007
1.40
×
10
-
20
0.99999
0.00670813126012938238
0.00670813126012938238
2.24
×
10
-
22(a) Exact and AQS solutions whenN=2 and h(t)=2t2-3t+5 and (b) absolute error between exact and AQS solution when N=2 and h(t)=2t2-3t+5.
(a)
(b)Example 2.
Consider the HSIs of the form(75)H2(h,x)=1π1+x1-x⨎-111-t1+th(t)t-x2dt,
where h(t)=2t2-3t+5. The exact solution is H2(h,x)=1+x/1-x(5-4x).The numerical results in Table2 show that the AQS (21) gives a very good result for the quadratic function h(t)=2t2-3t+5 for the case i=2 and N=2. Figure 2(a) indicates that the exact and AQS solutions are nearly the same. Further details of the accuracy of the absolute error of the proposed AQS are demonstrated in Figure 2(b).Table 2
Results of HSIs forw2(t).
N
x
Exact solution
Approximate solution
Absolute error
2
−0.99999
0.02012457266627356241
0.0201245726662735624
1.57
×
10
-
21
−0.91111
1.8643200698228626472
1.8643200698228626470
1.43
×
10
-
19
−0.71111
3.2232147018485970132
3.2232147018485970129
2.40
×
10
-
19
−0.31111
4.5263577087926478228
4.5263577087926478225
3.08
×
10
-
19
0.00000
5.0000000000000000000
4.9999999999999999997
3.00
×
10
-
19
0.31111
5.1810684429214884711
5.1810684429214884708
2.42
×
10
-
19
0.71111
5.2460548398163357508
5.2460548398163357508
3.79
×
10
-
20
0.91111
6.2854320328769430625
6.2854320328769430630
2.99
×
10
-
19
0.99999
447.23036596367022628
447.23036596367022632
4.47
×
10
-
17(a) Exact and AQS solutions whenN=2 and ht=2t2-3t+5 and (b) absolute error between exact and AQS solution when N=2 and h(t)=2t2-3t+5.
(a)
(b)Example 3.
Consider the HSIs of the form(76)H1(h,x)=1π1-x1+x⨎-111+t1-th(t)t-x2dt,
where h(t)=1/2+t. The exact solution is H1(h,x)=-1-x/1+x3/3(2+x)2.The numerical results in Table3 show that the AQS (21) is highly accurate for the rational function h(t)=1/2+t for the case i=1 and N=40. Figure 3(a) reveals that there js good agreement between the exact solution and AQS solution which can be seen from absolute error in Figure 3(b) when N=10. The absolute error is improved in Figure 4(b) when N=20 with the corresponding exact and AQS solution in Figure 4(a). Best result is obtained in Figures 5(a) and 5(b) when N=40.Table 3
Results of HSIs forw1(t).
N
x
Exact solution
Approximate solution
Absolute error
10
−0.99999
−258.19308036170334212
−258.17083250027430060
2.23
×
10
-
2
−0.91111
−2.2578118246552894121
−2.2578564435672151572
4.46
×
10
-
5
−0.77111
−1.0634637975870125656
−1.0634476360338990858
1.62
×
10
-
5
−0.33111
−0.2924243743021256602
−0.2924196578457308097
4.72
×
10
-
6
0.00000
−0.1443375672974064411
−0.1443397190188599320
2.15
×
10
-
6
0.33111
−0.0753154744680519215
−0.0753153016422631300
1.73
×
10
-
7
0.77111
−0.0270285335549129334
−0.0270280566695517436
4.77
×
10
-
7
0.91111
−0.0146928400579184367
−0.0146941562302624467
1.32
×
10
-
6
0.99999
−0.0001434451425475821
−0.0001431650832316927
2.80
×
10
-
7
20
−0.99999
−258.19308036170334212
−258.19308021411085378
1.48
×
10
-
7
−0.91111
−2.2578118246552894121
−2.2578118246107028644
4.46
×
10
-
11
−0.77111
−1.0634637975870125656
−1.0634637975433417458
4.37
×
10
-
11
−0.33111
−0.2924243743021256602
−0.2924243742855352783
1.66
×
10
-
11
0.00000
−0.1443375672974064411
−0.1443375672897319217
7.68
×
10
-
12
0.33111
−0.0753154744680519215
−0.0753154744695265617
1.48
×
10
-
12
0.77111
−0.0270285335549129334
−0.0270285335485592653
6.35
×
10
-
12
0.91111
−0.0146928400579184367
−0.0146928400450019102
1.29
×
10
-
11
0.99999
−0.0001434451425475821
−0.0001434451390347442
3.51
×
10
-
12
40
−0.99999
−258.19308036170334212
−258.19308036170338346
4.04
×
10
-
14
−0.91111
−2.2578118246552894121
−2.2578118246552894116
4.50
×
10
-
19
−0.77111
−1.0634637975870125656
−1.0634637975870125637
1.85
×
10
-
18
−0.33111
−0.2924243743021256602
−0.2924243743021256612
9.67
×
10
-
19
0.00000
−0.1443375672974064411
−0.1443375672974064412
1.13
×
10
-
19
0.33111
−0.0753154744680519215
−0.0753154744680519211
4.00
×
10
-
19
0.77111
−0.0270285335549129334
−0.0270285335549129336
1.72
×
10
-
19
0.91111
−0.0146928400579184367
−0.0146928400579184373
5.73
×
10
-
19
0.99999
−0.0001434451425475821
−0.0001434451425475826
4.86
×
10
-
19(a) Exact and AQS solutions whenN=10 and h(t)=1/(2+t) and (b) absolute error between exact and AQS solution when N=10 and h(t)=1/(2+t).
(a)
(b)(a) Exact and AQS solutions whenN=20 and h(t)=1/(2+t) and (b) absolute error between exact and AQS solution when N=20 and h(t)=1/(2+t).
(a)
(b)(a) Exact and AQS solutions whenN=40 and h(t)=1/(2+t) and (b) absolute error between exact and AQS solution when N=40 and h(t)=1/(2+t).
(a)
(b)Example 4.
Consider the HSIs of the form(77)H2(h,x)=1π1+x1-x⨎-111-t1+th(t)t-x2dt,
where h(t)=1/2+t. The exact solution is H2(h,x)=1+x/1-x3/(2+x)2.The numerical results in Table4 show that the AQS (21) is highly accurate for the rational function h(t)=1/2+t for the case i=2 and N=40. Figure 6(a) shows that there is good agreement between the exact solution and AQS solution which can be seen from absolute error in Figure 6(b) when N=10. The absolute error is reduced in Figure 7(b) when N=20 with the corresponding exact and AQS solution in Figure 7(a). The best result is obtained and illustrated in Figures 8(a) and 8(b) when N=40.Table 4
Results of HSIs forw2(t).
N
x
Exact solution
Approximate solution
Absolute error
10
−0.99999
0.00387291557000340015
0.00387141593292902562
1.50
×
10
-
6
−0.91111
0.31504763162812503076
0.31505448317559772604
6.85
×
10
-
6
−0.77111
0.41231131092313516294
0.41230879229200872013
2.52
×
10
-
6
−0.33111
0.44083450592426358350
0.44083359884530457966
9.07
×
10
-
7
0.00000
0.43301270189221932338
0.43301675286046974386
4.05
×
10
-
6
0.33111
0.44963976686376800332
0.44963389568653105901
5.87
×
10
-
6
0.77111
0.62742591722366873321
0.62741343576104182143
1.25
×
10
-
5
0.91111
0.94767578680690190767
0.94770492350476572039
2.91
×
10
-
5
0.99999
86.066655193121619385
86.053938870581623578
01.27
×
10
-
2
20
−0.99999
0.00387291557000340015
0.00387291555159212453
1.84
×
10
-
11
−0.91111
0.31504763162812503076
0.31504763156789346359
6.02
×
10
-
11
−0.77111
0.41231131092313516294
0.41231131089736085386
2.58
×
10
-
11
−0.33111
0.44083450592426358350
0.44083450592691374976
2.65
×
10
-
12
0.00000
0.43301270189221932338
0.43301270187830840205
1.39
×
10
-
11
0.33111
0.44963976686376800332
0.44963976684337128944
2.04
×
10
-
11
0.77111
0.62742591722366873321
0.62742591718912403402
3.46
×
10
-
11
0.91111
0.94767578680690190767
0.94767578677558019222
3.13
×
10
-
11
0.99999
86.066655193121619385
86.066655108152556362
8.50
×
10
-
8
40
−0.99999
0.00387291557000340015
0.00387291557000339911
1.04
×
10
-
18
−0.91111
0.31504763162812503076
0.31504763162812503215
1.40
×
10
-
18
−0.77111
0.41231131092313516294
0.41231131092313516515
2.28
×
10
-
18
−0.33111
0.44083450592426358350
0.44083450592426358289
6.14
×
10
-
19
0.00000
0.43301270189221932338
0.43301270189221932269
6.75
×
10
-
19
0.33111
0.44963976686376800332
0.44963976686376800195
1.31
×
10
-
18
0.77111
0.62742591722366873321
0.62742591722366873124
1.91
×
10
-
18
0.91111
0.94767578680690190767
0.94767578680690190858
1.31
×
10
-
18
0.99999
86.066655193121619385
86.066655193121618680
7.05
×
10
-
16(a) Exact and AQS solutions whenN=10 and h(t)=1/(2+t) and (b) absolute error between exact and AQS solution when N=10 and h(t)=1/(2+t).
(a)
(b)(a) Exact and AQS solutions whenN=20 and h(t)=1/(2+t) and (b) absolute error between exact and AQS solution when N=20 and h(t)=1/(2+t).
(a)
(b)(a) Exact and AQS solutions whenN=40 and h(t)=1/(2+t) and (b) absolute error between exact and AQS solution when N=40 and h(t)=1/(2+t).
(a)
(b)


## 6. Conclusion

In this paper, we have constructed the automatic quadrature schemes for the semibounded weighted Hadamard type hypersingular integrals using the truncated series of the third and fourth kinds of Chebyshev polynomials. The unknown coefficients are found by applying the discrete orthogonality condition and the choice of collocation points on(-1,1). The error estimations for the semibounded solutions of the hypersingular integrals are obtained. Numerical examples have shown the accuracy of the developed methods and agreed with theoretical findings.



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*Source: 387246-2014-12-01.xml*