Semi Bounded Solution of Hypersingular Integral Equations of the First Kind on the Rectangle
Zainidin Eshkuvatov1,∗, Massamdi Kommuji1, Rakhmatullo Aloev2, Nik Mohd Asri Nik Long3, Mirzoali Khudoyberganov2
1FacultyofScienceandTechnology,UniversitiSainsIslamMalaysia,Malaysia
2FacultyofMathematics,NationalUniversityofUzbekistan,Uzbekistan
3DepartmentofMathematics,FacultyofScience,UniversitiPutra,Malaysia
Received June 29, 2019; Revised September 17, 2019; Accepted September 23, 2019
Copyright c2020 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License
Abstract
A hypersingular integral equations (HSIEs) of the first kind on the interval[−1,1]with the assumption that kernel of the hypersingular integral is constant on the diagonal of the domain is considered. Truncated series of Chebyshev polynomials of the third and fourth kinds are used to find semi bounded (unbounded on the left and bounded on the right and vice versa) solutions of HSIEs of first kind. Exact calculations of singular and hypersingular integrals with respect to Chebyshev polynomials of third and forth kind with corresponding weights allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature formula is extended for regular kernel integrals. Three examples are provided to verify the validity and accuracy of the proposed method. Numerical examples reveal that approximate solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights.Keywords
Approximation, Chebyshev polynomials, Convergence, Hypersingular integral equations.1 Introduction
Hypersingular integral equations (HSIEs) of the first kind of the form 1
π = Z 1
−1
ϕ(t)
K(x, t)
(t−x)2 +L1(x, t)
dt=f(x), −1< x <1, (1) encounters in several physical problems such as aerodynamics, hydrodynamics, and elasticity theory (see [1]-[7]).
In 1985, Golberg [1] consider Eq. (1) with the kernelK(x, t) = 1and proposed projection method with the truncated series of Chebyshev polynomials of the second kind together with Galerkin and collacation methods. Uniform convergence and the rate of convergence of projection method are obtained in subspace of Hilbert space for HSIEs (1). In 1992, Martin [2] obtained the analytic solution to the simplest one-dimensional hypersingular integral equation i.e. the case ofK(x, t) = 1andL1(x, t) = 0 in Eq. (1). In 2006, Mandal and Bera have proposed a simple approximate method (Polynomial approximation) for solving a general hypersingular integral equation of the first kind (1) withK(x, x) 6= 0. The method is mostly concentrated with the bounded solution and illustrated proposed method by considering some simple examples. Mandal and Bhattacharya ([4], 2007) proposed approximate numerical solutions of some classes of singular integral equations including HSIEs (1) withK(x, t) = 1 by using Bernstein polynomials as basis. The method was explained with illustrative examples. Convergence of the method is referred to book of Golberg and Chan [5]. In 2009-2010, Boykov et al. ([6]-[7]) proposed spline-collocation method and its justification for the solution of one-dimensional hypersingular integral equations, poly-hypersingular integral equations, and multi-dimensional hypersingular integral equations. Proved convergence of the method and illustrative examples demonstrated to show the accuracy and efficiency of the developed method. Gulsu and Uzturk ([8], 2014) have purposed approximation method for hypersingular integro-differential equations in the most general form under the mixed conditions in terms of the second kind
Chebyshev polynomials. This method transforms mixed hypersingular integro-differential equations and the given conditions into matrix equation which corresponding to a system of linear algebraic equation. The error analysis and convergence for the proposed method is also obtained. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Novin and Fariborzi Araghi ([9], 2019) have applied modification of the homotopy perturbation method (HPM) to HSIEs (1) withK(x, t) = 1, L(x, t) = 0and compared with the standard homotopy perturbation method. the proposed method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed.
Different examples are included to prove so that proposed method gave exact solutions for all type of characteristic equations.
For singular integral equations many efficient methods are derived and proved convergence of the method as well as showed illustrative examples (see Chakrabarti at al. [10]-[11], Dardery and Allan [12], Capobianco et al. [13], Elliot [14]-[15] and so on).
In 2011, Abdulkawi et.al. [16], considered the finite part integral equation (1) with K(x, t) = 1 and used Chebyshev polynomials of 1st and 2nd kind to find bounded solution of Eq. (1). Exactness of the proposed method for the linear density function are showed and illustrated it with examples. Nik Long and Eshkuvatov [17] have used the complex variable function method to formulate the multiple curved crack problems into hypersingular integral equations of the first kind in more general case and these hypersingular integral equations are solved numerically for the unknown function, which are later used to find the stress intensity factor (SIF). In 2016, Eshkuvatov et al. [18], have used modifed homotopy perturbation method (HPM) to solve Eq. (1) on the interval [- 1, 1] with the assumption that the kernelK(x, t)of the hypersingular integral is constant on the diagonal of the domain. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM, reproducing kernel method and Chebyshev expansion method. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points. Eshkuvatov and Narzullaev ([19], accepted in 2018) have solved Eq. (1) using projection method together with Chebyshev polynomials of the first and second kinds to find bounded and unbounded solutions of HSIEs (1) respectively. Existence of inverse of hypersingular operator and exact calculations of hypersingular integral for Chebyshev polynomials allows us to obtain high accurate approximate solution for the case of bounded and unbounded solutions, where the kernelK(x, t)is a constant on the diagonal of the domainD= [−1,1]×[−1,1].
In this note, HSIEs (1) is considered for the cases of semi-bounded solutions and outlined the collocation method together with Gauss-Chebyshev quadrature formula for regular kernel expansions. The structure of the paper is arranged as follows: In section 2, all the necessary tools are outlined and in Section 3, the details of the derivation of the projection method is presented.
Section 4, discusses the existence and uniqueness of the solution in Hilbert space. Finally in Section 4, examples are provided to verify the validity and accuracy of the proposed method, followed by the conclusion in Section 5.
2 Preliminaries
In 2018, Ahdiaghdam ([20]) has solved by using four kind of Chebyshev polynomials for all four cases of solutions (bounded, unbounded, left and right bounded) of singular integral equations (SIEs) of the form
− Z 1
−1
ψ(t) (t−x)α +
Z 1
−1
K(x, t)ψ(t)dt=f(x), −1< x <1, α∈N, (2) whereK(t, x)andf(x)are given real valued Holder continuous functions andψ(t)is the unknown function to be determined.
Convergence of the proposed method is obtained forα={1,2}. On the other hand Ahdiaghdam [20] summarized the work in Mason and Handscomb [21] as follows. LetPr,j(t)be the Chebyshev polynomials of the first-forth kind given by
Pr,j(t) =
Tj(t) = cos(jθ), r= 1
Uj(t) = sin((j+ 1)θ)/sin(θ), r= 2 Vj(t) = cos
j+1
2
θ
/cos θ
2
, r= 3, Wj(t) = sin
j+1
2
θ
/sin θ
2
, r= 4,
(3)
wheret=cosθ. The functionPr,j(t)satisfies the following orthogonality properties
µri,j= 1
πhPr,i, Pr,jir=
0i6=j,
1, i=j= 0, r= 1 1/2, i=j6= 0, r= 1 1/2, i=j, r= 2 1, i=j, r=∈ {3,4},
(4)
with respect to inner product
hf, gir= Z 1
−1
wr(t)f(t)g(t)dt, wherewr(t), r∈ {1,2,3,4}, the weight function defined by
wr(t) = λr(t)
√1−t2, λr(t) =
1, r= 1, 1−t2, r= 2, 1 +t, r= 3, 1−t, r= 4,
(5)
In Mason and Handscomb [21] the following theorem have been proven.
Theorem 1. As a Cauchy principle value integral, we have
Sr,j(x) = 1 π −
Z 1
−1
wrPr,j(t) t−x dt=
Uj−1(x), r= 1,
−Tj+1(x) r= 2, Wj(x) r= 3, Vj(x) r= 4.
(6)
Ahdiaghdam [20] has proved the following statement.
Theorem 2. Form≥1derivative of Chebyshev polynomials has the form
d
dxPr.m(x) =
mUm−1(x), r= 1,
[m−1]/2
X
k=0
2(m−2k)Um−2k−1(x) r= 2,
m−1
X
k=0
(−1)k2(m−k)Um−k−1(x) r= 3,
m−1
X
k=0
2(m−k)Um−k−1(x) r= 4.
(7)
Darbery and Allan [12] summarized three term relations of four kind of Chebyshev polynomials which is given in Mason and Handscomb [21].
T0(x) = 1, T(x) =x,
Tn(x) = 2xTn−1−Tn−2(x), n≥2, (8)
U0(x) = 1, U(x) = 2x,
Un(x) = 2xUn−1−Un−2(x), n≥2, (9)
V0(x) = 1, V1(x) = 2x−1,
Vn(x) = 2xVn−1(x)−Vn−2(x), n≥2, (10)
W0(x) = 1, W(x) = 2x+ 1,
Wn(x) = 2xWn−1−Wn−2(x), n≥2. (11)
3 Description of the method
Since kernel in Eq. (1) is constant on the diagonal we can express it as follows
K(x, t) =c0+ (t−x)Q(x) + (t−x)2Q1(x, t), c06= 0. (12) whereQ(x)is smooth function andQ1(x, t)is square integrable kernel. Taking into account Eq. (12) we are able to write Eq.
(1) in the form c0
π = Z 1
−1
ϕ(t)
(t−x)2dt+Q(x)
π −
Z 1
−1
ϕ(t) t−xdt+ 1
π Z 1
−1
L(x, t)ϕ(t)dt=f(x), −1< x <1, (13)
whereL(x, t) =Q1(x, t) +L1(x, t), and the first integral in Eq. (13) is being understood as the Hadamard finite-part.
Main aim is to find semi-bounded solution of Eq. (13). Hence, we search solution in the form
ϕ(x) =wr(x)u(x), r∈ {3,4}, (14)
wherew3(x)andw4(x)are defined by (5). Substituting (14) into (13) yields c0
π Z 1
−1
wr(t)
(t−x)2u(t)dt+Q(x) π
Z 1
−1
wr(t) t−xu(t)dt + 1
π Z 1
−1
wr(x)L(x, t)u(t)dt=f(x), r∈ {3,4} −1< x <1, (15) Introducing notations
Hru=c0
π = Z 1
−1
wr(x)
(t−x)2u(t)dt, r∈ {3,4}
Cru=Q(x)
π −
Z 1
−1
wr(x)
t−xu(t)dt, r∈ {3,4} (16)
Lru= 1 π
Z 1
−1
L(x, t)wr(x)u(t)dt, r∈ {3,4}.
leads to the operator equation
Hru+Cru+Lru=f, r∈ {3,4}, f ∈L2ρ, u∈L1ρ, (17) where the spacesL2ρandL1ρare defined in Section 3.
It is known that the hypersingular operatorHgcan be considered as differential Cauchy operator i.e., Hgu= d
dxCgu= d dx
1 π −
Z 1
−1
ω(t) t−xu(t)dt
. (18)
Therefore, Eq. (17) may be viewed as an integro-differential Prandtl’s type equation (see [13]). On the other hand from (6) it follows that
CgVm(x) = 1 π −
Z 1
−1
r1 +t 1−t
Vm(t)dt
(t−x) =Wm(x), m= 0,2, ..., CgWm(x) = 1
π − Z 1
−1
r1−t 1 +t
Wm(t)dt
t−x =−Vm(x), m= 0,2, ...,
(19)
In Ahdiaghdam [20] it is shown that differentiating Eq. (19) leads to HgVm(x) = d
dxCgVm(x) = d
dxWm(x) =
m−1
X
k=0
2(m−k)Um−k−1(x), m= 1,2, ..., HgWm(x) = d
dxCgWm(x) =−d
dxVm(x) =
m−1
X
k=0
(−1)k+12(m−k)Um−k−1(x), m= 1,2, ...,
(20)
form= 0
HgV0(x) =HgW0(x) = 0, (21)
Moreover,
Tm+1(x) = 1
2[Um+1(x)−Um−1(x)], m= 0,1,2, ..., xUm(x) = 1
2[Um+1(x) +Um−1(x)], m= 0,1.2, ...
Vm(x) =Um(x)−Um−1(x), m= 0,1,2,· · ·
Wm(x) = (−1)mVm(−x) =Um(x) +Um−1(x)), m= 0,1,2,· · ·.
(22)
whereU−1(x) = 0andUn(−x) = (−1)nUn(x).
It can easily be shown from (16), (19), (20) and (22) that H3Vm(x) =c0
rπ 2
m−1
X
k=0
2(m−k)φm−k−1(x), m= 1,2,· · ·, H4Wm(x) =−c0
rπ 2
m−1
X
k=0
(−1)k2(m−k)φm−k−1(x), m= 1,2,· · · , C3Vm(x) =Q(x)
rπ 2
φm(x) +φm−1(x)
, n= 0,1,· · ·, C4Wm(x) =−
rπ 2
φm(x)−φm−1(x)
, n= 0,1,· · ·,
(23)
whereφ−1(x) = 0and
φm(x) = r2
πUm(x). (24)
Eqs. (23) -(24) are crucial to the rest of our analysis.
To find an approximate solution of Eq.(15),u(t)is approximated by u(t)∼=un,r(t) =
n
X
j=0
bj,rPj,r(t), r∈ {3,4}, (25)
which gives approximate solution of Eq. (13) as follows ϕ(x)≈ϕn,r(x) =ωr(x)
n
X
j=0
bj,rPj,r(x), r∈ {3,4}, (26)
where unknown coefficientsbj,rneed to be defined. To do this end we consider two cases:
• Letr = 3, i.e. the solution of Eq. (17) is bounded at the left end and unbounded on the right end of the interval[−1,1].
Substituting (25) into (15) and using Eqs. (19) - (22) yields b0,3(Q(x) +ψ0,3(x))
+
n
X
j=1
bj,3
( c0
j−1
X
k=0
2(j−k)Uj−k−1(x) +Q(x)(Uj(x) +Uj−1(x)) +ψj,3(x) )
=f(x),
(27)
whereUn(x)is the Chebyshev polynomials of the second kind withU−1(x) = 0and ψj,3(x) = 1
π Z 1
1
r1 +t
1−tL(x, t)Vj(t)dt. (28)
For the collocation method we choose the suitable collocation points{xi}ni=1such as roots ofVn+1(x)or(1−x2)Un−1(x).
Then Eq. (27) leads to a system of algebraic linear equations b0,3 Q(xi) +ψ0,3(xi)
+
n
X
j=1
bj,3 (
c0
j−1
X
k=0
2(j−k)Uj−k−1(xi) +Q(xi)(Uj(xi) +Uj−1(xi)) +ψj,3(xi) )
=f(xi), i= 0,1,· · ·,
(29)
Solving the system of Eq. (29) for the unknown coefficientsbj,3, j= 0,1, . . . , nand substituting the values ofbj,3into Eq. (26) yields the numerical solution of Eq. (13).
• Leti = 4, i.e. the solution of Eq. (17) is unbounded at the left end and bounded on the right end of the interval[−1,1].
Substitute (25) into (15) forr= 4and apply Eqs. (19) - (22) to get b0,4 Q(x) +ψ0,4(x)
+
n
X
j=1
bj,4
( c0
j−1
X
k=0
(−1)k2(j−k)Uj−k−1(x)−Q(x)(Uj(x)−Uj−1(x)) +ψj,4(x) )
=f(x), (30)
where
ψj,4(x) = 1 π
Z 1 1
r1−t
1 +tL(x, t)Wj(t)dt. (31)
To solve Eq. (30) for unknown parametersbj,2collocation method is used by choosing the suitable node points{xi}ni=1 such as roots ofWn+1(x)or(1−x2)Un−1(x). It reduces Eq. (30) to a system of linear equation
b0,4 Q(xi) +ψ0,4(xi) +
n
X
j=1
bj,4
( c0
j−1
X
k=0
(−1)k2(j−k)Uj−k−1(xi) +Q(xi)(Uj(xi) +Uj−1(xi)) +ψj,4(xi) )
=f(xi), i= 0,1,· · ·, n,
(32)
Solving the system of Eq. (32) for the unknown coefficientsbj,4, j = 0,1, . . . , nand substituting the values ofbj,4into Eq. (26) gives the approximate solution of Eq. (13).
4 Quadrature method
In the Section 3, we have obtained two types of weighted kernel integrals (28) and (31). It is known that many weighted kernel integrals have not exact solution. So that we need to derive suitable quadrature for numerical computation of weighted kernel integrals.
In this section, we develop Gauss-Chebyshev quadrature rule with Gauss-Lobotto nodes for weighted kernel integrals ((28) and (31)). In Kythe [22], states that the Gauss quadrature formula of the form
Z b a
w(x)f(x)dx=
n+1
X
i=1
Aif(xi), (33)
is exact for allf ∈P2n+1if the weightsAiand the nodesxican be found from orthogonal polynomials approximation off(x) and roots of orthogonal polynomials respectively on the interval[a, b].
In particular, if[a, b] = [−1,1]andw3(t)(x), w4(t)(x)are defined by (6) and orthogonal polynomials are the Chebyshev polynomialsVn(t), Wn(t)of the third and forth kind respectively then resulting formulas of Eq. (33) are known as Gauss- Chebyshev quadrature rule. To derive it let us define the nodesxi,1andxi,2as the zeros ofVn+1(x)andWn+1(x)respectively,
xk,1=cos
(2k−1)π 2n+ 3
, k= 1,2, ..., n+ 1 (34)
xk,2=cos 2kπ
2n+ 3
, i= 1,2, ..., n+ 1. (35)
Aghigh et al. [23] states that any Gauss qudrature formula with weight function on[a, b]can be written as follows Z b
a
w(x)f(x)dx≈
n+1
X
k=1
Akf(xk), wk= Kn+1 Kn
Rb
a Pn2(x)w(x)dx
Pn(xk)P0(xk−1), (36) wherePn(x)is a sequence of polynomials orthogonal with respect to weight functionw(x)on[a, b],Kn denotes the leading coefficients ofPn(x)and{xk}n+1k=1are the polynomial roots (Pn+1(x)).
Using (36) and taking into account (34)-(35), Gauss-Chebyshev quadrature rule can be constructed as follows Lemma 3. (Aghigh et al. [23]) Open Gauss-Chebyshev quadrature rule is given as
Z 1
−1
r1 +x
1−xf(x)dx=
n+1
X
k=1
Ak,1f(xk,1) +Rn+1,1(f). (37)
whereAk,1= 2π
2n+ 3(1 +xk,1)andxk,1is defined by(34). Similarly Z 1
−1
r1−x
1 +xf(x)dx=
n+1
X
k=1
Ak,2f(xk,2) +Rn+1,2(f). (38)
whereAk,2= 2π
2n+ 3(1−xk,2)andxk,2is defined by(35).
The word ”open” is used for not including endpoints. We usually omit ”open” since all Gaussian rules with positive weight function are of the open type.
Theorem 4. (Johnson and Riess 1977). Gaussian QF has precision2n+ 1only if the pointsxi, i= 0,1, ..., nare the zeros of φn+1(x), whereφn+1(x)are orthogonal polynomials.
Theorem 5. (Israilov [25] ) Iff ∈C2n+2[a, b], then the error of Gaussian QF is given by Rn(f) =Iab(f)−In(f) = f2n+2(ξ)
(2n+ 2)!
Z b a
w(x)P2n+1(x)dx, ξ∈[a, b] (39) wherePn+1(x)is the monic polynomials of degreen+ 1withn+ 1distinct zeros andw(x)is a weight function.
Due to Theorem 5 and referring to Eshkuvatov et al. [19] the following error estimation can be obtained.
Theorem 6. Iff ∈C2n+2[−1,1], then the error of Gauss-Chebyshev QFs(37)and(38)are given by Rn+1,1(f) =Rn+1,2(f) = π
22n+1
f2n+2(ξ1)
(2n+ 2)!, ξ1∈[−1,1], (40)
Now we extend Gauss-Chebyshev QF (37) - (38) for the weight kernel integrals (28) and (31) which are given in Section 3.
In many problems of HSIEs regular kernelL(x, t)will be given as convolution type L(x, t) =
m
X
i=1
ci(x)di(t). (41)
In the case of convolution type kernel (41), the Gauss-Chebyshev QF for the regular kernel in (28) has the form ψj,3(x) = 1
π Z 1
1
r1 +t
1−tL(x, t)Vj(t)dt= 1 π
m
X
i=1
ci(x) Z 1
1
r1 +t
1−tdi(t)Vj(t)dt
= 1 π
m
X
i=1 n+1
X
k=1
ci(x)Ak,1(di(tk,1)Vj(tk,1)), Ak,1= 2π
2n+ 3(1 +tk,1),
(42)
Similarly
ψj,4(x) = 1 π
Z 1 1
r1−t
1 +tL(x, t)Wj(t)dt= 1 π
m
X
i=1
ci(x) Z 1
1
r1−t
1 +tdi(t)Wj(t)dt
= 1 π
m
X
i=1 n+1
X
k=1
ci(x)Ak,2(di(tk,2)Wj(tk,2)), Ak,2= 2π
2n+ 3(1−tk,2),
(43)
wheretk,1 andtk,2 are defined by (34) and (35) respectively. For non convolution regular kernelL(x, t)case, we have the following Gauss-Chebyshev QF
ψj,3(x) = 1 π
Z 1 1
r1 +t
1−tL(x, t)Vj(t)dt
=
n+1
X
k=1
Ak,1f1(x, tk,1), Ak,1= 2
2n+ 3(1 +tk,1),
(44)
in the same way we obtain
ψj,4(x) = 1 π
Z 1 1
r1−t
1 +tL(x, t)Wj(t)dt
=
n+1
X
k=1
Ak,2f2(x, tk,2), Ak,1= 2
2n+ 3(1 +tk,2),
(45)
where
f1(x, tk,1) =L(x, tk,1)Vj(tk,1), f2(x, tk,2) =L(x, tk,2)Wj(tk,2).
5 Existence of the solution for the semi-bounded cases
First of all we introduce two type of spaces. First one is weighted Hilbert space. LetL2ρr(−1,1), r ∈ {3,4}denote the space of real-valued square integrable functions with respect to weighted functions
ρ3(x) =
r1 +x
1−x, ρ4(x) =
r1−x 1 +x. The spacesL2ρr, r∈ {3,4}are endowed with an inner product
hu, vir= Z 1
−1
ρr(t)u(t)v(t)dt, r∈ {3,4}, (46)
where the norm||u||r=p
hu, uir, r∈ {3,4}.
It is known thatφn, n= 0,1, . . . ,defined by (24) are orthonormal polynomials with
||φn||22ρ= Z 1
−1
ρ(t)φ2n(t)dt= 1, ρ(t) =p
1−t2, (47)
and the system{φk}∞k=0is a complete orthonormal basis forL2ρ, so that ifu∈L2ρthen u=
∞
X
k=0
hu, φkiφk, (48)
where the sum converges inL2ρ. Consequently, Parseval’s equality holds, i.e.
||u||22ρ=
∞
X
k=0
hu, φki2. (49)
Second space for the case ofr= 3is a subspace of the Hilbert space which is denoted byL1ρ⊆L2ρ3. It consists of allu∈L1ρ
such that
||u||21ρ=
∞
X
m=1 m−1
X
k=0
22(m−k)2hu, Vmi2<∞. (50) This subspace can be made themselves into Hilbert space if we define it’s inner product ofu∈L1ρandv∈L1ρby
hu, vi1ρ=
∞
X
m=1 m−1
X
k=0
2(m−k)hu, Vmihv, Vmi. (51)
and hence ifu∈L1ρthen
u=
∞
X
m=1
hu, Vmivm,k, vm,k=
m−1
X
k=0
2(m−k)Um−k−1. (52)
with the norm
kuk21ρ=hu, ui=
* ∞ X
m=0
hu, Vmivm,k,
∞
X
n=0
hu, Vnivn,j
+
=
* ∞ X
m=0
2(m+ 1)
∞
X
k=m+1
hu, Vk
! Um,
∞
X
n=0
2(n+ 1)
∞
X
j=n+1
hu, Vj
Un
+
=
∞
X
m=1 m−1
X
k=0
[2(m−k)]2
!
hu, Vmi2. (53)
We do proof that operators(Hr+Cr+Lr), r ∈ {3,4}in Eq. (17) are invertible. To do these end we consider each case separately.
• For the caser= 3, we extend the operatorH3defined by (16) as a bounded operator fromL1ρtoL2ρ3. Let u=
∞
X
m=0
hu, VmiVm, then using (20) we obtain
H3u=
∞
X
m=0
hu, VmiH3Vm=c0
∞
X
m=1 m−1
X
k=0
2(m−k)Um−k−1
!
hu, Vmi. (54) Restructuring the infinite sum in Eq. (54) and use orthogonality condition of Chebyshev polynomials of the 2nd kind we arrive at
H3u=c0
∞
X
m=0
2(m+ 1)
∞
X
j=m+1
hu, Vji
Um. (55) and observe that
kH3uk22ρ3 =c20
∞
X
m=0
[2(m+ 1)]2
∞
X
j=m+1
hu, Vji
2
=c20
∞
X
m=1 m−1
X
k=0
22(m−k)2hu, Vmi2=c20kuk21ρ. (56) From this it follows that
||Hu||2ρ3=|c0|||u||1ρ.
So thatH3is an isometry with constant multiplicity. It is not hard to show thatH3−1:L1ρ→L2ρ3exists and is given by H3−1u= 1
c0
∞
X
m=0
hu, Vmi
−(m+ 1)
Vm− 1 mUm−1
, (57)
henceH3is invertible.
Lemma 7. The norm of operatorH3−1:L1ρ→L2ρ3is
kH3−1k= 1
|c0|. (58)
Lemma 7 can be proven in a similar way as given in Eshkuvatov et al. [18].
Lemma 8. LetA, Bbe operators acting in Hilbert space. IfAis bounded andB is compact then the productsABand BAare compact.
Lemma 2 is proven in Reed and Simon [26, Theorem VI.12, pp. 200].
Lemma 9. The operatorsC3:L1,ρ→L2ρ3 andH3−1C3:L1ρ→L2ρ3are compact.
It can be proved by following Eshkuvatov et al. [18].
Assumption 10. λ= 1does not belong to the null space ofN(I+λH−1(C+L)) ={0}.
Lemma 11. Let the Assumption 4 is satisfied, then the operatorH3+C3+L3is invertible, and the main equation(17) for the caser= 3has a unique solution.
Proof of the Lemma 11: Consider more general HSIEs of the form.
[H3+λ(C3+L3)]u=f (59)
Forλ= 1it returns to Eq. (17). Let us rewrite Eq. (59) in the form
[I+λH3−1(C3+L3)]u=H3−1(f). (60)
Due to (57) operatorH3is invertible therefore it is bounded. If we are able to show that operatorI+λH3−1(C3+L3)are invertible then operator equation Eq. (60) has a unique solution.
Let us define operatorsTr, Tl:L1,ρ→L2ρ3 as T1u=
∞
X
m=0
hu, φmiφm, T2u=
∞
X
m=1
hu, φmiφm−1,
T3u=
∞
X
m=1
hu, φm−1iφm, T4u=
∞
X
m=1
hu, φm−1iφm−1. (61) From Eq. (23) it can be easily shown that
C3u=C3
∞
X
m=0
hu, VmiVm
!
= rπ
2Q(x)(T1+T2+T3+T4)u.
These operatorsTr, r ∈ {1,2,3,4}are bounded fromL1,ρ →L2ρ3. Moreover, boundedness and the compactly embed- dability ofTrfromL1,ρ →L2ρ3 (Berthold Berthold et al. (1992, Conclusion 2.3)) implies the compactness of operators Tr, r∈ {1,2,3,4}.
Since operatorsTrare compact, its linear combinations is also compact. As we knowQ(x)is a continues function on the closed interval[-1,1]andTrare compact, their productC3is also compact due to Lemma 8. Since operatorsC3andL3
are compact thenC3+L3:L1(ρ)→L1(ρ)is also compact. We know thatH3−1(C3+L3) :L1(ρ)→L1(ρ)is a compact operator due to Lemma 8.
Due to the Fredholm theorem Reed and Simon (1980, Teorem VI.14) the inverse operator(I+λH3−1(C3+L3))[-1]of the operator function(I+λH3−1(C3+L3)), exists for allλ∈C C1, whereC1is a discrete subset ofC(i.e. a setC1has no limit points inC) and forλ∈C1the null spaceN(I+H3−1(C3+L3))is finite, that isz =λ−1is the eigenvalue of H3(C3+L3)with finite multiplicity. These facts allows us to suppose thatλ= 1does not belong toC1. Thus we have proved that (15) is solvable and has unique solution for the caser= 3.
In the case ofr= 4corresponding changes can be done to provide the existence and uniqueness of the solutions of HSIEs (17).
6 Numerical Results
We very often need to use polynomial values of the Chebyshev polynomials for the numerical computation. Table 1 refers to the first few function of Chebyshev polynomials of firstTn(x)and second kindsUn(x)respectively.
Table 1:Chebyshev polynomials of the first and second kind
n Tn(x) Un(x)
0 1 1
1 x 2x
2 2x2−1 4x2−1
3 4x3−3x 8x3−4x
4 8x4−8x2+ 1 16x4−12x2+ 1 5 16x5−20x3+ 5x 32x5k−32x3+ 6x 6 32x6−48x4+ 18x2−1 64x6−80x4+ 24x2−1
Table 2 refers to the first few function of Chebyshev polynomials of thirdVn(x)and forth kindsWn(x).
6.1 Case 1 r = 3. Bounded solution on the left and unbounded solution on the right
Example 1:Let us investigate the following HSIEs.
1 π
Z 1
−1
(1 + 2(t−x))
(t−x)2 ϕ(t)dt+ 1 π
Z 1
−1
1 2e2xt3
ϕ(t)dt=f(x), (62)
where
f(x) =−32x3−32x2+ 24x+ 4 + 1 2e2x.
Table 2:Chebyshev polynomials of the third and forth kind
n Vn(x) Wn(x)
0 1 1
1 2x−1 2x+ 1
2 4x2−2x−1 4x2+ 2x−1
3 8x3−4x2−4x+ 1 8x3+ 4x2−4x−1
4 16x4−8x3−12x2+ 4x+ 1 16x4+ 8x3−12x2−4x+ 1 5 32x5−16x4+ 32x3+ 12x2+ 6x−1 32x5−16x4−32x3−12x2+ 6x+ 1 6 64x6−32x5−80x4+ 32x3+ 24x2−6x−1 64x6+ 32x5−80x4−32x3+ 24x2+ 6x−1
The exact solution of Eq. (62) is
ϕ(x) =
r1 +x
1−x(−16x3+ 24x2−12x+ 8). (63)
Solution:Comparing (62) with (13) we get
c0= 1, Q(x) = 2, L(x, t) = 1
2e2xt3. (64)
From (27)-(28) and (64) it follows that b0,3(2 +ψ0,3(x))
+
n
X
j=1
bj,3
(j−1 X
k=0
(−1)k2(j−k)Uj−k−1(x) + 2(Uj(x) +Uj−1(x)) +ψj,3(x) )
=−32x3−32x2+ 24x+ 4 + 1 2e2x.
(65)
whereU−1(x) = 0and
ψj,3(x) = 1 π
Z 1 1
1 2e2xt3
r1 +t
1−tVj(t)dt. (66)
It can be easily obtain that
t3= 3
8V0(t) +3
8V1(t) +1
8V2(t) +1
8V4(t). (67)
Using (66) and orthogonality conditions (4) we obtain ψ0,3(x) = 3
16e2x, ψ1,3(x) = 3
16e2x, ψ2,3(x) = 1
16e2x, ψ3,3(x) = 1
16e2x, ψj,3(x) = 0, j≥4. (68) Substituting (68) into (65) and equating like powers ofxleads to
b0,3= 8, b1,3=−6, b2,3= 4, b3,3=−2 (69) which leads to identical exact solution
ϕ(x) =1 +x
1−x[b0,3V0(x) +b1,3V1(x) +b2,3V2(x) +b3,3V3(x)]
=
r1 +x
1−x(−16x3+ 24x2−12x+ 8). (70)
To find approximate solution substitute (68) into (65) and choose collocation pointsxias the root ofVn+1(x)which are xi=cos(2i+ 1)π
2n+ 3 , i= 0, ..., n, (71)
it follows that the system of algebraic equations (65) has the form b0,3(2 + 3
16e2xi) +
n
X
j=1
bj,3
(j−1 X
k=0
(−1)k2(j−k)Uj−k−1(xi) + 2(Uj(xi) +Uj−1(xi)) +ψj,3(xi) )
=−32x3i −32x2i + 24xi+ 4 + 1
2e2xi, i= 0,1,· · ·n.
(72)
Solving Eq. (72) at the collocation points (71) for the different value ofn, we obtain the numerical solution of Eq. (62). The errors of numerical solution of Eq. (62) are summarized in Table 3. The results in Table 3 shows that method proposed is exact
Table 3:Numerical solution of Example 1
x ErrorsRn, withn= 5 ErrorsRn, forn= 6 ErrorsRn, forn= 8 ErrorsRn, forn= 10
-0.9999 0 0 7.7842×10−19 1.1469×10−19
-0.901 0 0 2.1755×10−17 3.2053×10−18
-0.725 0 0 3.2162×10−17 4.7386×10−18
-0.436 0 0 2.9765×10−17 4.3855×10−18
-0.015 0 0 3.1517×10−17 4.6436×10−18
0.015 0 0 3.1502×10−17 4.6414×10−18
0.436 0 0 3.3535×10−17 4.9410×10−18
0.725 0 0 4.1357×10−17 6.0937×10−18
0.901 0 0 6.2334×10−17 9.1842×10−18
0.9999 0 0 1.8389×10−15 2.7096×10−16
however the roots has many level of square roots(i.e : p 23 +√
2) it makes the equation more complicated and leads to a system of equations that cannot solve for largen. However, the numerical results are decreased asnincreased. Table 3 reveals that method proposed is very accurate and it coincides with exact solution when we choose five termsn= 5only.
6.2 Case 2, r = 4. Bounded solution on the right and unbounded solution on the left
Example 2:Consider the following HSIEs of the form 1
π Z 1
−1
ϕ(t)
(t−x)2dt+1 π
Z 1
−1
xsint
t ϕ(t)dt=f(x), (73)
wheref(x) =−4x2+ 2x−2and exact solution of Eq. (73) is not known.
Solution:Approximate solution is searched as
ϕ(x) =ϕn,4(x) =
r1−x 1 +x
3
X
j=0
bj,4Wj(x). (74)
Substitute (74) into (73) withn= 3yields
3
X
j=0
bj,4
1 π
Z 1
−1
1−t 1 +t
Wj(t) (t−x)2dt+ 1
π Z 1
−1
1−t 1 +t
xsint
t Wj(t)dt
=f(x). (75)
Using Maclaurin series forsintwe obtain
sint
t ≈1−t2 3!+t4
5!+t6
7!. (76)
Applying (19) - (22) and taking into account Table 2, we get b0,4ψ0,4(x) +b1,4
h−2 +ψ1,4(x)i +b2,4
h−(8x−2) +ψ2,4(x)i +b3,4
h−(24x2−8x−4) +ψ3,4(x)i
=f(x), (77)
where
ψj,4(x) = 1 π
Z 1
−1
x
1−t2 3!+t4
5!+t6 7!
r1−t
1 +tWj(t)dt (78)
Since
1−t2 3!+t4
5!+t6
7! =c0W0(t) +c1W1(t) +c2W2(t) +c3W3(t) +c4W4(t) +c5W5(t) +c6W6(t) (79) By equating the same powers oftwe find
c0= 0.9197, c1= 0.0396, c2=−0.0396, c3=−5.0223e−04
c4= 5.0223e−04, c5= 3.1002e−06, c6=−3.1002e−06. (80) Substitutions (79) - (80) into (78) and orthogonality conditions (4) yields
ψ0,4(x) = 0.9197x, ψ1,4(x) = 0.0396x, ψ2,4(x) =−0.0396x, ψ3,4(x) =−5.0223e−04x
ψ4,4(x) = 5.0223e−04x, ψ5,4(x) = 3.1002e−06x, ψ6,4(x) =−3.1002e−06x, ψj,4(x) = 0, j≥7. (81) By equating like powers ofxin (77) from both sides we find that
b0,4= 8.6985C+ 0.6676, b1,4=C+4
3, b2,4=C, b3,4=1
6, bj,4= 0, j≥4. (82) Exact solution is achieved when we substitute Eq. (82) into (74) i.e.
ϕ(x) =
r1−x 1 +x
8.6985C+ 0.6676 + (C+4
3)W1(x) +CW2(x) +1 6W3(x)
. (83)
The roots ofWn+1(x) = 0leads
xk =cos 2k
2n+ 3π
, k= 1,2, ..., n+ 1 (84)
Solving (77) at collocation points (84) yields
b0= 0.6675, b1= 4
3, b2= 0, b3= 1
6 (85)
The error of numerical solution of (73) are summarised in Table 4.
Table 4.Error term of Example 2
x Exact Approximate Error
-0.9999 -117.6502 -117.6530 2.8284×10−3 -0.901 -1.7606 -1.7607 8.7640×10−5
-0.725 0.5675 0.5674 5.0091×10−5
-0.436 1.5613 1.5613 3.1913×10−5
-0.015 1.8317 1.8317 2.0302×10−5
0.015 1.8367 1.8366 1.9702×10−5
0.436 1.8447 1.8447 1.2534×10−5
0.725 1.6541 1.6541 7.9855×10−6
0.901 1.1759 1.1759 4.5641×10−6
0.9999 0.0413 0.0413 1.4142×10−7
Example 3:Let us consider the following HSIEs.
1 π
Z 1
−1
ϕ(t)
(t−x)2dt+1 π
Z 1
−1
(tex)ϕ(t)dt=−4x+1
4ex, (86)
The exact solution of Eq. (86) given by Ahdiaghdam (2018) is ϕ(x) =p
1−x2(2x). (87)
which is in the form of Chebyshev polynomials of second kind. In this work, we obtain the solution in the form of Chebyshev polynomials of fourth kind.
Solution:The approximate solution is searched by choosing n=5 as follows ϕ(x) =ϕn,4(x) =
r1−x 1 +x
5
X
j=0
bj,4Wj(x). (88)
Subtitute Eq. (88) into Eq. (86) and from Eq. (30) - Eq. (31) it follows that b0,4 ψ0,4(x)
+
5
X
j=1
bj,4 (j−1
X
k=0
(−1)k2(j−k)Uj−k−1(x) +ψj,4(x) )
=−4x+1
4ex, (89)
where
ψj,4(x) = 1 π
Z 1 1
tex r1−t
1 +tWj(t)dt. (90)
Collocation method is used to solve Eq. (89) for unknown parametersbj,2. Collocation points{xi}5i=0are chosen as roots of Wn+1(x)or(1−x2)Un−1(x)and reduces Eq. (89) to a system of linear equation
b0,4 ψ0,4(xi) +
5
X
j=1
bj,4 (j−1
X
k=0
(−1)k2(j−k)Uj−k−1(xi) +ψj,4(xi) )
=−4xi+1
4exi, (91) Solving the system of Eq. (91) obtains
b0,4= 0, b1,4=1
2, b2,4=1
2, b3,4= 0, b4,4= 0, b5,4= 0. (92) Subtitute result in Eq. (88) obtains
ϕ(x) =ϕ5,4(x) =
r1−x
1 +x(2x2+ 2x), (93)
which is the solution for the Eq. (86) in the form of Chebyshev polynomials of fourth kind.
7 Conclusions
In this note, we have developed projection method for solving HSIEs of the first kind, where the kernelK(x, t)is constant on the diagonal of the rectangle regionD. Collocation method are used to obtain a system of algebraic equations for the unknown coefficients. Examples 1 and 2 verify that the developed method is very accurate and stable for HSIEs of the first kind. Numerical solution are obtained with the help of Matlab software.
Acknowledgment
This work was supported by Universiti Sains Islam Malaysia (USIM) under RMC Research Grant Scheme (FRGS, 2018).
Project code is USIM/FRGS/FST/055002/51118.
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