wavelet based numerical solution of second kind ... · where the hypersingular integrals are...

Applied Mathematical Sciences, Vol. 10, 2016, no. 54, 2687 - 2707HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2016.66194

Wavelet Based Numerical Solution of Second Kind

Hypersingular Integral Equation1

Swaraj Paul and M. M. Panja

Department of MathematicsVisva-Bharati, Santiniketan-731235

West Bengal, India

B. N. Mandal2

Physics and Applied Mathematics UnitIndian Statistical Institute

203, B T Road, Kolkata 700108, India

Copyright c© 2016 Swaraj Paul, M. M. Panja and B. N. Mandal. This article is distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

Abstract

A Legendre multiwavelet based method is developed in this paper to solve second kind

hypersingular integral equation by converting it into a Cauchy singular integro-differential

equation. Multiscale representation of the singular and differential operators is obtained

by employing Legendre multiwavelet basis. An estimate of the error of the approximate

solution of the integral equation is obtained. A number of examples are given to illustrate

the efficiency of the numerical method developed here.

Mathematics Subject Classification: 45E99, 65R20, 65T60

Keywords: Legendre multiwavelets, Hypersingular integral equation, Multiscaleapproximation, Error estimation

1This work is supported by a research grant from SERB(DST), No. SR/S4/MS:821/13.2Corresponding author

2688 Swaraj Paul, M. M. Panja and B. N. Mandal

1. Introduction

Integral equations with singular kernels arise in various areas of mathematicalphysics. The singularity of the kernel may be weak or strong. When the singularityis weak (e.g. logarithmic type, Abel type), then the associated integral is defined inthe sense of Riemann integral. If the singularity is strong, then the associated inte-gral no longer exists in the sense of Riemann. However, the associated integral canbe defined in some appropriate sense. For example, if the singularity is of Cauchytype, then it is defined in the sense of Cauchy principal value (CPV). If the singu-larity is stronger than Cauchy type, then also the associated integral can be definedin an appropriate manner. Singular integral equations, whose kernels have singular-ity stronger than Cauchy, are generally known as hypersingular integral equations,and the associated integral is defined in the sense of Hadamard finite part integral.These integral equations arise in a natural way in various problems of mathematicalphysics. Martin [17] gave a number of examples of hypersingular integral equationswhich arise in potential theory, hydrodynamics and elastostatics. Later Martin [18],Chakrabarti and Mandal [6] employed some elementary and straightforward meth-ods to solve the simple hypersingular integral equation in closed form involvinglogarithmically singular integral while Chakrabarti [7] developed a direct function-theoretic method to solve it in closed form involving hypersingular integral. Buhring[4] proposed a fully discretized quardrature method for the approximate solution ofthe simple hypersingular integral equation.

As application of fluid dynamics, Parsons and Martin [20, 21] formulated somewater scattering problems involving thin straight vertical or inclined plates or thincurved plates completely submerged or partially immersed in infinitely deep water,in terms of hypersingular integral equations of first kind. The kernel of such a hy-persingular integral equation has a hypersingular part and a non-singular smoothpart. Parsons and Martin [20,21] employed an expansion collocation method involv-ing Chebyshev polynomials of the second kind to solve the hypersingular integralequations numerically and presented very accurate numerical estimates for quanti-ties of physical interest for various geometrical configurations of the thin straightor curved plates. Later Midya et al.[19], Mandal and Gayen[15], Kanoria and Man-dal[12] studied a number of water wave scattering problems using hypersingularintegral equation formulations wherein the integral equations are of first kind with akernel having a hypersingular part and a regular part. Chan et al.[8] demonstratedthe applications of hypersingular integral equations of first kind to fracture mechan-ics in the theory of elasticity.

For solving hypersingular integral equations of second kind there exists severalanalytical as well as numerical methods in the literature. For example, the colloca-tion and Galerkin methods were used by Iokimidis[11] to solve numerically a secondkind hypersingular integral equation arising in elasticity for crack problems, the col-location method was used by Dragos[10] to solve Prandtl’s hypersingular integral

Solution of hypersingular integral equation 2689

equation arising in aerodynamics, the complex variable method related to Riemann-Hilbert boundary value problem was used by Chakrabarti et al.[5] to solve Prandtl’sequation in closed form. Mandal and Bera[16] solved this Prandtl’s equation using asimple method based on polynomial approximation and obtained the exact solution.De Klerk[9] earlier employed Lp-approximation method to solve hypersingular inte-gral equation of second kind wherein the problem of solving the integral equationwas formulated as solving a minimization problem.

In the collocation methods various types of orthogonal polynomials are used inthe expansion of the unknown function of an integral equation. However, the ap-proximate solution so obtained fails to provide local information such as smoothnessor regularity of the solution. It may be possible that a function satisfying a singularintegral equation may belong to the class Cα(0 < α < 1) where Cα denotes the classof continuous function with Holder exponent α(0 < α < 1). It is thus desirable tosearch for an appropriate method which can provide the local information in thenumerical solutions.

Multiresolution analysis (MRA) of function space in terms of refinable functionsand wavelets with compact support has emerged as an efficient mathematical tool foranalyzing phenomena arising in diverse fields of science and engineering. Numericalmethods based on expansion in wavelet basis of some MRA of the space of squareintegrable functions defined over a finite domain have been employed successfully toobtain numerical solutions of weakly singular integral equations of second kind(cf.Paul et al.[22]) which provide local informations about the function. This has alsobeen the case for Cauchy singular integral equations of second kind as observed inthe recent study by Paul et al.[23]. The success of this study on weakly singularand Cauchy singular integral equations of second kind has motivated us to extendthe method to hypersingular integral equations of second kind.

It may be noted that Alpert et al.[1] and Alpert et al.[2] are the pioneers in thedevelopment of generation of wavelets involving polynomials as refinable functions(called multiwavelets, also see Keinert[13]), and used them to obtain approximatesolution of second kind singular integral equations with logarithmic type kernel(weakly singular). Lakestani et al.[14] employed Legendre multiwavelets to solve aweakly singular Fredholm integro-differential equation with Abel-type kernel whilePaul et al.[22, 23] employed Legendre multiwavelets to solve singular integral equa-tions of second kind with Abel type kernel and Cauchy type kernel.

In this paper we consider Fredholm integral equations of second kind with hyper-singular kernel, of the two forms as given by

a(x)u(x) + b(x)

∫ 1

0

u(t)

(t− x)2dt = F (x) (1.1a)


and

a(x)u(x) +

∫ 1

0

b(t)u(t)

(t− x)2dt = F (x) (1.1b)

where the hypersingular integrals are defined in the sense of Hadamard finite part,a(x), b(x) and F (x) are known L2 [(0, 1)] functions and u(x) is an unknown whosesolution is sought in the class of L2 [(0, 1)] functions. The paper is organised asfollows. Reduction of the hypersingular integral equations (1.1a) and (1.1b) toequivalent Cauchy singular integro-differential equation is given in section 2. Prop-erties of Legendre multiwavelets, their two scale relations are presented in section3. The section 4 is concerned with multiscale approximation of a function and mul-tiscale representation of the derivative operator in the Legendre multiwavelet basis.Representation of Cauchy singular integro-differential operator is given in section5 while in section 6 representation of an operator A defined in section 2, is given.In section 7 the second kind Fredholm integral equation with hypersingular kernelis reduced to a system of linear equations and in section 8 error estimate for theapproximate solution together with an estimate of local Holder index is presented.A few illustrative numerical examples are given in section 9 while a short conclusionis given in section 10.

2. Reduction to Cauchy singular integro-differential equation

We consider the Fredholm integral equation of second kind with hypersinularkernel as given by equation (1.1a). Here the hypersingular integral is defined in thesense of Hadamard finite part of order 2 as given by

∫ 1

0

u(t)

(t− x)2dt = lim

ǫ→0

{∫ x−ǫ

0

u(t)

(t− x)2dt+

∫ 1

x+ǫ

u(t)

(t− x)2dt (2.1)

−u(x− ǫ) + u(x+ ǫ)

ǫ

}

, 0 < x < 1.

Following Boykov et al.[3], we can write∫ 1

0

u(t)

(t− x)2dt = −u(0)

x− u(1)

1− x+ lim

ǫ→0

{∫ x−ǫ

0

u′(t)

(t− x)dt (2.2)

+

∫ 1

x+ǫ

u′(t)

(t− x)dt

}

= −u(0)x

− u(1)

1− x+

∫ 1

0

u′(t)

t− xdt, 0 < x < 1

where the integral in the right side of (2.2) is defined in the sense of CPV integral.Thus the hypersingular integral equation (1.1a) can be expressed as

(A1u) (x) + (L1Du) (x) = f(x), 0 < x < 1. (2.3)

where

(A1u) (x) = x(1− x)a(x)u(x)− xb(x)u(1)− (1− x)b(x)u(0) (2.4a)


(L1u) (x) = ω1(x)

∫ 1

0

u(t)

t− xdt, ω1(x) = x(1− x)b(x) (2.4b)

(Du) (x) = u′(x) (2.4c)

f(x) = x(1− x)F (x). (2.4d)

It may be noted that (2.3) is a Cauchy singular integro-differential equation.

Similarly the integral equation (1.1b) can be reduced to another Cauchy singularintgro-differential equation.

a(x)u(x) +

∫ 1

0

b(t)u′(t)

t− xdt+

∫ 1

0

b′(t)u(t)

t− xdt− b(0)u(0)

x− b(1)u(1)

1− x= F (x), (2.5)

if the function b(x) vanishes at the x = 0 and x = 1, then the Eq.(2.5) reduces to

a(x)u(x) +

∫ 1

0

b(t)u′(t)

t− xdt+

∫ 1

0

b′(t)u(t)

t− xdt = F (x). (2.6)

Thus the hypersingular integral equation (1.1b) involving b(x) vanishing at endpoints can be expressed as

(A2u) (x) + (L2Du) (x) + (L3u) (x) = f(x), 0 < x < 1, (2.7)

where

(A2u) (x) = a(x)u(x) (2.8a)

(L2u) (x) =

∫ 1

0

ω2(t)u(t)

t− xdt, ω2(x) = b(x) (2.8b)

(Du) (x) = u′(x) (2.8c)

(L3u) (x) =

∫ 1

0

ω3(t)u(t)

t− xdt, ω3(x) = b′(x) (2.8d)

f(x) = F (x). (2.8e)

3. Legendre multiwavelets

We use the Legendre polynomials P0(x), P1(x), ..., PK−1(x) in the multiresolutionanalysis of L2([0, 1]) generated by the scale functions

φi(x) := (2i+ 1)12Pi(2x− 1), i = 0, 1, ..., K − 1; 0 ≤ x < 1. (3.1)

and wavelets ψi(x)(i = 0, 1, ..., K − 1). At a particular resolution j, these are givenby

φij,k(x) := 2j

2φi(2jx− k), j ∈ N ∪ {0}. (3.1a)

ψij,k(x) := 2j

2ψi(2jx− k), j ∈ N ∪ {0}. (3.1b)

For a given j(> 0), translation is denoted by the symbol k (k = 0, 1, ..., 2j − 1).Clearly the support of φij,k(x) and ψ

ij,k(x) is

[

k2j, k+1

2j

)

for all i = 0, ..., K − 1. The


functions φij,k(x) form an orthonormal basis for the approximation space V Kj in

which the inner product is defined by

< f, g >=

∫ 1

0

f(x)g(x)dx.

The two-scale relations among the scale functions between two consecutive scalesj, j + 1 are given by

φij,k(x) =1√2

K−1∑

r=0

(

h(0)i,r φ

rj+1,2k(x) + h

(1)i,r φ

rj+1,2k+1(x)

)

(3.2)

The wavelets ψij,k(x), (i = 0, 1, ..., K − 1; k = 0, 1, ..., 2j − 1) for given j can befound by using the relation

ψij,k(x) =1√2

K−1∑

r=0

(

g(0)i,r φ

rj+1,2k(x) + g

(1)i,r φ

rj+1,2k+1(x)

)

(3.3)

Here the symbols H = 1√2

(

h(0) ... h(1)

)

and G = 1√2

(

g(0) ... g(1)

)

with h(s) ≡[

h(s)i,r

]

, g(s) ≡[

g(s)i,r

]

(s = 0, 1) appearing in (3.2) and (3.3) are known as low- and

high-pass filters for the MRA generated by Legendre multiwavelets involving K

vanishing moments of their wavelets. The low- and high-pass filters H and G andexplicit forms of scale functions φl(x), wavelets ψl(x) (l = 0, 1, ..., K − 1) for K = 4and K = 5 can be found in Paul et al.[22].

4. Multiscale approximation

4.1. Representation of a function. For a function f ∈ L2[0, 1], the orthogonalprojection of f into the approximation space V K

J (the linear span of φiJ,k(x), i =

0, 1, . . . , K − 1, k = 0, 1, . . . 2J − 1) is

PV KJ

: L2[0, 1] → V KJ

so that

f(x) ≈ (PVKJf)(x) ≡

2J−1∑

k=0

K−1∑

i=0

ciJ,k φiJ,k(x). (4.1)

The decomposition

PVK0

⊕

(

J−1⊕

j=0WK

j

) : L2([0, 1]) → V K0

⊕

(

J−1⊕

j=0

WKj

)


of f(x) into the approximation space V K0 and detail spaces WK

j (0 ≤ j ≤ J − 1) isgiven by

f(x) ≈ fMSJ (x) ≡

PV K0

⊕

(

J−1⊕

j=0WK

j

)f

(x) (4.2)

=K−1∑

i=0

ci0,0 φi(x) +

J−1∑

j=0

2j−1∑

k=0

dij,k ψij,k(x)

where cij,k(

=< f(x), φij,k(x) >)

and dij,k(

=< f(x), ψij,k(x) >)

are the coefficients in

the expansion of f(x) in the basis of the approximation space V K0 and detail space

WKj respectively.

As in Paul et al.[22] the following notations are introduced for convenience.

Φj,k(x) =(

φ0j,k(x), φ

1j,k(x), . . . φ

K−1j,k (x)

)

,

Ψj,k(x) =(

ψ0j,k(x), ψ

1j,k(x), . . . ψ

K−1j,k (x)

)

.

The bases for approximation space V KJ , detail space WK

j andJ⊕

j=0

WKj are then

denoted byΦJ :=

(

ΦJ,0(x),ΦJ,1(x), . . .ΦJ,2J−1(x))

(4.3)

which is a vector having 2JK components,

Ψj :=(

Ψj,0(x),Ψj,1(x), . . .Ψj,2j−1(x))

(4.4)

which is a vector having 2jK components, and

JΨ := (Ψ0,Ψ1, . . .ΨJ) (4.5)

which is a vector with (2J+1 − 1)K components. Also we introduce the symbols

cJ,k :=(

c0J,k, c1J,k, ..., c

K−1J,k

)

,

dj,k :=(

d0j,k, d1j,k, ..., d

K−1j,k

)

,

cJ :=(

cJ,0, cJ,1, . . . cJ,2J−1

)

(4.6)

which is a vector with 2JK components,

dj :=(

dj,0,dj,1, . . .dj,2j−1

)

(4.7)

which is vector with 2jK components, and

Jd := (d0,d1, . . .dJ) (4.8)

which is vector having (2J+1 − 1)K components. Then (4.1) and (4.2) can be ex-pressed respectively as

(

PV KJf)

(x) = ΦJ cTJ (4.9)


and

f(x) ≈ fMSJ (x) ≡

PV K0

⊕

(

J−1⊕

j=0WK

j

)f

(x) = (Φ0, (J−1)Ψ)

cT0

(J−1)dT

(4.10)

4.2. Representation of the derivative of a function. Multiscale representationof the derivative of a function in Legendre multiwavelet basis is now given. Dueto the finite discontinuity of the elements of Legendre multiwavelet basis in theirdomain, representation of the derivative is defined in the weak sense. Let D

(

≡ ddx

)

denote the derivative operator. In order to construct the representation of D inthe Legendre multiwavelet basis, we consider the evaluation of integrals involvingproduct of elements in the basis and their images under D. Now we can write

(4.11)

(

Dφl1)

(x) =

K−1∑

l2=0

ρDl2,l1φl2(x) +

J−1∑

j2=0

2j2−1∑

k2=0

βDl2l1(j2, k2)ψ

l2j2,k2

(x)

,

(4.12)

(

Dψl1j1,k1)

(x) =

K−1∑

l2=0

(

αDl2l1(j1, k1)φ

l2(x)

+

J−1∑

j2=0

2j2−1∑

k2=0

γDl2l1(j2, k2; j1, k1)ψ

l2j2,k2

(x)

,

where

ρDl1l2=

∫ 1

0

φl1(x)d

dx

(

φl2(x))

dx, (4.13)

αDl1l2(j2, k2) =

∫ 1

0

φl1(x)d

dx

(

ψl2j2,k2(x))

dx,

βDl1l2(j1, k1) =

∫ 1

0

ψl1j1,k1(x)d

dx

(

φl2(x))

dx,

γDl1l2(j1, k1; j2, k2) =

∫ 1

0

ψl1j1,k1(x)d

dx

(

ψl2j2,k2(x))

dx.

Since the integrals in (4.13) involve derivative of discontinuous functions, these aredefined in the weak sense. Since the explicit forms of the Legendre multiwaveletbasis functions are known, the integrals in (4.13) can be evaluated by splitting theintegration domain at the points of discontinuity of the wavelets.


Thus the multiscale representation 〈 (Φ0, (J−1)Ψ), D(Φ0, (J−1)Ψ)〉 of D in thebasis (Φ0, (J−1)Ψ) can be written in the form

DMSJ =

ρρρD αααD (0) αααD (1) . . . . . . αααD (J − 1)βββD (0) γγγD (0, 0) γγγD (0, 1) . . . . . . γγγD (0, J − 1)βββD (1) γγγD (1, 0) γγγD (1, 1) . . . . . . γγγD (1, J − 1)

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

βββD (J − 1) γγγD (J − 1, 0) γγγD (J − 1, 1) . . . . . . γγγD (J − 1, J − 1)

(2JK)×(2JK)

(4.14)

where

ρρρD =[

ρDl1l2

]

K×K, (4.15)

αααD(j) =[

αDl1l2(j, k)

]

K×2jK,

βββD(j) =[

βDl1l2(j, k)

]

2jK×K,

γγγD(j1, j2) =[

γDl1l2(j1, k1; j2, k2)

]

2j1K×2j2K.

Now

D(Φ0, (J−1)Ψ) = (Φ0, (J−1)Ψ) 〈 (Φ0, (J−1)Ψ), D(Φ0, (J−1)Ψ)〉= (Φ0, (J−1)Ψ) DMS

J . (4.16)

We now show that DMSJ is a nilpotent matrix of order K. From (4.16) we find

DK(Φ0, (J−1)Ψ) = (Φ0, (J−1)Ψ)(

DMSJ

)K. (4.17)

Since the basis in the Legendre multiwavelet basis consists of piecewise continuouspolynomials of degree at most K − 1, all the elements of DK(Φ0, (J−1)Ψ) are zero

so that(

DMSJ

)K ≡ 0.

5. Evaluation of integrals

Here the integrals in the multiscale representation of the operators L1,L2 and L3

are evaluated.

5.1. Integrals involving scale functions. To evaluate the multiscale representa-tion of L1 given in (2.4b) we use the notation

ρ1;l1l2 =

∫ 1

0

∫ 1

0

ω1(x) φl1(x) φl2(t)

t− xdt dx, l1, l2 = 0, 1, .., K − 1, (5.2)

so that

ρ1;l1l2 =

∫ 1

0

ω1(x) φl1(x) C l2(x) dx,

where

C l2(x) =

∫ 1

0

φl2(t)

t− xdt (5.3)


=

ln| 1−xx

|, for l2 = 0,

√3(

2 + (2x− 1)ln| 1−xx

|)

, for l2 = 1,

√5(

6x− 3 + (6x2 − 6x+ 1)ln| 1−xx

|)

, for l2 = 2,

√73

(

60x2 − 60x+ 11 + (60x3 − 90x2 + 36x− 3)ln| 1−xx

|)

, for l2 = 3.

The values of ρ1;l1l2, l1, l2 = 0, 1, .., K− 1(K = 4) can be obtained either by elemen-tary integration or by using a appropriate quadrature rule depending on the formof b(x).

5.2. Integrals involving product of scale functions and wavelets. We denote

α1;l1l2(j, k) =

∫ 1

0

∫ 1

0

ω1(x) φl1(x) ψl2j,k(t)

t− xdt

so that

α1;l1l2(j, k) = 2j

2

K−1∑

l3=0

{

g(0)l2l3

∫ 1

0ω1(x) φ

l1(x) C l3(2j+1x− 2k)dx (5.4)

+g(1)l2l3

∫ 1

0ω1(x) φ

l1(x) C l3(2j+1x− 2k − 1)dx

}

.

where C l3(x) is defined in (5.3), g(0)l2l3, g

(1)l2l3

are the components of high-pass filter.The integrals in (5.4) can be evaluated explicitly if ω(x) is a polynomial.If we denote

β1;l1l2(j, k) =

∫ 1

0

∫ 1

0

ω1(x) ψl1j,k(x) φ

l2(t)

t− xdt dx,

then

β1;l1l2(j, k) =

∫ 1

0

ω1(x) ψl1j,k(x) C

l2(x) dx. (5.5)

which can be evaluated explicitly if ω1(x) is a polynomial. It may be noted thatif ω(x) is not a polynomial, then the integrals in (5.4) and (5.5) can be evaluatednumerically by using an appropriate quadrature rule.

5.3. Integrals involving product of wavelets. We denote

γ1;l1l2(j1, k1, j2, k2) =

∫ 1

0

∫ 1

0

ω1(x) ψl1j1,k1

(x) ψl2j2,k2(t)

t− xdt dx, j2 ≥ j1 ≥ 0.

Then using definition (3.1b) followed by transformation of variable one can obtain

γ1;l1l2(j1, k1, j2, k2) =

∫ 1

0

∫ 1

0

ω1(x+k12j1

)ψl1(x)ψl2j2−j1,r2(t)

r1 + t− xdt dx (5.6)

where

k2 − 2j2−j1k1 = r12j2−j1 + r2, r1 ∈ Z, r2 = 0, 1, .., 2j2−j1 − 1. (5.7)


For r1 6= 0, the integral in (5.6) can be evaluated by an appropriate quadrature rule,while for r1 = 0, it can be written in the form

γ1;l1l2(j1, k1, j2, k2) = 2j2−j1

2

K−1∑

l3=0

{

g(0)l2l3

∫ 1

0ω1

(

x+ k1

2j1

)

ψl1(x)C l3(2j2−j1+1x− 2r2)dx+

g(1)l2l3

∫ 1

0ω1

(

x+ k1

2j1

)

ψl1(x)C l3(2j2−j1+1x− 2r2 − 1)dx

}

.(5.8)

For j2 ≤ j1, (5.6) is of the form

γ1;l1l2(j1, k1, j2, k2) =

∫ 1

0

∫ 1

0

ω1(x+k22j2

)ψl1j1−j2,r2(x)ψl2(t)

r1 + t− xdt dx. (5.9)

where

k1 − 2j1−j2k2 = r12j1−j2 + r2, r1 ∈ Z, r2 = 0, 1, .., 2j1−j2 − 1. (5.10)

Also, for j2 ≤ j1, (5.8) is of the form

γ1;l1l2(j1, k1, j2, k2) = 2j1−j2

2

K−1∑

l3=0

{

g(0)l2l3

∫ 1

0

ω1

(

x+ k2

2j2

)

ψl1j1−j2,r2(x)Cl3(2x)dx+

g(1)l2l3

∫ 1

0

ω1

(

x+ k2

2j2

)

ψl1j1−j2,r2(x)Cl3(2x− 1)dx

}

.(5.11)

Thus all the integrals involved in the representation of L1 given in (2.4b) in Legendremultiwavelet basis are now evaluated.

In the representation of L2 and L3 given by (2.8b) and (2.8b), the quantities ρi;l1l2αi;l1l2(j, k), βi;l1l2(j, k), γi;l1l2(j1, k1, j2, k2) for i = 2, 3 can be obtained by using thefollowing formula.

ρi;l1l2 = −∫ 1

0

ωi(x) φl2(x) C l1(x) dx, (5.12a)

αi;l1l2(j, k) = −∫ 1

0

ωi(x) ψl2j,k(x) C

l1(x) dx. (5.12b)

βi;l1l2(j, k) = −2j

2

K−1∑

l3=0

{

g(0)l1l3

∫ 1

0

ωi(x) φl2(x) C l3(2j+1x− 2k)dx (5.12c)

+g(1)l1l3

∫ 1

0

ωi(x) φl2(x) C l3(2j+1x− 2k − 1)dx

}

.

(5.12d)


γi;l1l2(j1, k1, j2, k2) =

−∫ 1

0

∫ 1

0

ωi(x+k1

2j1)ψl1 (x)ψ

l2j2−j1,r2

(t)

r1+t−x dt dx

when j2 ≥ j1, r1 6= 0, k2 − 2j2−j1k1 = r12j2−j1 + r2

−2j2−j1

2

∑K−1l3=0

{

g(0)l1l3

∫ 1

0ωi(

x+k12j1

)

ψl2j2−j1,r2(x)Cl3(2x)dx+

g(1)l2l3

∫ 1

0ωi(

x+k12j1

)

ψl2j2−j1,r2(x)Cl3(2x− 1)dx

}

when j2 ≥ j1, r1 = 0 k2 − 2j2−j1k1 = r12j2−j1 + r2

−∫ 1

0

∫ 1

0

ωi(x+k22j2

)ψl1j1−j2,r2

(x)ψl2 (t)

r1+t−x dt dx

when j1 ≥ j2, r1 6= 0, k1 − 2j1−j2k2 = r12j1−j2 + r2

−2j1−j2

2

∑K−1l3=0

{

g(0)l1l3

∫ 1

0ωi(

x+k22j2

)

ψl2(x)C l3(2j1−j2+1x− 2k2)dx+

g(1)l2l3

∫ 1

0ωi(

x+k22j2

)

ψl2(x)C l3(2j1−j2+1x− 2k2 − 1)dx}

when j1 ≥ j2, r1 = 0, k1 − 2j1−j2k2 = r12j1−j2 + r2

We now denote the matrices with their dimensions in all three cases involving integraloperators Li(i = 1, 2, 3) as

ρρρi := [ρi;l1l2]K×K ,

αααi (j) := [αi;l1l2(j, k)]K×(2jK) ,

βββi (j) := [βi;l1l2(j, k)](2jK)×K ,

γγγi (j1, j2) := [γi;l1l2(j1, k1, j2, k2)](2j1K)×(2j2K) . (5.13)

5.4. Multiscale representation of L. To present the multiscale representation ofintegral operators Li given in (2.4b), (2.8b) and (2.8d), we write

(

Liφl1)

(x) =

K−1∑

l2=0

ρi;l1l2φl2(x) +

J−1∑

j2=0

2j2−1∑

k2=0

βi;l2l1(j2, k2)ψl2j2,k2

(x)

(5.14)

and

(

Liψl1j1,k1)

(x) =

K−1∑

l2=0

αi;l2l1(j1, k1)φl2(x) +

J−1∑

j2=0

2j2−1∑

k2=0

γi;l2l1(j2, k2, j1, k1)ψl2j2,k2

(x)

(5.15)

Using (5.14) and (5.15) the multiscale representation

〈 (Φ0, (J−1)Ψ), Li(Φ0, (J−1)Ψ)〉


of Li in the basis (Φ0, (J−1)Ψ) can be written in the form

LMSi J =

ρρρi αααi (0) αααi (1) . . . . . . αααi (J − 1)βββi (0) γγγi (0, 0) γγγi (0, 1) . . . . . . γγγi (0, J − 1)βββi (1) γγγi (1, 0) γγγi (1, 1) . . . . . . γγγi (1, J − 1). . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

βββi (J − 1) γγγi (J − 1, 0) γγγi (J − 1, 1) . . . . . . γγγi (J − 1, J − 1)

(2JK)×(2JK)

(5.16)

where the sub-matrices ρρρi,αααi,βββi, γγγi are given in (5.13). Thus

Li(Φ0, (J−1)Ψ) = (Φ0, (J−1)Ψ) LMSi J . (5.17)

6. Evaluation of integrals in the representation of ATo obtain multiscale representation of the operator A where A = A1 is defined in

(2.4a) and A = A2 in (2.8a), we write

(6.1)

(

Aφl1)

(x) =

K−1∑

l2=0

ρAl2l1φl2(x) +

J−1∑

j2=0

2j2−1∑

k2=0

βAl2l1(j2, k2)ψ

l2j2,k2

(x)

(6.2)

(

Aψl1j1 ,k1

)

(x) =K−1∑

l2=0

αAl2l1(j1, k1)φ

l2(x) +J−1∑

j2=0

2j2−1∑

k2=0

γAl2l1(j2, k2, j1, k1)ψ

l2j2,k2

(x)

where,ρρρA = [ρAl1l2

, l1, l2 = 0, 1, ..., K − 1]K×K

with

ρAl1l2=

∫ 1

0

φl1(x)(

Aφl2)

(x)dx, (6.3)

αααA(j) = [αααA(j, k), k = 0, 1, ..., 2j − 1]K×2jK

withαααA(j, k) = [αAl1l2

(j, k), l1, l2 = 0, 1, ..., K − 1]K×K

and

αAl1l2(j, k) =

∫ 1

0

φl1(x)(

Aψl2j,k)

(x)dx, (6.4)

βββA(j) = [βββA(j, k), k = 0, 1, ..., 2j − 1]2jK×K

withβββA(j, k) = [βAl1l2

(j, k), l1, l2 = 0, 1, ..., K − 1]K×K

and

βAl1l2(j, k) =

∫ 1

0

ψl1j,k(x)(

Aφl2)

(x)dx, (6.5)

γγγA(j1, j2) = [γγγA(j1, k1; j2, k2), k1 = 0, 1, ..., 2j1 − 1, k2 = 0, 1, ..., 2j2 − 1]2j1K×2j2K


with

γγγA(j1, k1; j2, k2) = [γAl1l2(j1, k1; j2, k2), l1, l2 = 0, 1, ..., K − 1]K×K

and

γAl1l2(j1, k1; j2, k2) =

∫ 1

0

ψl1j1,k1(x)(

Aψl2j2,k2)

(x)dx. (6.6)

Then the multiscale representation 〈 (Φ0, (J−1)Ψ), A(Φ0, (J−1)Ψ)〉 of A in the basis(Φ0, (J−1)Ψ) is

AAAMSJ =

ρρρA αααA (0) αααA (1) . . . . . . αααA (J − 1)βββA (0) γγγA (0, 0) γγγA (0, 1) . . . . . . γγγA (0, J − 1)βββA (1) γγγA (1, 0) γγγA (1, 1) . . . . . . γγγA (1, J − 1). . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

βββA (J − 1) γγγA (J − 1, 0) γγγA (J − 1, 1) . . . . . . γγγA (J − 1, J − 1)

(2JK)×(2JK)

(6.7)

Thus

A(Φ0, (J−1)Ψ) = (Φ0, (J−1)Ψ) AMSJ . (6.8)

7. Solution of the integral equation

We seek solution in the class of L2[0, 1] functions. Using the multiscale rep-resentation of u(x(satisfying the hypersingular integral equation (1.1a) or (1.1b) orequivalently the Cauchy singular integro-differential equation (2.3) or (2.7)) as givenby

u(x) ≈ uMSJ (x) = (Φ0, (J−1)Ψ)

aT0

J−1bT

(7.1)

and (4.10) in the equation (2.3), we obtain

(

A(Φ0, (J−1)Ψ) + L1D(Φ0, (J−1)Ψ))

aT0

J−1bT

= (Φ0, (J−1)Ψ)

cT0

J−1dT

.

Use (4.16),(5.17) and (6.8) followed by inner product with (Φ0, (J−1)Ψ)T producesthe linear system

(

AMSJ + LMS

1 JDMSJ

)

aT0

J−1bT

=

cT0

J−1dT

. (7.2)

The components of a0, (J−1)b are the coefficients of the multiscale expansion ofthe unknown solution at resolution J in the Legendre multiwavelet basis. Thecomponents of c0, (J−1)d are the coefficients of the multiscale expansion of the knownfunction f(x) at the same resolution J in (4.10) mentioned in section 4. Since the


matrix AMSJ +LMS

1 JDMSJ is well behaved, the unknown coefficients a0, (J−1)b can be

obtained from

aT0

J−1bT

=(

AMSJ + LMS

1 JDMSJ

)−1

cT0

J−1dT

. (7.3)

We can use the value of the components of (a0, J−1b) obtained in (7.3) to get thedesired approximate solution.

8. Estimation of regularity and error

An estimate for the local behaviour (regularity measured in terms of Holder expo-nents) of the solution at any point in [0, 1] can be obtained from a minor change inthe formula given in Paul et al.[22]. We choose the dyadic interval Ij,k =

[

k2j, k+1

2j

)

,and assume

u(x) ≈ constant

(

x− k

2j

)νj,k

for x ∈ Ij,k.

The proposed estimate of νj,k is given by

νj,k ≈

−12+ log2

∣

∣

∣

d0j,k

d0j+1,2k

∣

∣

∣for k

2j≤ x <

k+ 12

2j,

−12+ log2

∣

∣

∣

d0j,k

d0j+1,2k+1

∣

∣

∣for

k+ 12

2j≤ x < k+1

2j.

(8.1)

The L2- error ǫL2

J = ||u − uMSJ ||L2 in the approximate solution uMS

J given by (7.1)can be derived as in Paul et al.[22] and is given by

ǫL2

J =

K−1∑

l=0

∞∑

j=J

2j−1∑

k=0

|blj,k|2

12

. (8.2)

9. Illustrative examples

In this section several numerical examples are given to illustrate the efficiency ofthis proposed method.

Example 1. Consider the integral equation (1.1a)

(2x− 1)4u(x) + x(3− 2x)

∫ 1

0

u(t)

(t− x)2dt = F (x) (9.1)

where

F (x) = (2x−1)4(1−γ+2γx)−x(3−2x)

{

(1− γ + 2γx)

(

1

1− x+

1

x

)

− 2γln

∣

∣

∣

∣

1− x

x

∣

∣

∣

∣

}

and γ is a constant. The exact solution of (9.1) is given by[3]

u(x) = 1− γ + 2γx.


The Eq. (9.1) is of the form Eq.(1.1a) with

a(x) = (2x− 1)4, b(x) = x(3− 2x)

so that it can be reduced to integro-differential equation (2.3) with appropriate formsof the coefficients involved in the operators D,R and A given by Eq. (2.4a)-(2.4d).One can now use the representation of D,R and A given by (4.14),(5.16) and (6.7) torecast integro-differential equation for (9.1) to a system of linear algebraic equationsfor the unknown coefficients of a0, J−1b. Their values have been calculated forparameter γ = 0, 0.1, 0.5, 1, 5 at resolution J = 1(K = 4) and presented in Table 1.The multiscale approximation of the unknown solution uMS

1 obtained by using thecoefficients in formula (4.2) have also been calculated and presented the last columnof Table 1 against the value of γ in each rows.

γ a00 0 a10 0 a20 0 a30 0 b00 0 b10 0 b20 0 b30 0 Approximate solution

u1(x)(J = 1)

0 1 0 0 0 0 0 0 0 1

0.1 1 1

10√

30 0 0 0 0 0 .9 + .2x

0.5 1 1

2√

30 0 0 0 0 0 .5 + x

1 1 1√3

0 0 0 0 0 0 2x

5 1 5√3

0 0 0 0 0 0 −4 + 10x

10 1 10√3

0 0 0 0 0 0 −9 + 20x

Table 1. Components of a0, 0b and approximate solution for J = 1and γ = 0, 0.1, 0.5, 1, 5, 10

From the table it is found that all the wavelet coefficients of components of 0bare zero and components of a0 are found to coincide with the coefficients of themultiscale expansion of the exact solution u(x). Thus the present numerical methodrecovers the exact solution, as this method appears to be more efficient than thespline collocation method used by Boykov et al.[3] wherein errors exist even afterchoosing one thousand collocation points.

Example 2. We consider the integral equation

u(x)− 1

2β

√

x(1 − x)

∫ 1

0

u(t)

(t− x)2dt =

4πk

β

√

x(1− x), 0 < x < 1, (9.2)

with u(0) = 0 = u(1). This is known as the elliptic wing case of Prandtl’s equa-tion([5, 10, 16]). Here β is a known constant and has the exact solution

u(x) =8k

1 + 2βπ

√

x(1− x).


Comparing with (1.1a) we find

a(x) = 1, b(x) = −√

x(1 − x)

2β, F (x) =

4πk

β

√

x(1 − x).

Because of the end condition u(0) = 0, u(1) = 0, we can write

u(x) =√

x(1 − x)v(x), (9.3)

where v(x) is well-behaved unknown function of x ∈ (0, 1). Thus Eq.(9.2) reducesto

v(x)− 1

2β

∫ 1

0

√

t(1− t)v(t)

(t− x)2dt =

4πk

β, 0 < x < 1. (9.4)

The above equation is of the form Eq.(1.1b) with

a(x) = 1, b(x) = − 1

2β

√

x(1− x), F (x) =4πk

β.

Now applying the multiscale representation of (4.14),(5.16) and (6.7) the aboveintegral equation is transformed into the system of linear algebraic equations. Aftersolving the linear equations(in case of J = 1) we find that all the components of 0bare zeros. The components of the coefficients of a, 0b and the approximate solutionsv1(x)(J = 1) are presented in Table 2 for

a) k = 1 = β b) k =1

2= β c) k = 1 = 2β a) 2k = 1 = β

a00 0 a10 0 a20 0 a30 0 b00 0 b10 0 b20 0 b30 0 Approximate solution

v1(x)(J = 1)

k = 1 = β 8π2+π

0 0 0 0 0 0 0 8π2+π

k = 12= β 4π

1+π0 0 0 0 0 0 0 4π

1+π

k = 1 = 2β 8π1+π

0 0 0 0 0 0 0 8π1+π

2k = 1 = β 4π2+π

0 0 0 0 0 0 0 4π2+π

Table 2. Components of a0, 0b and approximate solution for J = 1

From this table it appears that the multiscale approximation(at resolution J =1) v1(x) coincides with the exact solution v(x) irrespective of the choice of theparameter k and β involved in Eq.(9.3). After putting the value of v(x) in (9.3),we get the exact solution which is the same as given in Chakrabarti et al.[5] for thedifferent values of k and β.


Example 3. We have considered here the integral equation

u(x) +√

x(1 − x)

∫ 1

0

u(t)

(t− x)2dt = G(x), 0 < x < 1, (9.5)

where

G(x) =

√

x(1 − x)

8

{

8x− 8x2 + 3π(1− 8x+ 8x2)}

.

The exact solution of Eq.(9.5) is given by

u(x) = {x(1− x)} 32 . (9.6)

The Eq.(9.5) is of the form (1.1a) with a(x) = 1, b(x) =√

x(1− x). The integralequation is reduced to a system of linear equation by using the multiscale representa-tion (4.14),(5.16) and (6.7) for K = 4, J = 6. We evaluate the approximate solutionu4(x) and absolute error, which are presented in Fig 1. We have also estimated theHolder exponent of the solution u(x) near the end points by using (8.1). The esti-mated values of γj,0(the exponent near x = 0) and γj,2j−1(the exponent near x = 1)are presented in Table 3 for j = 0, 1, 2, 3, 4. The sequences of {γj,0, j = 0, 1, 2, 3, 4}and {γj,2j−1, j = 0, 1, 2, 3, 4} in this table are found to be convergent and convergesto 1.5. From the absolute error presented in Fig 1 we observe that the absoluteerror is comparatively high near the end points. This may be attributed due to thefractional form of the Holder exponent near the end points.

Figure 1. The approximate solution uMS6 (x) and absolute error for

J = 6

j νj ,0 νj,2j−1

0 2.9 2.9

1 1.94 1.94

2 1.68 1.68

3 1.58 1.58

4 1.54 1.54

Table 3. Estimation of Holder exponent of u(x) around the end points


To reduce the error near the end points in the numerical solution, u(x) can beexpressed as

u(x) =√

x(1 − x)v(x). (9.7)

This representation will avoid the fractional part of the Holder exponent near theend point for v(x). Now using (9.7) in (9.5), the integral equation is converted into

v(x) +

∫ 1

0

√

t(1− t)v(t)

(t− x)2dt =

1

8

{

8x− 8x2 + 3π(1− 8x+ 8x2)}

. (9.8)

The Eq.(9.8) is of the form (1.1b) with

a(x) = 1, b(x) =√

x(1− x).

Now the integral equation reduces to a system of linear algebraic equation for K =4, J = 1. By solving the linear equations we find that all the components of

0b are zero. Therefore the approximate solution coincides with the exact solutionv(x) = x(1 − x). The components of a, 0b and the approximate solution for v(x)are presented in Table 4.

a00 0 a10 0 a20 0 a30 0 b00 0 b10 0 b20 0 b30 0 Approximate solution

v1(x)(J = 1)

16

0 − 1

6√

50 0 0 0 0 x(1− x)

Table 4. Components of a0, 0b and approximate solution ofv1(x)(J = 1) for J = 1 and Ex.3

From this table we find that the approximate solution of v(x) is

v1(x) = x(1− x).

Use of this result in (9.7) recovers the exact solution.

10. Conclusion

In this paper we have developed a numerical scheme based on multiresolutionanalysis of L2([0, 1]) functions generated by Legendre multiwavelets for solving hy-persingular integral equation of second kind. In the process of development of the

proposed scheme, the representation of derivative(≡ d(.)dx

) and the integro-differential

operator(≡∫ 1

0

ddt(.)

t−x dt) with Cauchy singular kernel in the Legendre multiwavelet ba-sis have been derived. These representations have been used to reduce the hyper-singular integral equation to a system of algebraic equations which can be solvedby any efficient numerical solver now available. An estimate of L2− error and lo-cal Holder exponent of unknown solution in terms of wavelet coefficients have beenpresented. The efficiency of the proposed scheme has been tested by considering afew examples. This test shows that our scheme is very efficient.


References

[1] B.K. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, Wavelet-like bases for the fast so-lution of second-kind integral equations, SIAM J. Sci. Comput., 14 (1993), 159-184.http://dx.doi.org/10.1137/0914010

[2] B.K. Alpert, G. Beylkin, D. Gines, L. Vozovoi, Adaptive solution of partial differential equa-tions in multiwavelet bases, J. Comp. Phys., 182 (2002), 149-190.http://dx.doi.org/10.1006/jcph.2002.7160

[3] I.V. Boykov, E.S. Ventsel and A.I. Boykova, An approximate solution of hypersingular integralequations, Appl. Numer. Math., 60 (2010), 607-628.http://dx.doi.org/10.1016/j.apnum.2010.03.003

[4] K. Buhring, A quadrature method for the hypersingular integral equation on an interval, J.Integral Equations Appl., 7 (1995), 263-301.http://dx.doi.org/10.1216/jiea/1181075880

[5] A. Chakrabarti, B.N. Mandal, U. Basu and S. Banerjea, Solution of a hypersingular integralequation of the second kind, Zeitschrift fur Angewandte Mathematik und Mechanik, 77 (1997),319-320. http://dx.doi.org/10.1002/zamm.19970770424

[6] A. Chakrabarti and B.N. Mandal, Derivation of the solution of a simple hypersingular integralequation, International Journal of Mathematical Education in Science and Technology, 29

(1998), 47-53. http://dx.doi.org/10.1080/0020739980290105[7] A. Chakrabarti, Solution of a simple hyper-singular integral equation, J. Integral Equations

Appl., 19 (2007), 465-471. http://dx.doi.org/10.1216/jiea/1192628619[8] Y. Chan and C.A. Fannjiang and G.H. Paulino, Integral equations with hypersingular kernels–

theory and applications to fracture mechanics, International Journal of Engineering Science,41 (2003), 683-720. http://dx.doi.org/10.1016/s0020-7225(02)00134-9

[9] De Klerk and H. Johan, Solving strongly singular integral equations by Lp approximationmethods, Appl. Math. Comput., 127 (2002), 311-326.http://dx.doi.org/10.1016/s0096-3003(01)00009-1

[10] L. Dragos, A collocation method for the integration of Prandtl’s equation, Zeitschrift fur

Angewandte Mathematik und Mechanik, 74 (1994), 289-290.http://dx.doi.org/10.1002/zamm.19940740716

[11] N.I. Ioakimidis, Application of finite-part integrals to the singular integral equations ofcrack problems in plane and three-dimensional elasticity, Acta Mechanica, 45 (1982), 31-47.http://dx.doi.org/10.1007/bf01295569

[12] M. Kanoria and B.N. Mandal, Water wave scattering by a submerged circular-arc-shapedplate, Fluid Dynamics Research, 31 (2002), 317-331.http://dx.doi.org/10.1016/s0169-5983(02)00136-3

[13] F. Keinert, Wavelets and Multiwavelets, CRC Press, 2003.http://dx.doi.org/10.1201/9780203011591

[14] M. Lakestani, B.N. Saray and M. Dehghan, Numerical solution for the weakly singular Fred-holm integro-differential equations using Legendre multiwavelets, J. Comp. Appl. Math., 235(2011), 3291-3303. http://dx.doi.org/10.1016/j.cam.2011.01.043

[15] B.N. Mandal and R. Gayen, Water-wave scattering by two symmetric circular-arc-shaped thin plates, Journal of Engineering Mathematics, 44 (2002), 297-309.http://dx.doi.org/10.1023/a:1020944518573

[16] B.N. Mandal and G.H. Bera, Approximate solution of a class of singular in-tegral equations of second kind, J. Comput. Appl. Math., 206 (2007), 189-195.http://dx.doi.org/10.1016/j.cam.2006.06.011

[17] P.A. Martin, End-point behaviour of solutions to hypersingular integral equations, Proceedingsof the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 432(1991), 301-320. http://dx.doi.org/10.1098/rspa.1991.0019


[18] P.A. Martin, Exact solution of a simple hypersingular integral equation, J. Integral EquationsAppl., 4 (1992), 197-204. http://dx.doi.org/10.1216/jiea/1181075681

[19] C.H. Midya, M. Kanoria and B.N. Mandal, Scattering of water waves by inclined thinplate submerged in finite-depth water, Archive of Applied Mechanics, 71 (2001), 827-840.http://dx.doi.org/10.1007/s004190100187

[20] N.F. Parsons and P.A. Martin, Scattering of water waves by submerged plates us-ing hypersingular integral equations, Applied Ocean Research, 14 (1992), 313-321.http://dx.doi.org/10.1016/0141-1187(92)90035-i

[21] N.F. Parsons and P.A. Martin, Scattering of water waves by submerged curvedplates and by surface-piercing flat plates, Applied Ocean Research, 16 (1994), 129-139.http://dx.doi.org/10.1016/0141-1187(94)90024-8

[22] S. Paul, M. M. Panja and B.N. Mandal, Multiscale approximation of the solution of weaklysingular second kind Fredholm integral equation in Legendre multiwavelet basis, J. Comput.

Appl. Math., 300 (2016), 275-289.http://dx.doi.org/10.1016/j.cam.2015.12.022

[23] S. Paul, M. M. Panja and B. N. Mandal, Use of Legendre multiwavelets to solve Carlemantype singular integral equations, preprint.

Received: July 1, 2016; Published: September 2, 2016

wavelet based numerical solution of second kind ... · where the hypersingular integrals are...

Documents