[for solving cauchy singular integral equations]

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[For solving Cauchy singular integral equations]Abdelaziz Mennouni

To cite this version:Abdelaziz Mennouni. [For solving Cauchy singular integral equations]. General Mathematics[math.GM]. Université Jean Monnet - Saint-Etienne; Université Mohamed Khider (Biskra, Algérie),2011. English. �NNT : 2011STET4005�. �tel-00691919�

https://tel.archives-ouvertes.fr/tel-00691919

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Université Mohamed Khider, Biskra, Algerie

Université Jean Monnet, Saint Etienne, France,

membre d'Université de Lyon

THÈSE

présentée pour obtenir le diplôme de

Doctorat Sciences Université Mohamed Khider Biskra,Algerie

Docteur de l'Université Jean Monnet,Saint Etienne, France

Discipline :Mathématiques Appliquées et Applications des Mathématiques

Spécialité :Analyse Numérique

Sur la Résolution des Equations Intégrales

Singulières à Noyau de Cauchy

par

Abdelaziz MENNOUNI

soutenue le 27 avril 2011après avis de

Lazhar Bougoffa Assoc Prof, Université d'Al Imam, Arabie Saoudite Rapporteur

Amina Benbernou Professeur, Université de Mostaganem Rapporteur

Devant le jury composé de :

Khaled Melkemi Professeur UMK, Biskra Examinateur

Brahim Mezerdi Professeur UMK, Biskra Examinateur

Laurence Grammont MC UJM, St-Etienne Examinateur

Said Guedjiba MC HDR UHL, Batna Directeur de thèse

Mario Paul Ahues Blanchait Professeur CE UJM, St-Etienne Directeur de thèse

Remerciements

Je tiens à remercier M. le Dr Said Guedjiba, pour son soutien exemplaire durantla préparation de ce projet. Je lui suis également reconnaissant pour les qualitésscienti�ques et pédagogiques de son encadrement et pour la disponibilité sans failledont il a fait preuve durant ma formation doctorale. Dès le début, sa con�ance m'adonné l'énergie et l'inspiration me permettant de surmonter toutes les di�cultés.

Je tiens également à remercier Monsieur le Professeur Mario Ahues, qui m'aaccueilli au sein du LaMUSE. Je le remercie pour son aide précieuse durant lapréparation des premières étapes de ce travail. Il a su me guider sur le chemin dela recherche.

Mes remerciements les plus respectueux vont à Monsieur Alain Largillier.Veuillez trouver ici l'expression de mes remerciements les plus sincères ainsi que lamarque de mon profond respect.

Je remercie Pr. Lazhar Bougoffa et Pr. Amina Benbernou qui ont bienvoulu être rapporteurs de cette thèse.

Ma sincère reconnaissance à tous les membres du jury pour l'honneur qu'ils mefont en acceptant de le présider et d'examiner ce travail.

Je tiens à remercier Madame Laurence Grammont, pour avoir accepté d'êtremembre de jury de cette thèse.

J'adresse toute ma gratitude à Monsieur le Professeur Brahim Mezerdi, qu'iltrouve ici l'expression de ma profonde reconnaissance pour avoir accepté d'examinercette thèse.

Je remercie très sincèrement Monsieur le Professeur Khaled Melkemi, d'avoiraccepté d'être président du jury.

J'adresse aussi mes sincères remerciements à M. Zouhir Mokhtari, Chef dudépartement de Mathématiques à l'université Mohamed Khider, Biskra.

Je tiens à remercier tous les membres du département de Mathématiques dela Faculté des Sciences et Techniques de Saint-Étienne, en particulier M. GrigoryPanasenko et M. Mahdi Boukrouche .

Je tiens à remercier mon ami Hamza et sa famille. Merci pour l'accueil chaleureuxque vous m'avez prodigué lors de mon arrivée.

Je tiens à remercier tout particulièrement mes amis Nazih, Fayçal, Hicham,Ahmed, Lamine, Nachite. Qu'ils trouvent ici mes chaleureux remerciements.

Quant à ma femme et mon �ls Abdeldjalil, je leur adresse mes plus chaleureuxremerciements. Leur soutien moral, indispensable pour moi, m'a permis de tenirbon et de ne pas abandonner dans les moments di�ciles. Ils ont su me soutenir etparfois me supporter. Qu'ils soient certains de toute ma reconnaissance et de toutmon amour.

Finalement, je tiens à remercier mes parents pour leur soutien exemplaire et leurs

sacri�ces loyaux durant ces longues années de quête sur la voie du savoir.

3

Dédicace

A Ma Femme

A Mon Fils Abdeldjalil

A Mes Parents...

A la mémoire d'Alain Largillier

5

Contents

List of Tables 9

List of Figures 10

Introduction 11

Preliminaries 16

1 Solving Cauchy Integral Equations of the First Kind by Iterations 22

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.2 Development of the Method . . . . . . . . . . . . . . . . . . . . . . . 221.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3.1 Case A: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.3.2 Case B: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Fourier Expansion for Cauchy Integral Equations of the Second

Kind 27

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Collocation Method for Cauchy Integral Equations Using Trigono-

metric Polynomials in L2([0, 2π] ,C) . . . . . . . . . . . . . . . . . . . 282.2.1 Description of The Method . . . . . . . . . . . . . . . . . . . . 282.2.2 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Direct Solution of Cauchy Integral Equation on the Real Line . . . . 31

3 Two Projection Methods for Skew-Hermitian Operator Equations 34

3.1 Introduction and Mathematical Background . . . . . . . . . . . . . . 343.2 Cauchy Integral Equations of the Second Kind . . . . . . . . . . . . . 35

3.2.1 Galerkin Approximation . . . . . . . . . . . . . . . . . . . . . 363.2.2 Kulkarni Approximation . . . . . . . . . . . . . . . . . . . . . 37

3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Collocation Method for Solving Integro-Di�erential Equations with

Cauchy Kernel 41

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 The Approximate Solution . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . 444.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6

5 Projection Methods for Integro-Di�erential Equations with Cauchy

Kernel 47

5.1 Introduction and Mathematical Background . . . . . . . . . . . . . . 475.2 A Projection Method for Integro-Di�erential Equations with Cauchy

Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Sloan Projection Method . . . . . . . . . . . . . . . . . . . . . . . . . 505.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Regularization and Projection Approximations for Cauchy Integral

Equations of the Second Kind 53

6.1 Introduction and Mathematical Background . . . . . . . . . . . . . . 536.2 Finite Rank Approximations and Regularization . . . . . . . . . . . . 556.3 Sloan Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.4 Galerkin Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Conclusion and perspective 60

Bibliographie 60

7

List of Tables

3.1 Example 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Example 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Example 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 Example 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

9

List of Figures

2.1 n = 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2 n = 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.3 n = 62 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10

Introduction

In the last two decades, Cauchy integral equations have assumed an increasing rel-evance. This is due to the great variety of problems arising in sciences, engineeringand technology, which can be described by such equations. Cauchy integral andIntegro-di�erential equations involve important mathematical techniques, becausethey are encountered by mathematicians, and physical and social scientists, in theirinvestigations. These equations were described in many available books, concerningtheory and applications ( cf. [2], [7], [9], [10], [16], [20], [34], [37], [39], [45], [70], [89],[92], [95], [108], [109], [117], [122], [125], [130], [139]-[141] ).Several books include the latest questions related to high technology on solving veryimportant theoretical and practical problems on solid mechanics, fracture mechan-ics, structural analysis, elastodynamics, �uid mechanis and aerodynamics, by usingsingular integral equation methods ( cf. [1], [13], [48], [71], [90], [96]-[98], [118], [119],[126] ). In ([132]-[138]), the author presented many papers relating to boundednessof singular integral operators with Cauchy kernel in weighted spaces.In ( [18], [14], [72]-[78] ), the authors have studied the Cauchy integral equationsin weigthted spaces of continuous functions, using Jacobi weights; they introduceda certain number of polynomial approximation spaces. In ( [50]-[69] ), the authorspresented analytical theories and numerical evaluation methods for solving Cauchyintegral equations, in an accessible manner for a variety of applications to problemsin the theory of three-dimensional elasticity. In the last decades several papers havebeen published on the convergence of the quadrature rules for evaluating Cauchysingular integrals, (see [21]-[31], [32], [102]-[107]). In ([80]-[88]), the author hasinvestigated the algebra of Cauchy integral operators with piecewise continuous co-e�cients on re�exive Orlicz spaces, and he has presented the necessary conditionsfor Fredholmness of singular integral operators in re�exive weighted rearrangement-invariant spaces.During the last 30 years, there has been a substantial increase in interest in the nu-merical solution of the Fredholm singular integral and integro-di�erential equationswith Cauchy kernel (cf. [35, 40, 41]). These equations have important applicationsin mathematical physics, applied mathematics, and numerical analysis. The math-ematical formulation of physical phenomena often involves Cauchy type, see, forexample the excellent book by Muskhelishvili (cf. [110]), and the references therein.Various of Fredholm singular integral equations with Cauchy kernel have been solvednumerically in recent times by several authors using approximate methods. Recently,Chakrabarti, and Martha, have developed a straightforward method, involving ex-pansion of the unknown function of a Fredholm integral equation of the secondkind, in terms of polynomials, and have used the method of least-squares (cf. [19]).Eshkuvatov et al have described a special approximate method for solving Fredholm

12

integral equation of the �rst kind, with Cauchy type (cf. [36]). Elliott, described aclassical collocation method for singular integral equations having a Cauchy kernel,and showed that, under reasonable conditions, the approximate solutions convergeto the solution of the original equation (cf. [35]). Golberg, analyzed and obtainedconvergence proofs of some numerical methods, for solving several classes of Cauchysingular integral equations (cf. [40, 41, 42]). In 2007 Mandal and Bera employeda simple method based on polynomial approximation of a function to obtain ap-proximate solution of a class of singular integral equations of the second kind (cf.[100]). Also Many di�erent methods have been developed to obtain an approximatesolution of these equations (cf. [12, 91, 123, 131]). Other techniques for solvingFredholm singular integro-di�erential equations with Cauchy kernel have been pre-sented in several works. Badr, presented a Galerkin approach for solving the linearintegro-di�erential equation of the second kind with Cauchy kernel by using the or-thogonal basis of Legendre polynomials (cf. [11]). In 2006 Maleknejad and Arzhanghave presented a Taylor-series expansion method for a class of Fredholm singularintegro-di�erential equation with Cauchy kernel, and used the truncated Taylor-series polynomial of the unknown function and transform the integro-di�erentialequation into an nth order linear ordinary di�erential equation with variable co-e�cients (cf. [99]). In 2008 Subhra Bhattacharya, and Mandal have presenteda method based on polynomial approximation using Bernstein polynomial basis,to obtain approximate numerical solutions of singular integro-di�erential equationswith Cauchy kernel, and compared their numerical results with those obtained byvarious Galerkin methods (cf. [100]). So the polynomial approximation play animportant role in the numerical computation of the integral and integro-di�erentialequations (cf. [35, 40, 41, 120, 121]).The purpose of this thesis, is to develop and illustrate various new methods forsolving many classes of Cauchy singular integral and integro-di�erential equations.This work is organized as follows:

In the beginning, we brie�y recall a few basic concepts from general theoreticalframework, such as bounded and compact operators, Hilbert spaces and adjoint op-erators, spectral theory framework, convergence of operators, approximation basedon projections, and some classi�cation of integral equations.

In chapter one, we study the successive approximation method for solving aCauchy singular integral equations of the �rst kind in the general case. We provethe convergence of the method in this general case. The proposed method has beentested for two kernels which are particularly important in practice.

In chapter two, we present two methods for solving Cauchy integral equation ofthe second kind: Firstly we present a collocation method based on trigonometricpolynomials combined with a regularization procedure, for solving Cauchy integralequations of the second kind, in L2([0, 2π] ,C). A system of linear equations isinvolved. We prove the existence of the solution for a double projection scheme, andwe perform the error analysis. Numerical examples illustrate the theoretical results.Secondly we solve directly Cauchy integral equation on the real line using Fourierexpansion in Sobolev spaces.

The purpose of chapter three, is to approximate the solution of an operator equa-tion involving a non compact bounded operator in Hilbert spaces, using projectionmethods. We prove the existence of the solution for the approximate equation, and

13

we perform the error analysis. We apply the method for solving the Cauchy integralequations in L2 (0, 1) for two cases : Galerkin projections and Kulkarni projectionsrespectively, using a sequence of orthogonal �nite rank projections. Numerical ex-amples illustrate the theoretical results.

In chapter four, we introduce a collocation method for Cauchy integro-di�erentialequations, using airfoil polynomials of the �rst kind. According to this method, weobtain a system of linear equations. We give some su�cient conditions for the con-vergence of this method. In the end, we investigate the computational performanceof our approach through some numerical examples. w In chapter �ve, we proposetwo methods for solving integro-di�erential equations with Cauchy kernel: First, wepresent a modi�ed projection method based on Legendre polynomials. A system oflinear equations is solved. Second, we present a Sloan projection method for solvingintegro-di�erential equations with Cauchy kernel, using Legendre polynomials. Wegive numerical examples.

The last chapter deals with regularization for Cauchy integral equation of thesecond kind. We apply three projection methods to the regularized equation. Firstwe use Kantorovich projection, and perform the error analysis. After we study theSloan projection and prove some results about the error analysis. Finally Galerkinprojection is established and its error analysis is discussed.

14

Preliminaries

We begin by recalling brie�y a few basic concepts from general theoretical frame-work, such as bounded and compact operators, Hilbert spaces and adjoint operators,spectral theory framework, convergence of operators, approximation based on pro-jections, and some classi�cation of integral equations.

Linear Operators

Let X be a Banach space, and let T be a linear operator de�ned on X. By thenullspace of T , N (T ) we mean the space of vectors annihilated by T , so

N (T ) := {ϕ ∈ X : Tϕ = 0} .

The image (or range) of T is de�ned by

R(T ) := {Tϕ : ϕ ∈ X} .

A linear operator T from a normed space X into a normed space Y is called boundedif there exists a positive number M such that

‖Tx‖ ≤M ‖x‖ for all x ∈ X,

and‖T‖ := sup

‖x‖≤1

‖Tx‖ ,

is the norm of T . BL(X, Y ) will denote the space of bounded linear operators fromX into Y , and BL(X) those from X into itself.

T is called compact if it maps each bounded set in X into a relatively compactset in Y . That is T is compact if the set {Tx : ‖x‖ ≤ 1} has compact closure inY .

We recall that a subset U of a normed space X is called compact if every opencovering of U contains a �nite subcovering.

A subset of a normed space is called relatively compact if its closure is compact.

Theorem 1. A bounded subset of a �nite-dimensional normed space is relativelycompact.

Proof. See [91].

We have also:

1. Compact linear operators are bounded.

16

2. A linear operator T from a �nite-dimensional normed space X into a normedspace Y is bounded.

3. The identity operator I : X → X is compact if and only if X has �nitedimension.

4. A bounded operator T from a normed space X into a normed space Y , with�nite-dimensional range T (X) is compact. Such an operator is called a �nite-rank one.

5. Linear combinations of compact operators are compact.

6. If (Tn) is a sequence of compact operators and ‖Tn − T‖ → 0 as n→∞, thenT is compact.

Theorem 2. Let X be a Banach space, and let T be a bounded linear operator fromX into X, with

‖T‖ < 1.

Then the I − T has a bounded inverse on X that is given by the Neumann series

(I − T )−1 =∞∑j=0

T j,

and satis�es ∥∥(I − T )−1∥∥ ≤ 1

1− ‖T‖.

Proof. See [91].

A Banach space X is called a Hilbert space if the norm on X is induced by aninner product, that is, by a Hermitian positive de�nite sesquilinear form 〈., .〉, asfollows:

‖x‖ := 〈x, x〉1/2 for x ∈ X.Let X be a Hilbert space and T ∈ BL(X). The adjoint of T is the unique operatorT ∗ ∈ BL(X) such that

〈Tϕ, ψ〉 = 〈ϕ, T ∗ψ〉 for all ϕ, ψ ∈ X.

We say that T is normal if T ∗T = TT ∗, and that T is selfadjoint if T ∗ = T .The folowing results hold in a Hilbert space X:

1. Schwarz Inequality: For all x, y ∈ X,

|〈x, y〉| ≤ ‖x‖ ‖y‖ .

2. Let T ∈ BL(X), then‖T‖ = ‖T ∗T‖1/2 .

3. Let T ∈ BL(X), then

N (T ) = R(T ∗)⊥ and R(T ) = N (T ∗)⊥.

17

4. Suppose that T ∈ BL(X) is compact. The linear equation

(T − zI)x = y,

has a unique solution x ∈ X in the case where the corresponding homogenousequation

(T − zI)x = 0,

has only the trivial solution.

Spectral Theory Framework

Let T ∈ BL(X), we recall the following de�nitions:The resolvent of T is de�ned by

re (T ) := {z ∈ C : T − zI is invertible} .

For z ∈ re (T ), the resolvent operator of T at z is de�ned by

R(T, z) := (T − zI)−1.

The spectrum of T is the set

sp (T ) := {z ∈ C : z /∈ re (T )} .

If X is �nite dimensional, then sp (T ) consists of the eigenvalues of T .The spectral radius of T is de�ned by

ρ(T ) := sup {|λ| : λ ∈ sp (T )} .

We have:

1. If z ∈ re (T ), then R(T, z) ∈ BL(X).

2. If T ∗ = T , then sp (T ) ⊆ R.

3. If T ∗ = −T , then sp (T ) ⊆ iR.

4. If T ∈ BL(X) is normal, then ρ(T ) = ‖T‖.

5. Let T ∈ BL(X). If z ∈ C is given, then for every y ∈ X, the linear equation

(T − zI)x = y,

has a unique solution x ∈ X determined by y if and only if z ∈ re (T ).

18

Sequences of Operators

Let (Tn)n≥1 be a sequence of bounded linear operators on a Banach space X. LetT ∈ BL(X). Unless otherwise mentioned, the convergence is as n→∞.

Let us consider three well-known modes of convergence.The pointwise convergence, denoted by Tn

p→ T :

‖Tnx− Tx‖ → 0 for every x ∈ X.

The norm convergence, denoted by Tnn→ T :

‖Tn − T‖ → 0.

The collectively compact convergence, denoted by Tncc→ T :

Tnp→ T , and for some positive integer n0, the set

∪n≥n0 {(Tn − T )x : x ∈ X, ‖x‖ ≤ 1} ,

is a relatively compact subset of X.If Tn

n→ T or Tncc→ T , then clearly Tn

p→ T . But the converse is not true.In [4], the authors have studied a new mode of convergence called: the ν-

convergence, denoted by Tnν→ T :

(‖Tn‖) is bounded , ‖(Tn − T )T‖ → 0, ‖(Tn − T )Tn‖ → 0.

Theorem 3. 1. If Tnn→ T , then Tn

ν→ T . Conversely, if 0 /∈ sp (T ) and Tnν→ T ,

then Tnn→ T .

2. If Tncc→ T and T is compact, then Tn

ν→ T .

3. Let Tnν→ T and Un

ν→ U . Then Tn+Unν→ T+U if and only if (Tn−T )U

n→ 0.

Proof. See [4]

Approximation Based on Projections

Let (πn) be a sequence of bounded projections de�ned on X, that is each πn isbounded linear operator and π2

n = πn, hence ‖πn‖ ≥ 1.The following three conditions are equivalent one with each other if πn is a

bounded projection de�ned on Hilbert space X:

π∗n = πn, ‖πn‖ ≤ 1, N (πn) = R(πn)⊥.

If one of these conditions is satis�ed, then πn is called an orthogonal projection.De�ne

T Pn := πnT, T Sn := Tπn, TGn := πnTπn, TKn := πnT + Tπn − πnTπn.

The bounded operators T Pn , TSn , T

Gn , T

Kn , are known as the projection approximation

of T , the Sloan approximation of T , the Galerkin approximation of T , and theKulkarni approximation of T , respectively.

19

Theorem 4. Let T ∈ BL(X) and πnp→ I. Then

1. T Pnp→ T , T Sn

p→ T and TGnp→ T .

2. If T is a compact operator, then T Pnn→ T , T Sn

ν→ T , TGnν→ T .

3. If T is a compact operator and π∗np→ I∗, then T Sn

n→ T , TGnn→ T .

Proof. See [4]

Classi�cation of Integral Equations

An integral equation is an equation for an unknown fuction ϕ, where ϕ appears alsounder the integral sign. The classi�cation of integral equations centres on manybasic characteristics:

1. Limits of integration

• Both �xed: Fredholm equation.

• One variable: Volterra equation.

2. Placement of the unknown function

• Only inside integral: First kind.

• Both inside and outside integral: Second kind.

The Fredholm integral equation of the �rst kind is represented by∫ b

a

k(s, t)ϕ(t)dt = g(s), a ≤ s ≤ b.

The Volterra integral equation of the second kind is represented by

ϕ(s) =

∫ s

a

k(s, t)ϕ(t)dt+ g(s), a ≤ s ≤ b.

3. Nature of the known function

• Identically zero: Homogeneous.

• Not identically zero: Inhomogeneous.

For example, the equation

ϕ(s) =

∫ b

a

k(s, t)ϕ(t)dt, a ≤ s ≤ b,

is referred to as the homogeneous Fredholm integral equation of the secondkind.

4. Linearity: The equation is linear with respect to the unknown function or not.

20

• Linear integral equations.

• Nonlinear integral equations.


ϕ(s)−∫ s

a

k(s, t, ϕ(t))dt = g(s), a ≤ s ≤ b,

is a nonlinear Volterra integral equation of the second kind, where k is not alinear function of its third variable.

5. Depending on the kind of the integral

• Regular integral equations.

• Singular integral equations.

A special and important example of a singular integral equation is the Cauchy orstrongly singular integral equation. In this case the integral must be understood asthe Cauchy principal value.

An example of a Cauchy singular kernel is

k(s, t) =1

s− t, s 6= t.

Sometimes, integral equations occur with additional derivatives of the unknownfunction (under the integral or ouside). In this case, the problem is called an integro-di�erential equation.


ϕ′(s) +

∮ b

a

ϕ(t)

t− sdt = f(s), a ≤ s ≤ b,

is an integro-di�erential equation with Cauchy kernel.

21

Chapter 1

Solving Cauchy Integral Equations ofthe First Kind by Iterations

1.1 Introduction

The successive approximation method is applied for the �rst time by N.I. Ioakimidis(cf. [49]), to solve practical cases of a Cauchy singular integral equation: the airfoilone (cf. [131]). In this chapter we study a more general case. We prove the conver-gence of the method in this general case. The proposed method has been tested fortwo kernels which are particularly important in practice.Cauchy singular integral equations of the �rst kind are often encountered in con-tact and fracture problems in solid mechanics. Sokhottski, Harnack, Mushkelishvili(cf. [110]), Privalov, Magnaradze, Mikhlin, Khvedelidze, Vekua, Kupradze, Gakhov,Golberg, Elliott, Srivastav, Sesko, Erdogan, Junghannes, Linz, Ioakimidis and oth-ers have investigated such type of equation. The solutions of these problems may beobtained analytically using the theory developed by Mushkelishvili. Cauchy integralequations are usually di�cult to solve analytically, and it is required to obtain ap-proximate solutions. So many di�erent methods have been developed to obtain anapproximate solution of a Cauchy integral equation (cf. [12], [91], [123]). In 1988,Ioakimidis solved the airfoil equation with the successive approximation method forthe �rst time. In this chapter this method is applied for solving a Cauchy singu-lar integral equations of the �rst kind in the general case. The convergence of themethod is investigated.

1.2 Development of the Method

We consider the Cauchy integral equation of the �rst kind

1

π

∮ 1

−1

v(t)ϕ(t)

t− xdt = g(x), −1 < x < 1, (1.1)

22

where v and g are known functions and ϕ is the unknown. We shall assume that ghas derivatives of all orders on [−1, 1], and that

1

π

∮ 1

−1

v(t)

t− xdt = 1, −1 < x < 1, (1.2)

1

π

∫ 1

−1

|v(t)|dt ≤ 1, −1 < x < 1, (1.3)

1

π

∮ 1

−1

1

v(t)(t− x)dt = 1, −1 < x < 1, (1.4)

1

π

∫ 1

−1

1

|v(t)|dt ≤ 1, −1 < x < 1.

We substract the singularity of equation (1.1) at t = x as follows:

1

π

∮ 1

−1

v(t)ϕ(x)

t− xdt+

1

π

∮ 1

−1

v(t) [ϕ(t)− ϕ(x)]

t− xdt = g(x), −1 < x < 1. (1.5)

Using (1.2) we rewrite equation (1.5) as:

ϕ(x) +1

π

∮ 1

−1

v(t) [ϕ(t)− ϕ(x)]

t− xdt = g(x), −1 < x < 1.

We obtain

ϕ(x) = g(x)− 1

π

∮ 1

−1

v(t) [ϕ(t)− ϕ(x)]

t− xdt, −1 < x < 1.

Now, we apply to this equation the successive approximation method:

ϕn+1(x) = g(x)− 1

π

∮ 1

−1

v(t) [ϕn(t)− ϕn(x)]

t− xdt, −1 < x < 1.

Using (1.2), we have:

ϕn+1(x) = ϕn(x) + g(x)− 1

π

∮ 1

−1

v(t)ϕn(t)

t− xdt, −1 < x < 1, (1.6)

where

ϕ0(x) = 0, −1 < x < 1.

Let

Rn(x) = ϕ(x)− ϕn(x) (1.7)

From (1.6) we obtain, for −1 < x < 1,

ϕn+1(x)− ϕ(x) = ϕn(x)− ϕ(x) + g(x)

− 1

π

∮ 1

−1

v(t) [ϕn(t)− ϕ(t)]

t− xdt

− 1

π

∮ 1

−1

v(t)ϕ(t)

t− xdt.

23

From (1.7) and (1.1),

−Rn+1(x) = −Rn(x) +1

π

∮ 1

−1

v(t)Rn(t)

t− xdt, −1 < x < 1.

Then

Rn+1(x) = Rn(x)− 1

π

∮ 1

−1

v(t)Rn(t)

t− xdt, −1 < x < 1. (1.8)

Now, using (1.2), equation (1.8) becomes:

Rn+1(x) = − 1

π

∮ 1

−1

v(t) [Rn(t)−Rn(x)]

t− xdt, −1 < x < 1,

and hence

R′n+1(x) = − 1

π

∮ 1

−1

v(t)Rn(t)−Rn(x)− (t− x)R′n(x)

(t− x)2dt, −1 < x < 1,

But following Taylor's theorem with integral remainder,

Rn(t)−Rn(x)− (t− x)R′n(x) =

∫ 1

0

(1− s)R′′n(x+ s(t− x))(t− x)2ds.

Hence,

‖R′n+1‖ ≤1

2‖R′′n‖.

Recursively,

‖R(j)n+1‖ ≤

1

j + 1‖R(j+1)

n ‖, j ∈ N.

Thus

‖R(j)n−j+1‖ ≤

1

j + 1‖R(j+1)

n−j ‖, j ∈ N.

Multiplying memberwise for j ∈ [[0, n ]], we get

‖Rn+1‖ ≤1

(n+ 1)!‖R(n+1)

0 ‖,

but from (1.7),

‖R(n+1)0 ‖ = ‖ϕ(n+1)‖.

So

‖Rn‖ ≤1

n!‖ϕ(n)‖, n ∈ N.

24

From (1.1) and by the Sohngen inversion formula,

ϕ(x) = − 1

π

∮ 1

−1

g(t)

v(t) (t− x)dt, −1 < x < 1. (1.9)

Equation (1.9) takes the form

ϕ(x) = −g(x)− 1

π

∮ 1

−1

g(t)− g(x)

v(t) (t− x)dt, −1 < x < 1.

By standard calculus,

‖ϕ(n)‖ ≤ ‖g(n)‖+1

n+ 1‖g(n+1)‖.

Using (1.4),

‖Rn‖ ≤1

n!‖g(n)‖+

1

(n+ 1)!‖g(n+1)‖, n ∈ N.

Hence, if

limn→∞

(1

n!‖g(n)‖+

1

(n+ 1)!‖g(n+1)‖

)= 0,

then the successive approximation method converges.

1.3 Numerical Experiments

The proposed method has been tested for the two following kernels which are par-ticularly important in practice:

1.3.1 Case A:

Let

v(t) =

√1 + t

1− t.

All the hypotheses on v are satis�ed. From (1.6) we obtain

ϕn+1(x) = ϕn(x) + g(x)− 1

π

∮ 1

−1

ϕn(t)

t− x

√1 + t

1− tdt, −1 < x < 1. (1.10)

But (cf. [94]),

1

π

∮ 1

−1

ϕ(t)

t− xi

√1 + t

1− tdt =

m∑j=1

2 (1 + tj)

2m+ 1

ϕ(tj)

tj − xi,

where the points tj and xi are given by

tj = cos

(2j − 1

2m+ 1π

), j ∈ [[1,m ]],

xi = cos

(2iπ

2m+ 1

), i ∈ [[1,m ]].

25

From (1.10),

ϕn+1(xi) = ϕn(xi) + g(xi)−m∑j=1

2 (1 + tj)

2m+ 1

ϕn(tj)

tj − xi, i ∈ [[1,m ]],

ϕn+1(tk) = ϕn(tk) + g(tk)−m∑p=1

2 (1 + xp)

2m+ 1

ϕn(xp)

xp − tk, k ∈ [[1,m ]].

1.3.2 Case B:

Let

v(t) =

√1− t1 + t

.

All the hypotheses on v are satis�ed. This case has been investigated by Ioakimidisin 1988. From (1.6),

ϕn+1(x) = ϕn(x)− g(x) +1

π

∮ 1

−1

ϕn(t)

t− x

√1− t1 + t

dt, −1 < x < 1. (1.11)

But

1

π

∮ 1

−1

ϕ(t)

t− xi

√1− t1 + t

dt =n∑j=1

2 (1− tj)2m+ 1

ϕ(tj)

tj − xi,

where the points tj and xi are given by

tj = cos

(2j

2m+ 1π

), j ∈ [[1,m ]],

xi = cos

(2i− 1

2m+ 1π

), i ∈ [[1,m ]].

From (1.11),

ϕn+1(xi) = ϕn(xi)− g(xi) +m∑j=1

2 (1− tj)2m+ 1

ϕn(tj)

tj − xi, i ∈ [[1,m ]],

ϕn+1(tk) = ϕn(tk)− g(tk) +m∑p=1

2 (1− xp)2m+ 1

ϕn(xp)

xp − tk, k ∈ [[1,m ]].

26

Chapter 2

Fourier Expansion for CauchyIntegral Equations of the SecondKind

2.1 Introduction

In the �rst section of this chapter we present a collocation method based on trigono-metric polynomials combined with a regularization procedure, for solving Cauchyintegral equations of the second kind, in L2([0, 2π] ,C). A system of linear equationsis involved. We prove the existence of the solution for a double projection scheme,and we perform the error analysis. Some numerical examples illustrate the theoreti-cal results. In second section we present a direct method for solving Cauchy integralequation on the real line.Cauchy integral equations appear in many applications in scienti�c �elds such asunsteady aerodynamics and aero elastic phenomena, visco elasticity, �uid dynamics,electrodynamics. There is a theoretical study on some kind of Cauchy integral equa-tions in [110]. Many Cauchy integral equations are di�cult to solve analytically, andit is required to obtain approximate solutions. In ([128]), the author has studied areduction of some class of singular integral equations to regular Fredholm integralequations in Lp([−1, 1] ,C). The purpose of this chapter is �rstly to approximatethe solution of a Cauchy integral equation of the second kind in L2([0, 2π] ,C), usingcollocation, trigonometric polynomials and a regularization procedure, secondly tosolve directly Cauchy integral equation on the real line using Fourier expansion inSobolev spaces.

27

2.2 Collocation Method for Cauchy Integral Equa-

tions Using Trigonometric Polynomials in L2([0, 2π] ,C)

2.2.1 Description of The Method

For each nonzero real constant µ, and each real function f , consider the problem of�nding a function ϕ, such that

µϕ(s)−∮ 2π

0

ϕ(t)

t− sdt = f(s), 0 ≤ s ≤ 2π, (2.1)

where the integral is understood to be the Cauchy principal value:∮ 2π

0

ϕ(t)

t− sdt = lim

ε→0

[ ∫ s−ε

0

ϕ(t)

t− sdt+

∫ 2π

s+ε

ϕ(t)

t− sdt].

Equation (2.1) is a Cauchy integral equation of the second kind. Letting

Tϕ(s) :=

∮ 2π

0

ϕ(t)

t− sdt, 0 ≤ s ≤ 2π,

equation (2.1) reads asµϕ− Tϕ = f.

Theorem 5. For each f ∈ L2([0, 2π] ,C), equation (2.1) has a unique solution ϕ ∈L2([0, 2π] ,C), and the Cauchy integral operator T is bounded and skew-Hermitianfrom L2([0, 2π] ,C) into itself.

Proof. See [124].

Let X := L2([0, 2π] ,C), and Xn denote the space spanned by the �rst 2n+ 1 oftrigonometric polynomials. De�ne σn to be the orthogonal projection from X ontoXn. Hence, for ψ ∈ X,

limn→∞

‖σnψ − ψ‖2 = 0.

Let be 0 ≤ sn,1 < sn,2 < . . . < sn,2n+1 ≤ 2π. For each i ∈ [[1, 2n + 1 ]] consider thehat function en,i in C0([0, 2π] ,C), such that, for each j ∈ [[1, 2n+ 1 ]],

en,i(sn,j) = δi,j.

Let Yn be the space spanned by these hat functions, which has dimension 2n + 1.De�ne the interpolatory projection operator πn from C0([0, 2π] ,C) onto Yn:

πnh(s) :=2n+1∑j=1

h(sn,j)en,j(s), h ∈ C0([0, 2π] ,C).

We recall that (see [4, 8]):limn→∞

‖πnh− h‖∞ = 0.

28

De�ne the regularized operator Tε for ε > 0:

Tεϕ(s) :=

∮ 2π

0

(t− s)ϕ(t)

(t− s)2 + ε2dt, 0 ≤ s ≤ 2π,

which is compact and skew-Hermitian from L2([0, 2π] ,C) into itself. Let ϕε be thesolution of the regularized integral equation

(µI − Tε)ϕε = f,

and consider the approximate operator

Tε,n := πnTεσn.

Theorem 6. For n large enough, the operator µI − Tε,n is invertible, the constant

βε := supn‖(µI − Tε,n)−1‖

is �nite, and the solution ψε,n of the equation

(µI − Tε,n)ψε,n = f,

converges to the solution ϕ of equation (2.1) if, �rst, n→∞ and then ε→ 0.

Proof. Since Tε is compact, the theory developped in [4, 8] shows that the inverseoperator (I − Tε,n)−1 exists and is uniformly bounded for n large enough. Since

ψε,n − ϕε = [(µI − Tε,n)−1 − (µI − Tε)−1]f

= (µI − Tε,n)−1[Tε − Tε,n](µI − Tε)−1f

= (µI − Tε,n)−1[Tε − Tε,n]ϕε,

we get‖ψε,n − ϕε‖2 ≤ βε‖(Tε − Tε,n)ϕε‖2 → 0 as n→∞.

Since Tε is skew-Hermitian,

‖(µI − Tε)−1‖ ≤ 1

|µ|,

independently of ε. Hence, the constant

γ := supε‖(µI − Tε)−1‖

is �nite and fromϕε − ϕ = (µI − Tε)−1[T − Tε]ϕ

we get‖ϕε − ϕ‖2 ≤ γ‖[T − Tε]ϕ‖2 → 0 as ε→ 0.

Hence‖ψε,n − ϕ‖2 ≤ ‖ϕε − ϕ‖2 + ‖ψε,n − ϕε‖2 → 0,

if, �rst, n→∞, and then ε→ 0.

The collocation method leads to the following linear system

(µI − Tεσn)ψε,n(sn,i) = f(sn,i), i ∈ [[1, 2n+ 1 ]].

29

Figure 2.1: n = 9

Figure 2.2: n = 23

2.2.2 Numerical Example

Let µ = −1 and, for s ∈ [0, 2π] ,

f(s) :=−s[sin s+Si (s) cos s−sin(s)Ci (s)−Si (s−2π) cos s+Ci (2π−s) sin s],

where Si is the sine integral function, and Ci is the cosine integral function. Theexact solution of equation (2.1) is then

ϕ(s) := s sin s, 0 ≤ s ≤ 2π.

For the regularization process, take ε = 10−4. For the numerical approximation taken = 9, n = 23 and n = 62. The results are exhibited in �gures 2.1, 2.2 and 2.3respectively.

30

Figure 2.3: n = 62

2.3 Direct Solution of Cauchy Integral Equation on

the Real Line

We consider the Cauchy integral equation of the second kind on the real line

ϕ(x) +1

π

∮ +∞

−∞

ϕ(t)

t− xdt = g(x), x ∈ R. (2.2)

We assume that g is 2π-periodic.Let

φm(t) = eimt, m ∈ Z.

Theorem 7. For m ∈ Z, we have∮ +∞

−∞

φm(t)

t− xdt = −iπφm(x), x ∈ R.

Proof. We have ∮ +∞

−∞

φm(t)

t− xdt = m

∮ +∞

−∞

φm(t)

mt−mxdt,

hence ∮ +∞

−∞

φm(t)

t− xdt =

∮ +∞

−∞

eiy

y − sdy.

It is well known (cf.[79]) that∮ +∞

−∞

eiy

y − sdy = −iπeis,

so that ∮ +∞

−∞

φm(t)

t− xdt = −iπeimx = −iπφm(x).

31

Let

ϕ(x) =+∞∑−∞

amφm(x).

Soient p > 0 et Hp([0, 2π] ,C) l'espace de Sobolev classique, for 0 ≤ p < ∞, theSobolev space of all functions ϕ ∈ L2([0, 2π] ,C) such that

+∞∑−∞

(1 +m2)p |am|2 <∞.

We introduce the following norm in Hp([0, 2π] ,C):

‖ϕ‖p =

{+∞∑−∞

(1 +m2)p |am|2} 1

2

.

Let

Aϕ(x) =1

π

∮ +∞

−∞

ϕ(t)

t− xdt.

Theorem 8. If p > q the operator A is bounded from Hp([0, 2π] ,C) into Hq([0, 2π] ,C).

Proof. If

ϕ(x) =+∞∑−∞

amφm(x),

we get

Aϕ(x) =+∞∑−∞

amπ

∮ +∞

−∞

φm(t)

t− xdt.

But

Aφm(x) = −iφm(x),

so that

‖Aϕ‖q =

{+∞∑−∞

(1 +m2)q |am|2} 1

2

.

Since

(1 +m2)q ≤ (1 +m2)p,

we obtain+∞∑−∞

(1 +m2)q |am|2 ≤+∞∑−∞

(1 +m2)p |am|2 .

Thus

‖Aϕ‖q ≤ ‖ϕ‖p .

32

Theorem 9. If p > q then Hp([0, 2π] ,C) is dense in Hq([0, 2π] ,C), with compactimbedding from Hp([0, 2π] ,C) into Hq([0, 2π] ,C).

Proof. (see [91]).

We rewrite equation (2.2) as:

+∞∑−∞

am

{φm(x) +

1

π

∮ +∞

−∞

φm(t)

t− x

}= g(x).

Hence

+∞∑−∞

am {φm(x)− iφm(x)} = g(x),

that is

+∞∑−∞

am (1− i)φm(x) = g(x).

On the space L2([0, 2π] ,C), the inner product

〈ϕ, ψ〉 =

∫ 2π

0

ϕ(t)ψ(t)dt.

{φm}m∈Z is an orthogonal system, so

+∞∑−∞

am (1− i) 〈φm, φk〉 = 〈g, φk〉 , k ∈ Z.

But

〈φm, φk〉 =

{2π m = k0 otherwise.

Hence

ak (1− i) 〈φk, φk〉 = 〈g, φk〉 .

Thus

ak =1

2π (1− i)〈g, φk〉 .

33

Chapter 3

Two Projection Methods forSkew-Hermitian Operator Equations

3.1 Introduction and Mathematical Background

In this chapter we present a projection method for solving an operator equationswith bounded operator in Hilbert spaces. We prove the existence of the solution forthe approximate equation, and we perform the error analysis. We apply the methodfor solving the Cauchy integral equations in L2([0, 1],C) for two cases: Galerkinprojections and Kulkarni projections respectivly, using a sequence of orthogonal�nite rank projections. Numerical examples illustrate the theoretical results.

Since 1980, many papers have been dedicated to the numerical solution of opera-tor equations in the compact case, using Galerkin and other projection methods. In[4], the authors have studied some �nite rank approximations using bounded �niterank projections. In [3], the authors have used a projection approximation for solvingweakly singular Fredholm integral equations of the second kind. In [93], the authorhas proposed a more accurate approximation for compact operator equations. Thegoal of this chapter is to apply two projection methods to an integral equation withsingular kernel. The abstract framework is that of bounded but noncompact skew-Hermitian operators in a Hilbert space. The application will deal with a Cauchyintegral equation in L2([0, 1],C) with two discretizations: the classical Galerkin andthe new Kulkarni approximations built upon a sequence of orthogonal �nite ranksequence of projections.

Let H be a Hilbert space, and T a bounded operator from H into itself. For agiven function f ∈ H, we consider the problem of �nding a function ϕ ∈ H suchthat

ϕ− Tϕ = f. (3.1)

Let T ∗ be the adjoint of T . We assume that equation (3.1) has a unique solutionϕ ∈ H, and that T is skew-Hermitian: is T ∗ = −T . Let (Tn)n≥1 be a sequence ofskew-Hermitian operators from H into itself.

Theorem 10. For all n, the operator I − Tn is invertible, and

‖(I − Tn)−1‖ ≤ 1.

34

Proof. Since(iTn)∗ = −iT ∗n = iTn,

the operator iTn is self-adjoint, and hence sp (Tn) ⊆ iR, where sp denotes thespectrum. This shows that 1 /∈ sp (Tn) and hence the operator I − Tn is invertible.On the other hand, for all x ∈ H,

Re 〈(I − Tn)x, x〉 =1

2

[〈(I − Tn)x, x〉+ 〈(I − Tn)x, x〉

]= 〈x, x〉 ,

hence

‖x‖2 ≤ |Re 〈(I − Tn)x, x〉| ≤ |〈(I − Tn)x, x〉| ≤ ‖(I − Tn)x‖ ‖x‖ ,

so ‖(I − Tn)−1‖ ≤ 1.

3.2 Cauchy Integral Equations of the Second Kind

Set H := L2([0, 1],C), and consider the following Cauchy integral equation of thesecond kind

ϕ(s)−∮ 1

0

ϕ(t)

t− sdt = f(s), 0 ≤ s ≤ 1, (3.2)

where f is a known function. The above integral is understood to be the Cauchyprincipal value: ∮ 1

0

ϕ(t)

t− sdt = lim

ε→0

[∫ s−ε

0

ϕ(t)

t− sdt+

∫ 1

s+ε

ϕ(t)

t− sdt

].

Letting

Tϕ(s) :=

∮ 1

0

ϕ(t)

t− sdt, 0 ≤ s ≤ 1,

equation (3.2) is equivalent to the equation (3.1). We recall that for each f ∈ H,equation (3.2) has a unique solution ϕ ∈ H, and the Cauchy integral operator T isbounded from H into itself, further T ∗ = −T (see [124]). Let (sn,j)

nj=0 be a grid on

[0, 1] such that

0 ≤ sn,0 < sn,1 < . . . < sn,n ≤ 1.

Sethn,i := sn,i − sn,i−1, i ∈ [[1, n ]], hn := (hn,1, hn,2, . . . , hn,n).

Let us consider (Πn)n≥1, a sequence of bounded projections each one of �nite rank,such that

Πnx :=n∑j=1

〈x, en,j〉 en,j,

where

en,j :=φn,j√hn,j

, φn,j(s) :=

{1 for s ∈]sn,j−1, sn,j[,0 otherwise.

35

LetJn := {sn,j, j ∈ [[0, n ]]} .

De�ne the modulus of continuity of the function ψ ∈ H relative to hn as follows:

ω2(ψ, Jn) := sup0≤δ≤hn

(∫ 1

0

|ψ(τ + δ)− ψ(τ)|2 dτ) 1

2

.

All functions are extended by 0 outside [0, 1]. We recall that

ω2(ψ, Jn)→ 0 as n→∞ for all ψ ∈ H,

and that, for all ψ ∈ H (cf. [5]),

‖(I − Πn)ψ‖2 ≤ ω2(ψ, Jn). (3.3)

3.2.1 Galerkin Approximation

In the past two decades, several results have been established for solving compactoperator equations using the Galerkin method. In this section we use the Galerkinmethod for approximate the solution of our bounded equation. Since Π∗n = Πn weget T ∗n = −Tn, with Tn = TGn := ΠnTΠn, hence the following Galerkin approximateequation

ϕGn − TnϕGn = Πnf, (3.4)

has a unique solution ϕGn , given by

ϕGn =n∑j=1

xn,jen,j

for some scalars xn,j. Equation (3.4) reads as

n∑j=1

xn,j [en,j − ΠnTen,j] = Πnf,

so that

n∑j=1

xn,j

[en,j −

n∑i=1

〈Ten,j, en,i〉 en,i

]=

n∑i=1

〈f, en,i〉 en,i,

that is to say, the coe�cients xn,j are obtained by solving the following linear system

(I − An)xn = bn,

where

An(k, j) :=1√

hn,jhn,k

∫ sn,k

sn,k−1

∮ n,sj

sn,j−1

dt

t− sds,

bn(k) :=1√hn,k

∫ sk

sk−1

f(s)ds.

36

Theorem 11. The following estimate holds:

‖ϕGn − ϕ‖2 ≤ ω2(f, Jn) + ω2(Tϕ, Jn) + πω2(ϕ, Jn).

Proof. In fact

ϕGn − ϕ = (I − TGn )−1Πnf − (I − T )−1f

= (I − TGn )−1Πnf − (I − TGn )−1f + (I − TGn )−1f − (I − T )−1f

= (I − TGn )−1(Πn − I)f + (I − TGn )−1[(I − T )− (I − TGn )

](I − T )−1f

= (I − TGn )−1[(Πn − I)f + (TGn − T )ϕ

].

It is proved in Theorem 1 that∥∥(I − TGn )−1

∥∥ ≤ 1. Since

(TGn − T )ϕ = (Πn − I)Tϕ+ ΠnT (Πn − I)ϕ,

and since ‖Πn‖ = 1 and ‖T‖ ≤ π (cf. [124]), then using (3.3), we get the desiredresult.

3.2.2 Kulkarni Approximation

In [93] the author has proposed to approximate a linear operator T by the following�nite rank operator

TKn := ΠnT + TΠn − ΠnTΠn.

Theory has been developped for the compact case. In this section, we propose toapproximate our noncompact bounded operator T by this �nite rank operator. LetϕKn be the approximate solution of the equation (3.2) using TKn . As in [93], let

un := ΠnϕKn .

Since Πnun = un, there exist scalars cn,j such that

un =n∑j=1

cn,jen,j.

Following [93],

un − [ΠnTΠn + ΠnT (I − Πn)TΠn]un = Πnf + ΠnT (I − Πn)f,

so that

n∑j=1

cn,j [en,j − (ΠnTen,j + ΠnT (I − Πn)Ten,j)] = Πnf + ΠnT (I − Πn)f,

and hence

n∑j=1

cn,j

[en,j −

n∑k=1

(〈Ten,j, en,k〉+ 〈T (I − Πn)Ten,j, en,k〉)en,k

]=

37

=n∑k=1

〈f, en,k〉 en,k +n∑k=1

〈T (I − Πn)f, en,k〉 en,k.

Performing the inner product with en,i, we obtain the linear system

cn,i −n∑j=1

cn,j [〈Ten,j, en,i〉+ 〈T (I − Πn)Ten,j, en,i〉] = 〈f, en,i〉+ 〈T (I − Πn)f, en,i〉 , i ∈ [[1, n ]],

which becomes

cn,i −n∑j=1

[〈Ten,j, en,i〉+

⟨T 2en,j, en,i

⟩−

n∑k=1

〈Ten,j, en,k〉〈Ten,k, en,i〉

]cn,j

= 〈f, en,i〉+ 〈Tf, en,i〉 −n∑k=1

〈f, en,k〉〈Ten,k, en,i〉 , i ∈ [[1, n ]]. (3.5)

The following computations are needed:

〈Ten,j, en,i〉 =1√

hn,jhn,i

∫ si

si−1

∮ sj

sj−1

dt

t− sds,

⟨T 2en,j, en,i

⟩=

1√hn,jhn,i

∫ si

si−1

∮ 1

0

1

t− s

∮ sj

sj−1

dτ

τ − tdtds,

〈Tf, en,i〉 =1√hn,i

∫ si

si−1

∮ 1

0

f(t)

t− sdtds,

〈f, en,i〉 =1√hn,i

∫ si

si−1

f(s)ds.

Once the system (3.5) is solved, the solution ϕKn is built through

ϕKn = un + (I − Πn)Tun + (I − Πn)f.

Hence

ϕKn (s) = un(s)+

∮ 1

0

un(t)

t− sdt−

n∑k=1

φn,k(s)

hn,k

∫ sk

sk−1

∮ 1

0

un(τ)

τ − tdτdt+f(s)−

n∑k=1

φn,k(s)

hn,k

∫ sk

sk−1

f(t)dt.

Theorem 12. The following estimate holds:

‖ϕKn − ϕ‖2 ≤[2ω2(ϕ, Jn)

∥∥(I − Πn)T 2 (I − Πn)ϕ∥∥

2

] 12 .

Proof. On one hand

ϕKn − ϕ = (f + TKn ϕKn )− (f + Tϕ) = TKn (ϕKn − ϕ) + (TKn − T )ϕ.

Thus(I − TKn )(ϕKn − ϕ) = (TKn − T )ϕ,

so thatϕKn − ϕ = (I − TKn )−1(TKn − T )ϕ,

38

which lends to ∥∥ϕKn − ϕ∥∥2≤∥∥(I − TKn )−1

∥∥∥∥(TKn − T )ϕ∥∥

2.

On the other hand,(TKn − T

)ϕ = [ΠnT (I − Πn)− T (I − Πn)]ϕ = − (I − Πn)T (I − Πn)ϕ.

Since∥∥(I − TKn )−1

∥∥ ≤ 1,∥∥ϕKn − ϕ∥∥2≤ ‖(I − Πn)T (I − Πn)ϕ‖2 ≤ ω2(T (I − Πn)ϕ, Jn).

But

ω42(T (I − πn)ϕ, Jn) = sup

0≤δ≤hn〈T (I − πn) (ϕ(.+ δ)− ϕ), T (I − πn) (ϕ(.+ δ)− ϕ)〉2

= sup0≤δ≤hn

⟨ϕ(.+ δ)− ϕ,− (I − Πn)T 2 (I − Πn) [ϕ(.+ δ)− ϕ]

⟩2≤ sup

0≤δ≤hn

∫ 1

0

|ϕ(τ + δ)− ϕ(τ)|2 dτ ×∫ 1

0

∣∣(I − Πn)T 2 (I − Πn) [ϕ(τ + δ)− ϕ(τ)]∣∣2 dτ

≤ 4ω22(ϕ, Jn)

∥∥(I − Πn)T 2 (I − Πn)ϕ∥∥2

2,

and we get the desired result.

3.3 Numerical Examples

Example 3.1

We consider the following Cauchy integral equation

ϕ(s)−∮ 1

0

ϕ(t)

t− sdt = s2 − 2s+

1

2+ (s2 − s) ln

(s

1− s

), 0 < s < 1.

The right hand side has been built so that the exact solution to this equation be

ϕ(s) = s2 − s.

We present in table (3.1) the corresponding absolute errors for this example.

Example 3.2

We consider the following Cauchy integral equation

ϕ(s)−∮ 1

0

ϕ(t)

t− sdt =

4 + ln s+ 2 ln 2 + πs− 4 ln (1− s)4 (s2 + 1)

, 0 < s < 1.

The right hand side has been built so that the exact solution to this equation be

ϕ(s) =1

s2 + 1.

We present in table (3.2) the corresponding absolute errors for this example.

39

n∥∥ϕ− ϕGn∥∥2

∥∥ϕ− ϕKn ∥∥2

3 6.69e-2 5.83e-35 3.82e-2 3.17e-37 2.64e-2 2.54e-315 1.17e-2 7.17e-425 6.88e-3 1.75e-4

Table 3.1: Example 3.1

n∥∥ϕ− ϕGn∥∥2

∥∥ϕ− ϕKn ∥∥2

4 4.04e-2 6.44e-36 2.63e-2 3.12e-310 1.55e-2 1.00e-312 1.29e-2 9.54e-420 7.70e-3 3.35e-4


3.4 Conclusions

This work extends the application of projection methods to singular integral equa-tions of Cauchy type. As it has already established for su�ciently di�erentiablekernels (see [93]), the Kulkarni approximation gives more accurate results than theclassical Galerkin approximation. In exchange, from a computational point of view,the complexity of Kulkarni approximation doubles Galerkin's one since one moreevaluation of the integral operator is needed to build each coe�cient of the matrixassociated to the auxiliary linear system.

40

Chapter 4

Collocation Method for SolvingIntegro-Di�erential Equations withCauchy Kernel

4.1 Introduction

This chapter investigates the numerical solution for a class of integro-di�erentialequations with Cauchy kernel by using airfoil polynomials of the �rst kind. Accord-ing to this method, we obtain a system of linear algebraic equations. We give somesu�cient conditions for the convergence of this method. In the end, we investigatethe computational performance of our approach through some numerical examples.The last two decades have been witnessing a strong interest among physicists, en-gineers and mathematicians for the theory and numerical modeling of integral andintegro-di�erential equations. These equations are solved analytically see, for exam-ple the excellent book by Muskhelishvili (cf. [110]), and the references therein. Butonly special cases of these equations are solved analytically, so we should solve otherclasses of these equations by using numerical methods, several methods have beenrecently developed for the numerical solution of the integro-di�erential equations.Speci�cally, in [11], Badr presented a Galerkin approach for solving the integro-di�erential equation of the second kind with Cauchy kernel by using the orthogo-nal basis of Legendre polynomials. In [46], the authors have considered a methodbased on projector-splines for the numerical solution of integro-di�erential equa-tion. In [99], the authors presented a Taylor-series expansion method for a classof Fredholm singular integro-di�erential equation with Cauchy kernel, and usedthe truncated Taylor-series polynomial of the unknown function, and transform theintegro-di�erential equation into a linear ordinary di�erential equation of order nwith variable coe�cients. In [15], we �nd a method based on polynomial approxi-mation using Bernstein polynomial basis, to obtain approximate numerical solutionof a singular integro-di�erential equation with Cauchy kernel, and compared the nu-merical results obtained with those obtained by various Galerkin methods. A greatdeal of e�ort has been made in the development of numerical techniques for the ap-proximate solution of Cauchy integral equations, (cf. [12], [19], [35], [36], [40], [41],[42], [43], [91], [94], [100], [123], [131]). Several methods use polynomial techniques,(cf. [8], [35], [40], [42], [44], [120], [121]). So di�erent kinds of polynomials play

41

an essential role in approximation theory, and have many interesting applications,particularly they may be applied to solve integro-di�erential equations.

In this chapter, we will propose to employ a method based on the airfoil polyno-mials of the �rst kind, for solving the Fredholm singular integro-di�erential equationwith Cauchy kernel.

The chapter is organized as follows: In the next section we will discuss airfoilpolynomials and their key properties. In section 2 we give the description anddevelopment of the method, and we discuss estimates for the rate of convergence ofthe method. In section 3 for showing e�ciency of this method, we use numericalexamples. Section 4 is devoted to the conclusion of this chapter.

We recall that the so-called airfoil polynomials are used as expansion functionsto compute the pressure on an airfoil in steady or unsteady subsonic �ow.

The airfoil polynomail tn of the �rst kind is de�ned by

tn(x) =cos[(n+ 1

2) arccosx]

cos(12

arccosx).

The airfoil polynomail un of the second kind is de�ned by

un(x) =sin[(n+ 1

2) arccosx]

sin(12

arccosx).

4.2 The Approximate Solution

Given a function f and a constant λ, consider the problem of �nding a function ϕsuch that

ϕ′(x) +λ

π

∮ 1

−1

ϕ(t)

t− xdt = f(x), −1 < x < 1. (4.1)

The above equation called Fredholm integro-di�erential equation with Cauchy ker-nel. We will propose an approximate solution for equation (4.1). For this purpose,we will introduce an approximation using the airfoil polynomials of the �rst kind tnas

ϕn(x) = ω(x)n∑i=0

aiti(x),

where

ω(x) =

√1 + x

1− x.

The formula (cf. [33])

(1 + x)t′i(x) = (i+1

2)ui(x)− 1

2ti(x),

gives

ϕ′n(x) =n∑i=0

ai{ω′(x)ti(x) +ω(x)

1 + x[(i+

1

2)ui(x)− 1

2ti(x)]}. (4.2)

42

On the other hand (cf. [33]),

1

π

∮ 1

−1

√1 + t

1− tti(t)

t− xdt = ui(x). (4.3)

Consider the set of n+ 1 collocation points xj, which are the zeros of un+1:

xj = − cos2j − 1

2n+ 3π, j ∈ [[0, n ]].

Let us introduce the following notations

(Aϕ)(x) := ϕ′(x), −1 < x < 1.

(Tϕ)(x) :=λ

π

∮ 1

−1

ϕ(t)

t− xdt, −1 < x < 1.

Equation (4.1) can be written in the following operator form

Aϕ+ Tϕ = f.

Denote by C0,λ([−1, 1],R) the space of all functions ϕ de�ned on [−1, 1] satisfyingthe following Hölder condition: ∃M ≥ 0 such that

∀x1, x2 ∈ [−1, 1] , |ϕ(x1)− ϕ(x2)| ≤M |x1 − x2|λ ,

where 0 < λ ≤ 1.Let

H :={ϕ ∈ L2([−1, 1],R) : ϕ′ ∈ L2([−1, 1],R), ϕ(−1) = 0

}.

The operator T is bounded from L2([−1, 1],R) into itself and also from C0,λ([−1, 1],R)into itself (cf.[110]). We recall that

(A−1y)(s) =

∫ s

−1

y(t)dt,

and that A−1 : L2[−1, 1]→ H is compact (cf.[17]).Consider hat functions e0, e1, e2, . . . , en in C0([−1, 1],R) such that

ej(xk) = δj,k.

De�ne the projection operators Pn from C0([−1, 1],R) into the space of continuousfunctions by

Png(x) :=n∑j=0

g(xj)ej(x).

Let us de�ne the operators

Dn := A−1PnT, D := A−1T.

Consider the following approximate equation in the unknown ϕn:

ϕn +Dnϕn = A−1f.

43

Theorem 13. Assume that f ∈ C0([−1, 1],R). There exists a positive constant M ,such that

‖ϕn − ϕ‖∞ ≤ M‖Dnϕ−Dϕ‖∞.

for n large enough.

Proof. It is well-known that ‖Png − g‖∞ → 0, for all g ∈ C0([−1, 1],R). Since A−1

is compact, it is clear that D is compact. In (cf. [8] and [91]) it is shown that theinverse operator (I+Dn)−1 exists and is uniformly bounded for n large enough. Onthe other hand,

ϕn − ϕ =[A−1f −Dnϕn

]−[A−1f −Dϕ

],

hence

ϕn − ϕ = [Dϕ−Dnϕn] .

This leads to

ϕn − ϕ = [(D −Dn)ϕ−Dn(ϕn − ϕ)] .

Thus

(I +Dn)(ϕn − ϕ) = (D −Dn)ϕ.

Consequently

ϕn − ϕ = (I +Dn)−1 [(D −Dn)ϕ] ,

‖ϕn − ϕ‖∞ ≤ M‖(D −Dn)ϕ‖∞,

where

M := supn≥N

∥∥(I +Dn)−1∥∥ ,

which is �nite.

Finally, the following system follows:

Aϕn(xj) + Tϕn(xj) = f(xj), j ∈ [[0, n ]].

By (4.2) and (4.3),

n∑i=0

ai{ω′(xj)ti(xj) +ω(xj)

1 + xj[(i+

1

2)ui(xj)−

1

2ti(xj)] + λui(xj)} = f(xj), j ∈ [[0, n ]].

4.3 Numerical Results and Discussion

In order to illustrate the performance of our method, we report in this section,numerical results of some examples, selected integro-di�erential equations, solvedby the method of this study. In these numerical computations each table shows thenumerical error of our approximate solution.

44

x n = 5 n = 20 n = 116-0.8 0.1640e-1 0.183e-2 0.16e-3-0.6 0.1855e-1 0.185e-2 0.15e-3-0.4 0.2204e-1 0.181e-2 0.2e-4-0.2 0.2119e-1 0.184e-2 0.12e-30.0 0.189e-1 0.19e-2 0.2e-40.2 0.1856e-1 0.171e-2 0.3e-40.4 0.1992e-1 0.179e-2 0.8e-40.6 0.2044e-1 0.174e-2 0.8e-40.8 0.2148e-1 0.177e-2 0.9e-4


x n = 5 n = 49 n = 135-0.8 0.737e-2 0.21001e-3 0.40012e-4-0.6 0.927e-2 0.23e-3 0.80002e-4-0.4 0.1439e-1 0.17e-3 0.60007e-4-0.2 0.1171e-1 0.23001e-3 0.90003e-40.0 0.59827e-2 0.17379e-3 0.45706e-40.2 0.52100e-2 0.14001e-3 0.60001e-40.4 0.1147e-1 0.21004e-3 0.60024e-40.6 0.1616e-1 0.26002e-3 0.6e-40.8 0.282e-2 0.22001e-3 0.30065e-4


Example 4.1

Let us �rst consider the following integro-di�erential equation

ϕ′(x) +1

π5

∮ 1

−1

ϕ(t)

t− xdt =

1

π4x2 + 2(

1

π5+ 1)x− 1

π4

The exact solution is

ϕ(x) = x2 − 1.

Table (4.1) gives the numerical results for Example 4.1.

Example 4.2

In this example we consider the following integro-di�erential equation

ϕ′(x) +

∮ 1

−1

ϕ(t)

t− xdt = πx3 − 5x2 − πx+

7

3+ (x3 − x) ln(x+ 1)− (x3 − x) ln(x− 1).

The exact solution for this equation is

ϕ(x) = −x3 + x.

Table (4.2) shows the rate of convergence of the method. The results con�rm theconvergence properties proved above.

45

4.4 Concluding Remarks

Cauchy kernel are important in many �elds of applied mathematics. The method canbe developed and applied to other class of integral and integro-di�erential equations.The advantage of this method is that we can eliminate the singularity, and computean approximate solution through a system of linear equations.

46

Chapter 5

Projection Methods forIntegro-Di�erential Equations withCauchy Kernel


In this chapter we present two methods for solving Cauchy integro-di�erential equa-tions. First, we present a projection method based on Legendre polynomials, forsolving integro-di�erential equations with Cauchy kernel, in L2([−1, 1],C). The pro-posed numerical procedure leads to solve a system of linear equations. We prove theexistence of the solution for the approximate equation, and we perform the erroranalysis. Numerical examples illustrate the theoretical results. Next, we proposea Sloan projection method for the approximate solution of an integro-di�erentialequations with Cauchy kernel in L2([−1, 1] × [−1, 1],C) using Legendre polynomi-als. A system of linear equations is to be solved.The theory of integro-di�erential equations with Cauchy kernel has important ap-plications in the mathematical modelling of many scienti�c �elds such as �uid dy-namics, electrodynamics, elasticity. Many integro-di�erential equations need to besolved numerically. Several authors have been studied projection approximationsfor solving integral equations with di�erent numerical procedures, the theory of pro-jection approximations is developed in [4]. In [4], the authors have studied some�nite rank approximations using bounded �nite rank projections. Projection ap-proximation methods play an essential role in approximation theory, and have manyinteresting applications, particularly to solve integral equations. In [3], the authorshave used a projection approximation for solving weakly singular Fredholm integralequations of the second kind. Let H := L2([−1, 1],C), be the space of complex-valued Lebesgue square integrable (classes of) functions on [−1, 1]. The purposeof this chapter is �rstly to introduce a projection method based on the Legendrepolynomials, for solving integro-di�erential equations with Cauchy kernel in H. Thepurpose of the second method is to approximate the solution of integro-di�erentialequations with Cauchy Kernel using Sloan projection in the �rst time.

Let the universe of our discours be the Hilbert space H. Set

D := {ϕ ∈ H : ϕ′ ∈ H, ϕ(−1) = 0} ,

47

and consider the integro-di�erential equation with Cauchy kernel

ϕ′(s) +

∮ 1

−1

ϕ(t)

t− sdt = f(s), −1 < s < 1, (5.1)

where the integral is understood as the Cauchy principal value:∮ 1

−1

ϕ(t)

t− sdt = lim

ε→0

(∫ s−ε

−1

x(t)

t− sdt+

∫ 1

s+ε

x(t)

t− sdt

).

Letting

Tϕ(s) :=

∮ 1

−1

ϕ(t)

t− sdt, −1 < s < 1,

Aϕ(s) := ϕ′(s), −1 < s < 1,

the operator T is bounded from H into itself and

A−1y(s) =

∫ s

−1

y(t)dt, −1 < s < 1,

is compact. Equation (5.1) can be rewritten as

ϕ+ A−1Tϕ = A−1f.

Let

K := A−1T

which is compact. We assume that −1 is not an eigenvalue of K.Let (Ln)n≥0 be the sequence of Legendre polynomials which is an orthogonal basisfor H:

〈Lj, Lk〉 = δjk2

2j + 1,

and

ej :=

√2j + 1

2Lj,

the corresponding normalized sequence.Let (πn)n≥0 be the sequence of bounded �nite rank orthogonal projections de�ned

by

πnx :=n−1∑j=0

〈x, ej〉 ej.

Hence, for ψ ∈ H,limn→∞

‖πnψ − ψ‖ = 0.

48

5.2 A Projection Method for Integro-Di�erential Equa-

tions with Cauchy Kernel

Let Hn denote the space spanned by the �rst n of Legendre polynomials. It is clearthat A−1(Hn) = Hn+1. The approximate problem is the following equation for ϕn:

ϕn + A−1πnTϕn = A−1πnf.

Clearly ϕn ∈ D ∩Hn+1. We introduce the following notations:

K := A−1T, Kn := A−1πnT, g := A−1f, gn := A−1πnf,

and we assume that −1 is not an eigenvalue of K. Hence the equation

(I +K)ϕ = g,

is approximated by

(I +Kn)ϕn = gn.

For all x ∈ H,limn→∞

‖Knx−Kx‖ = 0,

and since A−1 is compact,

limn→∞

‖ (Kn −K)K‖ = 0, limn→∞

‖ (Kn −K)Kn‖ = 0.

Writing

ϕn =n∑j=0

xn,jej,

the n+ 1 unknowns xn(j) solven∑j=0

xn(j)[e′j + πnTej

]= πnf,

n∑j=0

xn(j)ej(−1) = 0.

This leads to a linear system

Anxn = bn,

where, for i ∈ [[0, n− 1 ]] and j ∈ [[0, n ]],

An(i, j) :=

√2i+ 1

2

√2j + 1

2

[∫ 1

−1

L′j(s)Li(s)ds+

∫ 1

−1

(

∮ 1

−1

Lj(τ)

τ − sdτ)Li(s)ds

],

An(n, j) := ej(−1),

bn(i) :=

√2i+ 1

2

∫ 1

−1

f(s)Li(s)ds,

bn(n) := 0.

49

Since K is compact, the theory developped in [4] shows that for n large enough, theoperator I + Kn is invertible, and its inverse is uniformly bounded with respect ton.

Let s > 0 and Hs([−1, 1],C) be the classical Sobolev space, and let ‖.‖s denoteits norm. (For details, see [10].) Remark that

(I + A−1T )(Hs([−1, 1],C)) = Hs([−1, 1],C).

We recall that (cf. [10]) there exists c > 0 such that, for all ψ ∈ Hs([−1, 1],C),

‖(I − πn)ψ‖ ≤ cn−s‖ψ‖s. (5.2)

Theorem 14. Assume that f ∈ Hs([−1, 1],C) for some s > 0. Then, there existsα > 0 such that

‖ϕn − ϕ‖ ≤ α[n1−s‖Tϕ‖s−1 + n−s‖f‖s].

Proof. We have

ϕn − ϕ =[(I +Kn)−1 gn − (I +K)−1 g

]+ (I +Kn)−1 g − (I +Kn)−1 g

= (I +Kn)−1 [(K −Kn)ϕ+ gn − g] ,

and hence‖ϕn − ϕ‖ ≤ C

[‖ (K −Kn)ϕ‖+ ‖A−1‖‖ (I − πn) f‖

].

On the other hand,(K −Kn)ϕ = A−1 (I − πn)Tϕ.

But f ∈ Hs([−1, 1],C), so ϕ ∈ Hs([−1, 1],C) and Tϕ ∈ Hs−1([−1, 1],C). Using(5.2), the desired result follows.

5.3 Sloan Projection Method

Consider the approximate problem of �nding ϕSn ∈ D such that

ϕSn +KπnϕSn = A−1f. (5.3)

Clearly, if such a function exists, it belongs to D.Applying the operator πn to both sides of equation (5.3) we get

πnϕSn + πnKπnϕ

Sn = πnA

−1f,

or, equivalently,

n−1∑j=0

⟨ϕSn, ej

⟩ej +

n−1∑j=0

⟨ϕSn, ej

⟩πnKej =

n−1∑j=0

⟨A−1f, ej

⟩ej,

and performing the inner product with ei we get the following system:

⟨ϕSn, ei

⟩+

n−1∑j=0

⟨ϕSn, ej

⟩〈πnKej, ei〉 =

⟨A−1f, ei

⟩, i ∈ [[0, n− 1 ]].

50

Since π∗n = πn, and πnei = ei,

⟨ϕSn, ei

⟩+

n−1∑j=0

⟨ϕSn, ej

⟩〈Kej, ei〉 =

⟨A−1f, ei

⟩, i ∈ [[0, n− 1 ]]. (5.4)

Since K is compact, (In+An)−1 exists for n large enough (see [4]). Once the system(5.4) is solved, ϕSn is recovered as

ϕSn(s) =

∫ s

−1

f(t)dt−n−1∑j=0

xn(j)

√2j + 1

2

∫ s

−1

∮ 1

−1

Lj(τ)

τ − tdτdt.

Let

M := supn≥N

∥∥(I +Kπn)−1∥∥ ,

which is �nite.

Theorem 15. Assume that f ∈ Hs([−1, 1],C) for some s > 0. Then, there existsβ > 0 such that

‖ϕSn − ϕ‖ ≤Mβ ‖K‖n−s‖ϕ‖s.

Proof. We have

ϕSn − ϕ =(A−1f −KπnϕSn

)−(A−1f −Kϕ

)= K (I − πn)ϕ+Kπn

(ϕ− ϕSn

),

and hence (ϕSn − ϕ

)= (I +Kπn)−1K (I − πn)ϕ.

But f ∈ Hs([−1, 1],C), so ϕ ∈ Hs([−1, 1],C). Using (5.2), the desired resultfollows.

5.4 Numerical Example

In this section, we present a numerical example to illustrate the theoretical resultsobtained in the above sections. Tables 5.1 shows the absolute error as a function ofn.

Example 5.1

Let f be de�ned so that the exact solution be

ϕ(s) =s+ 1

s2 + 1.

51

n ‖ϕ− ϕn‖2∥∥ϕ− ϕSn∥∥2

4 2.40e-2 2.15e-25 9.88e-3 7.92e-36 4.08e-3 2.59e-37 1.68e-3 9.85e-48 6.99e-4 3.44e-49 2.89e-4 1.32e-410 1.20e-4 4.81e-511 4.96e-5 1.88e-512 2.05e-5 6.98e-6


52

Chapter 6

Regularization and ProjectionApproximations for Cauchy IntegralEquations of the Second Kind


In this chapter, we derived the regularization to the solution of Cauchy integralequation, and we apply the projection to the obtained equation. First we us Kan-torovich projection, and we perform the error analysis. After we study the Sloanprojection and we prove some results about the error analysis. In the end of thischapter Galerkin projection is estabished and its error analysis is discussed.

Let C0,α([−1, 1] ,C), 0 < α ≤ 1 be the space of all α-Hölder continuous functions.Let us denote by H∗([−1, 1] ,C), 0 < α ≤ 1 the space of all functions ϕ which satisfythe following conditions:

• ϕ is α-Hölder continuous on every closed subinterval of (−1, 1),

•

ϕ(t) =ϕ∗(t)

(t− c)µ, 0 ≤ µ < 1,

near c = ±1. ϕ∗ is Hölder continuous function.

Consider the following Cauchy integral equation of the second kind

aϑ(x) +b

π

∮ 1

−1

ϑ(t)

t− xdt− µ

∫ 1

−1

k(x, t)ϑ(t)dt = g(x), −1 < x < 1, (6.1)

where a, b are constants, such that a2 +b2 = 1. We assume that g ∈ C0,α([−1, 1] ,C).Following [110],

ϑ(x) = ω(x)ϕ(x),

where the function ϕ is a Hölder continuous function, and ω is a weight functionde�ned as

ω(x) := (1− x)α(1 + x)β,

53

where α and β are given by

α :=1

2πilog

a− iba+ ib

+ ν, β := − 1

2πilog

a− iba+ ib

+ ν ′, −1 < α, β < 1.

ν and ν ′ are integers related to the following index

κ := −(α + β) = −(ν + ν ′).

Hence

aω(x)ϕ(x) +b

π

∮ 1

−1

ω(t)ϕ(t)

t− xdt− µ

∫ 1

−1

k(x, t)ω(t)ϕ(t)dt = g(x), −1 < x < 1.

Let K0 be the operator de�ned by

K0φ(x) := aω(x)φ(x) +b

π

∮ 1

−1

ω(t)φ(t)

t− xdt, φ ∈ C0,α([−1, 1] ,C), −1 < x < 1,

and K1 be the operator de�ned by

K1φ(x) :=

∫ 1

−1

k(x, t)ω(t)φ(t)dt, φ ∈ C0,α([−1, 1] ,C), −1 < x < 1.

First, we will discuss the solution of the following Cauchy integral equation of the�rst kind

K0ϕ(x) = h(x), h ∈ C0,α([−1, 1] ,C), −1 < x < 1.

The solution of this equation has the following boundary behavior in [−1, 1]:

• If −1 < α < 0 and 0 < β < 1 : The solution is continuous in (−1, 1), boundedat x = −1 and may have a weak singularity at x = 1.

• If 0 < α < 1 and −1 < β < 0 : The solution is continuous in (−1, 1), boundedat x = 1 and may have a weak singularity at x = −1.

• If 0 < α < 1 and 0 < β < 1 : The solution is continuous in (−1, 1), andbounded at both ends x = −1 and x = 1.

• If −1 < α < 0 and −1 < β < 0 : The solution is continuous in (−1, 1), andmay have a weak singularities both at x = −1 and x = −1.

According to [47], the Cauchy integral equation of the �rst kind has the solution

ϕ(x) = ah(x)

ω(x)− b

π

∮ 1

−1

h(t)

ω(t)

dt

t− x+b

πC,

where

• For κ = 1, the solution is unbounded at both ends x = −1 and x = 1, and notunique since C is an arbitrary constant.

• For κ = 0, the solution is unique since C = 0.

54

• For κ = −1, the equation has a solution if and only if∫ 1

−1

h(t)

ω(t)dt = 0,

and in this case the solution is unique with C = 0.

Let K be the operator de�ned by

Kφ(x) := aω∗(x)φ(x)− b

π

∮ 1

−1

ω∗(t)φ(t)

t− xdt, φ ∈ C0,α([−1, 1] ,C), −1 < x < 1,

where

ω∗(x) :=1

ω(x).

Ifh := g + µK1φ,

then we get the following equivalent equation with a similar discussion as above

ϕ− µKK1ϕ = Kg +b

πC.

Let

f :=−1

µ(Kg +

b

πC), λ :=

1

µ, T := KK1.

Then ϕ solves

(T − λI)ϕ = f. (6.2)

Using the Proposition 1.1.2 in [47], if we assume that k ∈ C1([−1, 1]2 ,C), g ∈C1([−1, 1] ,C), then the operator T is compact from X := C0([−1, 1] ,C) into X.

6.2 Finite Rank Approximations and Regulariza-

tion

In this section we introduce a grid (xn,j)nj=0 on [−1, 1] such that

−1 < xn,0 < xn,1 < . . . < xn,n−1 < xn,n < 1.

Consider hat functions e0, e1, e2, . . . , en in C0([−1, 1] ,C) such that

ej(xn,k) = δj,k.

De�ne the projection πn from C0([−1, 1] ,C) into itself by

πng(x) :=n∑j=0

g(xn,j)ej(x).

55

We recall that by [4, 8]

limn→∞

‖πng − g‖∞ = 0.

The solution ϕ of (6.2) satis�es

ϕ =1

λ(Tϕ− f).

Letψ := Tϕ = λϕ+ f.

We get

ϕ =1

λ(ψ − f),

so that

ψ =1

λ(Tψ − Tf). (6.3)

Hence

πnψ =1

λ(πnTψ − πnTf). (6.4)

Let us approximate the solution of (6.3) by ψPn (P for Kantorovich) such that

ψPn =1

λ(πnTψ

Pn − πnTf). (6.5)

Theorem 16. The inverse operator (I− 1λπnT )−1 exists and it is uniformly bounded

for n large enough, and ∥∥ψ − ψPn ∥∥ ≤M0 ‖ψ − πnψ‖ .

Proof. In fact

(I − 1

λπnT )(ψ − ψPn ) = ψ − 1

λπnTψ − ψPn +

1

λπnTψ

Pn .

From (6.5),

(I − 1

λπnT )(ψ − ψPn ) = ψ − 1

λπnTψ +

1

λπnTf.

Thus

(I − 1

λπnT )(ψ − ψPn ) = ψ − 1

λ(πnTψ − πnTf).

Using (6.4), we get

(I − 1

λπnT )(ψ − ψPn ) = ψ − πnψ,

and the resulat follows, with

M0 := supn

∥∥∥∥(I − 1

λπnT )−1

∥∥∥∥ .Since T is compact, limn→∞ ‖πnT − T‖∞ = 0, and hence M0 <∞.

56

Let

ϕ̃Pn :=1

λ(ψPn − f).

Theorem 17. The iterated approximation ϕ̃Pn satis�es∥∥ϕ̃Pn − ϕ∥∥ ≤ C1 ‖T − πnT‖ .∥∥ϕ̃Pn − ϕ∥∥ ≤ M0

|λ|‖ψ − πnψ‖ .

Proof. We have

(πnT − λI)ϕ̃Pn =1

λ(πnT − λI)(ψPn − f),

so

(πnT − λI)ϕ̃Pn =1

λ(πnTψ

Pn − λψPn − πnTf + λf).

But (6.5) impliesπnTψ

Pn − λψPn = πnTf.

Hence(πnT − λI)ϕ̃Pn = f.

Thus

ϕ̃Pn − ϕ = (πnT − λI)−1 f − (T − λI)−1 f

= (πnT − λI)−1 (T − λI)−1 [(T − λI)− (πnT − λI)] f

= (πnT − λI)−1 (T − λI)−1 [T − πnT ] f.

Finally, we have ∥∥ϕ̃Pn − ϕ∥∥ ≤ C1 ‖T − πnT‖ .

WhereC0 := sup

n

∥∥(πnT − λI)−1∥∥ ,

C1 :=∥∥(T − λI)−1

∥∥ ‖f‖C0.

Also

ϕ̃Pn − ϕ =1

λ(ψPn − f)− 1

λ(ψ − f)

=1

λ(ψPn − ψ).

Using the above Theorem we get the desired bound.

6.3 Sloan Projection

Using (6.3),

ψSn =1

λ(Tπnψ

Sn − Tf). (6.6)

57

Theorem 18. The inverse operator (Tπn − λI)−1 exists, it is uniformaly boundedfor n large enough, and ∥∥ψ − ψSn∥∥ ≤M1 ‖T‖ ‖πnψ − ψ‖ ,where

M1 := supn

∥∥(Tπn − λI)−1∥∥ .

Proof. We remark that

(Tπn − λI)(ψ − ψSn ) = Tπnψ − λψ − TπnψSn + λψSn .

It follows from (6.6) that

(Tπn − λI)(ψ − ψSn ) = Tπnψ − λψ − Tf.By (6.6),

(Tπn − λI)(ψ − ψSn ) = Tπnψ − Tψ,and the result follows. Since T is compact, limn→∞ ‖πnT − T‖∞ = 0, and hence M1

is �nite.

6.4 Galerkin Projection

By using Galerkin projection from (6.3),

ψGn =1

λ(πnTπnψ

Gn − πnTf) (6.7)

Theorem 19. The inverse operator (I− 1λπnT )−1 exists and it is uniformaly bounded

for n large enough, and ∥∥ψ − ψGn ∥∥ ≤ γ ‖(πnψ − ψ‖ ,where

γ := supn

∥∥∥∥(I − 1

λπnT )−1

∥∥∥∥ .Proof. We have

(I − 1

λπnT )(ψ − πnψGn ) = ψ − 1

λπnTψ − πnψGn +

1

λπnTψ

Gn .

From (6.7),

(I − 1

λπnT )(ψ − πnψGn ) = ψ − 1

λπnTψ +

1

λπnTf

= ψ − 1

λ(πnTψ − πnTf).

By (6.4),

(I − 1

λπnT )(ψ − πnψGn ) = ψ − πnψ.

Hence,

ψ − πnψGn = (I − 1

λπnT )−1(ψ − πnψ),

and we get the desired result. Since T is compact, limn→∞ ‖πnT − T‖∞ = 0, andhence γ <∞.

58

Conclusions and perspectives

In this thesis, new numerical schemes based on the projection and collocation meth-ods have been constructed and justi�ed for approximate solutions of Cauchy integraland integro-di�erential equations. We have developed a projection method for solv-ing an operator equation with bounded noncompact operator in Hilbert spaces.

This work may be extended to other type of Cauchy integral and integro-di�erentialequations.

60

Bibliography

[1] R.P. Agarwal, D. O. Regan, Singular Di�erential and Integral Equations withApplications, Springer, 2000.

[2] R. P. Agarwal, D. O. Regan, Integral and Integrodi�erential Equations, CRCPress, 2000.

[3] M. Ahues, A. Amosove, A. Largillier, O. Titaud, Lp Error estimates for pro-jection approximations, Applied mathematics letters, 18 (2005), 381-386.

[4] M. Ahues, A. Largillier, B. V. Limaye, Spectral Computations for BoundedOperators, CRC, Boca Raton, 2001.

[5] A. Amosov, M. Ahues, A. Largillier, Superconvergence of some projection ap-proximations for weakly singular integral equations using general grids, SIAMJournal on Numerical Analysis, 47 (2008), 646-674.

[6] G. B. Arfken, Mathematical Methods for Physicists, �fth edition, Harcourt Aca-demic Press, 2001.

[7] K.E. Atkinson, A Survey of Numerical Methods for the Solution of FredholmIntegral Equations of the Second Kind, SIAM, 1976.

[8] K. Atkinson, The Numerical Solution of Integral Equations of the Second Kind,Cambridge university press, 1997.

[9] K.E. Atkinson, An Introduction to Numerical Analysis, Wiley Sons, Inc, 1989.

[10] K.E. Atkinson, Han, Theoretical Numerical Analysis: A Functional AnalysisFramework, 3rd edition Springer-Verlag, 2009.

[11] A. A. Badr, Integro-di�erential equation with Cauchy kernel, Journal of Com-putational and Applied Mathematics, 134 (2001), 191�199.

[12] C. T. H. Baker, The Numerical Treatment Of Integral Equations, ClarendonPress, Oxford, 1978.

[13] E. H. Bareiss, Singular Integrals and Singular Integral Equations with a CauchyKernel and the Method of Symmetric Pairing , University of Michigan Library,1965.

[14] D. Berthold, P. Junghanns, New error bounds for the quadrature method forthe solution of Cauchy singular integral equations, SIAM Journal on NumericalAnalysis, 30 (1993), 1351-1372.

61

[15] S. Bhattacharya, B. N. Mandal Numerical solution of a singular integro-di�erential equation, Applied Mathematics and Computation, 195 (2008), 346-350.

[16] A. Böttcher, M. A. Kaashoek, A. B. Lebre, and A. F. D. Santos, SingularIntegral Operators, Factorization and Applications, Birkhäuser Basel, 2003.

[17] H. Brezis, Analyse Fonctionnelle, Dunod, Paris, 1999.

[18] M. R. Capobianco, G. Criscuolo, P. Junghanns, A fast algorithm for Prandtl'sintegro-di�erential equation, Journal of Computational and Applied Mathemat-ics, 97 (1997), 103-128.

[19] A. Chakrabarti, S.C. Martha, Approximate solutions of Fredholm integral equa-tions of the second kind, Applied Mathematics and Computation, 211 (2009),459-466.

[20] M. Christ, Lectures on Singular Integral Operators, American MathematicalSociety, 1991.

[21] G. Criscuolo, G. Mastroianni, On the convergence of the Gauss quadrature rulesfor the Cauchy principal value integrals, Ricerche di Matematica, 35 (1986), 45-60.

[22] G. Criscuolo, G. Mastroianni, Convergence of Gauss-Christo�el formula withpreassigned node for Cauchy principal-value integrals, Journal of ApproximationTheory, 50 (1987), 326-340.

[23] G. Criscuolo, G. Mastroianni, Convergence of Gauss type product formulas forthe evaluation of two-dimensional Cauchy principal value integrals, BIT, 27(1987), 72-84.

[24] G. Criscuolo, G. Mastroianni, On the convergence of an interpolatory productrule for evaluating Cauchy principal value integrals, Mathematics of Computa-tions, 48 (1987), 725-735.

[25] G. Criscuolo, G. Mastroianni, Convergenza di formule gaussiane per il calcolodelle derivate di integrali a valor principale secondo Cauchy, Calcolo, 24 (1987),179-192.

[26] G. Criscuolo, G. Mastroianni, On the uniform convergence of modi�ed Gaus-sian rules for the numerical evaluation of derivatives of Principal Value inte-grals, Numerical Methods and Approximation Theory III (Nis, 1987), 139-147,University of Nis, Nis, 1988.

[27] G. Criscuolo, G. Mastroianni, On the approximation of the derivatives of P.V.integrals, Constructive Theory of Functions (Varna ,1987), 92-96, Publ. House.Bulgar. Acad. Sci., So�a, 1988.

[28] G. Criscuolo, G. Mastroianni, On the convergence of product formulas for thenumerical evaluation of derivatives of Cauchy principal value integrals, SIAMJ. Numer. Anal, 25 (1988), 713-727.

62

[29] G. Criscuolo, G. Mastroianni, Mean convergence of a product rule to evaluatethe �nite Hilbert transform, Numerical and Applied Mathematics, Part I (1988),43-45.

[30] G. Criscuolo, G. Mastroianni, On the uniform convergence of Gaussian quadra-ture rules for Cauchy principal value integrals, Numer. Math, 54 (1989), 445-461.

[31] G. Criscuolo, G. Mastroianni, On the uniform convergence of Gaussian quadra-ture rules for Cauchy principal value integrals, Numer. Math, 54 (1989), 445-461.

[32] G. Criscuolo, G. Mastroianni, E. Santi, Convergence of Gauss type formulaswith preassigned nodes, Facta Univ. Ser. Math. Inform., No. 3 (1988), 61-72.

[33] R. N. Desmarais, S. R. Bland, Tables of Properties of Airfoil Polynomials, Nasareference publication 1343, september 1995.

[34] A. Dzhuraev, Methods of Singular Integral Equations, Bellhaven Press, 1992.

[35] D. Elliott, The classical collocation method for singular integral equations havinga Cauchy kernel, SIAM J. Nwner. Anal. 19 (1982), 816-831.

[36] Z. K. Eshkuvatov, N.M.A. Nik Long, M. Abdulkawi, Approximate solution ofsingular integral equations of the �rst kind with Cauchy kernel, Applied Math-ematics Letters, 22 (2009), 651-657.

[37] R. Estrada, R P. Kanwal, Singular Integral Equations, Birkhäuser, Boston,1999.

[38] M. P. George, Iterpolation and Approximation by Polynomials, Springer, 2003.

[39] A. Gerasoulis, The stability of the Gauss-Chebyshev method for Cauchy singularintegral equations, Rutgers University, 1986.

[40] M. A. Golberg, Discrete projection methods for Cauchy singular integral equa-tions with constant coe�cients, Applied Mathematics and Computation, 33(1989), 1-41.

[41] M. A. Golberg, Numerical Solution of Integral Equations, Dover, New York,1990.

[42] M. A. Golberg, The convergence of a collocation method for a class of Cauchysingular integral equations , Journal of Mathematical Analysis and Applications,100 (1984), 500-512.

[43] M. A. Golberg, The convergence of several algorithms for solving integral equa-tions with fFinite part integrals, Applied Mathematics and Computation, 21(1987), 283-293.

[44] M. A. Golberg, The discrete Sloan iterate for the generalized airfoil equation,Applied Mathematics and Computation, 57 (2003), 103-115.

63

[45] S. Gong, Integrals of Cauchy Type on the Ball, International Press of Boston,1994.

[46] L. Gori, E. Santi, M.G. Cimoroni, Projector-splines in the numerical Solutionof Integro-Di�erential Equations, Computers math. Applic. 35 (1998), 107-116.

[47] N. Guessous, Numerical solution of a Cauchy singular integral equation by iter-ative methods, PhD thesis, Université Joseph-Fourier - Grenoble France, 1984.

[48] D. B. Ingham, M. A. Kelmanson , Boundary Integral Equation Analyses ofSingular, Potential and Biharmonic Problems, Springer, 1984.

[49] N.I. Ioakimidis, The successive approximation method for the airfoil equation,Journal of computational and applied mathematics, 21 (1988), 213-238.

[50] N. I. Ioakimidis, The numerical solution of crack problems in plane elasticity inthe case of loading discontinuities, Engineering Fracture Mechanics, 13 (1980),709-716.

[51] N. I. Ioakimidis, One more approach to the computation of Cauchy principalvalue integrals, Journal of Computational and Applied Mathematics, 7 (1981),289-291.

[52] N. I. Ioakimidis, A natural approach to the introduction of �nite-part integralsinto crack problems of three-dimensional elasticity, Engineering Fracture Me-chanics, 16 (1982), 669-673.

[53] N. I. Ioakimidis, On the numerical solution of Cauchy type singular integralequations by the collocation method, Applied Mathematics and Computation,12 (1983), 49-60.

[54] N. I. Ioakimidis, A remark on the solution of the integral equation of pla-nar cracks in three-dimensional elasticity, Engineering Fracture Mechanics, 18(1983), 1199-1200.

[55] N. I. Ioakimidis, A modi�cation of the quadrature method for the direct nu-merical solution of singular integral equations, Computer Methods in AppliedMechanics and Engineering, 14 (1984), 1-13.

[56] N. I. Ioakimidis, A remark on the application of interpolatory quadrature rules tothe numerical solution of singular integral equations, Journal of Computationaland Applied Mathematics, 11 (1984), 267-276.

[57] N. I. Ioakimidis, A new interpretation of Cauchy type singular integrals withan application to singular integral equations, Journal of Computational andApplied Mathematics, 14 (1986), 271-278.

[58] N. I. Ioakimidis, Mangler-type principal value integrals in hypersingular integralequations for crack problems in plane elasticity, Engineering Fracture Mechan-ics, 5 (1988), 895-898.

64

[59] N. I. Ioakimidis, Remarks on the gaussian quadrature rule for �nite-part inte-grals with a second-order singularity, Computer Methods in Applied Mechanicsand Engineering, 69 (1988), 325-343.

[60] N. I. Ioakimidis, Generalized Mangler-type principal value integrals with an ap-plication to fracture mechanics, Journal of Computational and Applied Math-ematics, 30 (1990), 227-234.

[61] N.I. Ioakimidis, On the quadrature methods for the numerical solution of sin-gular integral equations, Journal of Computational and Applied Mathematics,8 (1982), 81-86.

[62] N.I. Ioakimidis, Simple bounds for the stress intensity factors by the method ofsingular integral equations, Engineering Fracture Mechanics, 18 (1983), 1191-1198.

[63] N.I. Ioakimidis, Simple bounds for the stress intensity factors by the method ofsingular integral equations, Engineering Fracture Mechanics, 18 (1983), 1191-1198.

[64] N.I. Ioakimidis, Validity of the hypersingular integral equation of crack problemsin three-dimensional elasticity along the crack boundaries, Engineering FractureMechanics, 26 (1987), 783-788.

[65] N.I. Ioakimidis, Construction of singular integral equations for interactingstraight cracks by using reduce, Engineering Fracture Mechanics, 40 (1991),1179-1184.

[66] N.I. Ioakimidis, Application of computer algebra to the iterative solution ofsingular integral equations, Computer Methods in Applied Mechanics and En-gineering, 94 (1992), 229-237.

[67] N. I. Ioakimidis, G. T. Anastasselos, Direct Taylor-series solution of singularintegral equations with MAPLE, Computers and Structures, 45 (1992), 613-617.

[68] N. I. Ioakimidis, K. E. Papadakis, A new class of quite elementary closed-formintegral formulae for roots of nonlinear equations, Applied Mathematics andComputation, 29 (1989), 185-196.

[69] N.I. Ioakimidis, P.S. Theocaris, On the selection of collocation points for the nu-merical solution of singular integral equations with generalized kernels appearingin elasticity problems, Computers and Structures, 11 (1980), 289-295.

[70] A.A. Ivanov, R.S. Anderssen, D. Elliott, The Theory of Approximate Methodsand Their Applications to the Numerical Solution of Singular Integral Equa-tions, Springer, 1976.

[71] A. J. Jerri, Introduction to Integral Equations with Applications, Wiley-Interscience, 1999.

65

[72] P. Junghanns, Optimal control for a parametrized family of nonlinear Cauchysingular integral equations, Journal of Computational and Applied Mathematics, 164-165 (2004), 431-454.

[73] P. Junghanns, U. Luther, Cauchy singular integral equations in spaces of con-tinuous functions and methods for their numerical solution, Journal of Compu-tational and Applied Mathematics , 77 (1997), 201-237.

[74] P. Junghanns, K. Müller, A collocation method for nonlinear Cauchy singularintegral equations, Journal of Computational and Applied Mathematics , 115(2000), 283-300.

[75] P. Junghanns, D. Oestreich, Application of nonlinear singular integral equationsto the numerical solution of a symmetric electrochemical machining problem,Computers Structures, 44 (1992), 409-417.

[76] P. Junghanns, G. Semmler, U. Weber, Non-linear singular integral equationson a �nite interval, Mathematical Methods in the Applied Sciences, 24, (2001)1275�1288.

[77] P. Junghanns, B. Silbermann, Numerical analysis of Newton projection methodsfor nonlinear singular integral equations, Journal of Computational and AppliedMathematics, 55 (1994), 145-163.

[78] P. Junghanns, B. Silbermann, Numerical analysis for one-dimensional Cauchysingular integral equations, Journal of Computational and Applied Mathemat-ics, 125 (2000), 395-421.

[79] R. Kanwal, Linear Integral Equations, Academic Press, 1971.

[80] A. Karlovich, Algebras of singular integral operators with piecewise continuouscoe�cients on re�exive Orlicz spaces, Mathematische Nachrichten, 179 (1996),187-222.

[81] A. Karlovich, Singular integral operators with piecewise continuous coe�cientson re�exive Orlicz spaces, Doklady Mathematics, 54 (1996), no. 1, 490-492.

[82] A. Karlovich, Singular integral operators with regulated coe�cients on re�exiveOrlicz spaces, Siberian Math. J., 38 (1997), 253-266.

[83] A. Karlovich, On the index of singular integral operators in re�exive Orliczspaces, Mathematical Notes, 64 (1998), 330-341.

[84] A. Karlovich, Singular integral operators with piecewise continuous coe�cientsin re�exive rearrangement-invariant spaces,Integral Equations and OperatorTheory, 32 (1998), 436-481.

[85] A. Karlovich, On the essential norm of the Cauchy singular integral operatorin weighted rearrangement-invariant spaces, Integral Equations and OperatorTheory, 38 (2000), 28-50.

66

[86] A. Karlovich, Algebras of singular integral operators with PC coe�cients inrearrangement-invariant spaces with Muckenhoupt weights, Journal of OperatorTheory, 47 (2002), 303-323.

[87] A. Karlovich, Algebras of singular integral operators on rearrangement-invariantspaces and Nikolski ideals, New York Journal of Mathematics, 8 (2002), 215-234.

[88] A. Karlovich, Fredholmness of singular integral operators with piecewise con-tinuous coe�cients on weighted Banach function spaces, Journal of IntegralEquations and Applications, 15 (2003), 263 -320.

[89] A. C Kaya, On the solution of integral equations with strongly singular ker-nels, National Aeronautics and Space Administration, Langley Research Cen-ter, 1986.

[90] V. G. Kravchenko, G. S. Litvinchuk, Introduction to the Theory of SingularIntegral Operators with Shift, Springer, 1994.

[91] R. Kress, Linear Integral Equations, Springer-Verlag, Göttingen, 1998.

[92] R. Kress,Numerical Analysis. Graduate Texts in Mathematics 181, Springer-Verlag, New York, (1998).

[93] R. Kulkarni, A Superconvergence Result For Solutions of Compact OperatorEquations, Bull. Austral. Math. Soc. 68 (2003), 517-528.

[94] P.K. Kyhte and P. Puri, Computational Methods For Linear Integral Equations,Birkhauser-Verlag, Boston, 2002.

[95] L. Lerer, V. Olshevsky, I. M. Spitkovsky, O. Karlovych, Convolution Equationsand Singular Integral Operators, Birkäuser, 2010.

[96] I. K. Lifanov, L.N. Poltavskii, MG.M. Vainikko, Hypersingular Integral Equa-tions and Their Applications: Di�erential and Integral Equations and TheirApplications, CRC Press, 2003.

[97] A. M. Linkov, Boundary Integral Equations in Elasticity Theory, Springer, 2002.

[98] G. S. Litvinchuk, Solvability Theory of Boundary Value Problems and SingularIntegral Equations with Shift, Springer, 2000.

[99] K. Maleknejad, A. Arzhang, Numerical solution of the Fredholm singularintegro-di�erential equation with Cauchy kernel by using Taylor-series expan-sion and Galerkin method, Applied Mathematics and Computation 182 (2006),888-897.

[100] B. N. Mandal, G. H. Bera, Approximate solution of a class of singular integralequations of second kind, Journal of Computational and Applied Mathematics,206 (2007), 189-195

[101] J.C. Mason D.C. Handscomb, Chybechev polynomials, A CRC Press Company2003.

67

[102] G. Mastroianni, On the saturation of a quadrature rule for evaluating Cauchysingular integrals, Ricerche di Matematica, 36 (1987), 59-62.

[103] G. Mastroianni, On the convergence of product formulas for the numericalevaluation of derivatives of Cauchy principal value integrals, Mathematics ofComputation, 52 (1989), 95-101.

[104] G. Mastroianni, M. R. Occorsio, An algorithm for the numerical evaluation ofa Cauchy principal value integral, Ricerche Mat, 33 (1984), 3-18.

[105] G. Mastroianni, S. Prössdorf, Some nodes matrices appearing in the numericalanalysis for singular integral equations, BIT, 34 (1994), 120-128.

[106] G. Mastroianni, M.G. Russo, W. Themistoclakis, The boundedness of aCauchy integral operator in weighted Besov type spaces with uniform norms,Integral Equations Operator Theory, 42 (2002), 57-89.

[107] G. Mastroianni, M.G. Russo, W. Themistoclakis, Numerical methods forCauchy singular integral equations in spaces of weighted continuous functions,Recent advances in operator theory and its applications, 311-336, OperatorTheory: Advanced and Applications,160 , Birkäuser, Basel, 2005.

[108] L. G Mikhailov, A new class of singular integral equations and its applicationto di�erential equations with singular coe�cients, Wolters-Noordho� 1970.

[109] SG Mikhlin and S. Prössdorf, Singular Integral Operators, Akademie-Verlag,Berlin, 1986.

[110] N. I. Mushkelishvili, Singular Integral Equations, Noordho�, Groningen, 1953.

[111] M.T. Nair, A modi�ed projection method for equtions of the second kind , Bull.Austral. Math.Soc. 36 (1987), 485-492.

[112] M.T. Nair, A modi�ed projection method for equtions of the second kind-Corrigendum and Addendum , Bull. Austral. Math.Soc. 40 (1989), 487-487.

[113] M.T. Nair, On uniform convergence of approximation methods for operatorequations of the second kind , Numer.Funct.Anal. Optim. 13 (1992), 69-73.

[114] M.T. Nair, On accelerated re�nement methods for operator equations of thesecond kind , J. Indian Math. Soc. 59 (1993), 135-140.

[115] M.T. Nair, Linear Operator Equations: Approximation and Regularization,World Scienti�c Pub Co Inc, 2009.

[116] M.T. Nair, R.S Anderssen, Superconvergence of modi�ed projection method forintegral equations of the second kind , J. Integral Equations and Appl. 3 (1991)255-269.

[117] V. Nazaikinskii, A. Yu. Savin, B. Schulze, and B. Y. Sternin, Elliptic Theory onSingular Manifolds: Di�erential and Integral Equations and Their Applications,Chapman and Hall, 2005.

68

[118] L. Nickelson, Singular Integral Equation's Methods For The Analysis Of Mi-crowave Structures, Walter de Gruyter, 2005.

[119] V.Z. Parton, P.I. Perlin,Integral Equations in Elasticity, Mir Publisher,Moskow, 1982.

[120] R. Piessens, Computing integral transforms and solving integral equations us-ing Chebyshev polynomial approximations, Journal of Computational and Ap-plied Mathematics 121 (2000), 113-124

[121] R. Piessens, M. Branders, On the computation of Fourier transforms of sin-gular functions, journal of computational physics 21 (1976), 178-196.

[122] A. C. Pipkin, A Course on Integral Equations, Springer, 1991.

[123] A. D. Polyanin and A. V. Manzhirov, Handbook of iIntegral Equations, CRCPress, Boca Raton, 1998.

[124] D. Porter, D. Stirling, Integral Equations: A Practical Treatment, from Spec-tral Theory to Applications, Cambridge University Press, 1990.

[125] S. Prössdorf, B. Silbermann, Numerical Analysis for Integral and Related Op-erator Equations, Akademie-Verlag, Berlin 1991.

[126] M. Rahman, Integral Equations and their Applications, WIT Press, 2007.

[127] Robert Piessens, Computing integral transforms and solving integral equationsusing Chebyshev polynomial approximations, Journal of Computational and Ap-plied Mathematics 121 (2000), 113-124.

[128] A. Sengupta, A Note On a Reduction of Cauchy Singular Integral Equation toFredholm Equation in Lp, Applied Mathematics and Computation, 56 (1993),97-100 .

[129] I.H. Sloan, V. Thomee, Superconvergence of the Galerkin iterates for integralequations of the second kind, J. Integral Equations, 9 (1985), 2-23.

[130] E. M. Stein, Singular Integrals and Di�erentiability Properties of Functions,Princeton University Press, 1971.

[131] F. G. Tricomi, Integral Equations, Interscience Publishers, New York, 1957.

[132] K. M. Vakhtang, On boundedness of singular integral operators in Lp spaceswith weight, Soobshch. Akad. Nauk Gruzin. SSR 64(1971), No. 1, 17-20.

[133] K. M. Vakhtang, On boundedness of singular integral operators in weightedspaces, In: Proc. of Symposium on ContinuumMechanics and Related Problemsof Analysis (Tbilisi, 23-29. IX. 1971), vol. 1, Metsniereba, Tbilisi, 1973, 125-141.

[134] K. M. Vakhtang, On boundedness of singular operators with Cauchy kernels inweighted spaces, Soobshch. Akad. Nauk Gruzin. SSR 77 (1975), No. 3, 529-532.

69

[135] K. M. Vakhtang, On singular integrals and maximal operators with Cauchykernel, Dokl. Akad. Nauk SSSR 223 (1975), No. 3, 555-558.

[136] K. M. Vakhtang, On maximal singular integral operator with Cauchy kernel,Trudy Tbiliss. Mat. Inst. im. Razmadze Akad. Nauk Gruzin. SSR 55 (1977),38-58.

[137] K. M. Vakhtang, On weighted inequalities for singular integrals with Cauchykernel on smooth contours, Soobshch. Akad. Nauk Gruzin. SSR 90( 1978), No.3, 537-540.

[138] K. M. Vakhtang, Singular integral operators in weighted spaces, In: ColloqiaMath. Soc. J. Boyai 35, Functions, Series, Operators. Budapesht, 1980, 707-714.

[139] G. Vainikko, Multidimensional Weakly Singular Integral Equations, Springer,1993.

[140] A. M. Wazwaz, A First Course in Integral Equations, World Scienti�c Pub CoInc, 1997.

[141] U. Weber, Weighted Polynomial Approximation Methods for Cauchy SingularIntegral Equations in the Non-Periodic Case, Logos, 2000.

70

RESUME

L'objectif de ce travail est la résolution des équations intégrales singulières à noyauCauchy. On y traite les équations singulières de Cauchy de première espèce par laméthode des approximations successives. On s'intéresse aussi aux équations inté-grales à noyau de Cauchy de seconde espèce, en utilisant les polynômes trigonométriqueset les techniques de Fourier.

Dans la même perspective, on utilise les polynômes de Tchebychev de quatrièmedegré pour résoudre une équation intégro-di�érentielle à noyau de Cauchy.

En suite, on s'intéresse à une autre équation intégro-di�érentielle à noyau deCauchy, en utilisant les polynômes de Legendre, ce qui a donné lieu à développerdeux méthodes basées sur une suite de projections qui converge simplement versl'identité.

En outre, on exploite les méthodes de projection pour les équations intégralesavec des operateurs intégraux bornés non compacts et on a appliqué ces méthodesà l'équation intégrale singulière à noyau de Cauchy de deuxième espèce.

Mots clés: Equations intégrales, approximations successives, méthodes de projec-tion, méthodes de collocation, noyau de Cauchy.

Summary

The purpose of this thesis is to develop and illustrate various new methods forsolving many classes of Cauchy singular integral and integro-di�erential equations.

We study the successive approximation method for solving Cauchy singular in-tegral equations of the �rst kind in the general case, then we develop a collocationmethod based on trigonometric polynomials combined with a regularization proce-dure, for solving Cauchy integral equations of the second kind.

In the same perspective, we use a projection method for solving operator equationwith bounded noncompact operators in Hilbert spaces.

We apply a collocation and projection methods for solving Cauchy integro-di�erential equations, using airfoil and Legendre polynomials.

Keywords: Integral equations, successive approximations, projection methods, col-location methods, Cauchy kernel.

72

Thèse de Doctorat Cotutelle Internationale

Présenté par : Abdelaziz MENNOUNI

Sur la Résolution des Equations Intégrales Singulières à Noyau de Cauchy

Résumé

À Biskra, le 27 avril 2011

Le domaine des équations intégrales, aussi vaste qu'il le soit, a connu ces dernières années une attention con-sidérable. En e�et, les équations intégrales interviennent dans le traitement de di�érents problèmes apparraissantdans les domaines des sciences physiques et de la technologie tels que: le transfert radiatif, la di�usion, l'élasticité,où les équations intégrales singulières à noyau Cauchy trouvent leur place.

Concernant notre thème: équations intégrales singulières à noyau Cauchy, on s'intéresse à la résolution de cetype d'équations par di�érentes méthodes telles que celles de projection et collocation.

On vise en premier lieu à généraliser une méthode des approximations successives pour résoudre une équationintégrale singulière à noyau de Cauchy de première espèce, appliquée à deux cas d'intérêt pratique.

Puis, en se servant des méthodes de projection, on traite des équations fonctionnelles avec des opérateurs bornés(non compacts); notamment la méthode de Galerkin et la méthode de Kulkarni.

D'autre part on traite les équations intégro-di�érentielles à noyau de Cauchy via la méthode de collocation etla méthode de projection.

Un autre objectif réalisé est le traitement des équations intégrales singulières à noyau de Cauchy de deuxièmeespèce en utilisant les polynômes trigonométriques et une procédure de régularisation.

Le présent travail est organisé comme suit: On commence par un rappel général de la théorie des opérateursbornés, puis un rappel de théorie spectrale, en�n on donne brièvement une classi�cation des équations intégraleset une introduction à notre thème.

Dans le premier chapitre on traite une équation intégrale singulière à noyau de Cauchy de première espèce, enutilisant la méthode des approximations successives.

Soit l'équation intégrale singulière à noyau de Cauchy de première espèce

1π

∮ 1

−1

v(t)ϕ(t)t− x

dt = g(x), −1 < x < 1,

1

où v et g sont deux fonctions connues, et ϕ est l'inconnue. On suppose que

1π

∮ 1

−1

v(t)t− x

dt = 1, −1 < x < 1,

1π

∫ 1

−1

|v(t)|dt ≤ 1, −1 < x < 1,

1π

∮ 1

−1

1v(t)(t− x)

dt = 1, −1 < x < 1,

1π

∫ 1

−1

1|v(t)|

dt ≤ 1, −1 < x < 1.

Par la méthode des approximations successives on obtient

ϕn+1(x) = g(x)− 1π

∮ 1

−1

v(t) [ϕn(t)− ϕn(x)]t− x

dt, −1 < x < 1.

Considérons

Rn(x) = ϕ(x)− ϕn(x).

On a

‖Rn‖ ≤1n!‖g(n)‖+

1(n+ 1)!

‖g(n+1)‖, n ∈ N.

Si

limn→∞

(1n!‖g(n)‖+

1(n+ 1)!

‖g(n+1)‖)

= 0,

alors la méthode des approximations successives converge.La méthode proposée a été testée pour les deux cas suivants:

v(t) =

√1 + t

1− t,

et

v(t) =

√1− t1 + t

,

respéctivement.Le deuxième chapitre traite des equations intégrales singulières à noyau de Cauchy de deuxième éspèce en

utilisant les polynômes trigonométriques. Ce chapitre contient deux sections:Dans la première section, nous présentons une méthode de collocation basée sur les polynômes trigonométriques

combinée à une procédure de régularisation, pour résoudre l'équation intégrale de Cauchy de seconde espèce

µϕ(s)−∮ 2π

0

ϕ(t)t− s

dt = f(s), 0 ≤ s ≤ 2π.

2

Cette équation s'écritµϕ− Tϕ = f,

où

Tϕ(s) :=∮ 2π

0

ϕ(t)t− s

dt, 0 ≤ s ≤ 2π.

Soient X := L2([0, 2π] ,C), et Xn l'espace engendré par les polynômes trigonométriques de degré ≤ 2n + 1. Soitσn la projection orthogonale de X sur Xn. Alors, pour tout ψ ∈ X,

limn→∞

‖σnψ − ψ‖2 = 0.

Soient 0 ≤ sn,1 < sn,2 < . . . < sn,2n+1 ≤ 2π. Pour tout i ∈ [[1, 2n + 1 ]] considérons la fonction chapeau en,i dansC0([0, 2π] ,C), telle que, pour chaque j ∈ [[1, 2n+ 1 ]],

en,i(sn,j) = δi,j .

Soit Yn l'espace engendré par ces fonctions chapeau, qui est de dimension 2n + 1. Nous dé�nissons la projectioninterpolation πn de C0([0, 2π] ,C) sur Yn:

πnh(s) :=2n+1∑j=1

h(sn,j)en,j(s), h ∈ C0([0, 2π] ,C).

Dé�nissons l'opérateur de régularisation Tε pour ε > 0:

Tεϕ(s) :=∮ 2π

0

(t− s)ϕ(t)(t− s)2 + ε2

dt, 0 ≤ s ≤ 2π,

qui est compact et sesqui-hermitien de X dans lui-même.Soit ϕε la solution de l'équation intégrale régularisée:

(µI − Tε)ϕε = f,

et considérons l'opérateur d'approximationTε,n := πnTεσn.

Théorème 0.1 Pour n assez grand, l'operateur µI − Tε,n est inversible, la constante

βε := supn‖(µI − Tε,n)−1‖

est �nie et la solution ψε,n de l'equation

(µI − Tε,n)ψε,n = f,

converge vers la solution ϕ si, en premier lieu, n→∞ puis ε→ 0.

La méthode de collocation conduit au système linéaire

(µI − Tεσn)ψε,n(sn,i) = f(sn,i), i ∈ [[1, 2n+ 1 ]].

3

Dans la deuxième section, nous présentons une méthode directe basée sur les polynômes trigonométriques pourrésoudre l'équation intégrale singulière de Cauchy de deuxième espèce

ϕ(x) +1π

∮ +∞

−∞

ϕ(t)t− x

dt = g(x), x ∈ R.

Supposons que g est 2π-periodique.Soit

φm(t) = eimt, m ∈ Z.

Alors

ϕ(x) =+∞∑−∞

amφm(x),

où

ak =1

2π (1− i)〈g, φk〉 .

Dans le troisième chapitre on applique les méthodes de projection à des opérateurs bornés non compacts.Soit H un espace de Hilbert, et T un opérateur borné de H dans lui-même. Pour une fonction donnée f ∈ H,

le problème est de trouver une fonction ϕ ∈ H telle que

ϕ− Tϕ = f.

On suppose que cette équation admet une solution unique ϕ ∈ H, et que T est sesqui-hermitien: i.e T ∗ = −T . Soit(Tn)n≥1 une suite d'opérateurs sesqui-hermitiens de H dans lui-même.

Théorème 0.2 Pour tout n, l'opérateur I − Tn est inversible et

‖(I − Tn)−1‖ ≤ 1.

Soit H := L2([0, 1],C), considérons l'équation intégrale singulière de Cauchy

ϕ(s)−∮ 1

0

ϕ(t)t− s

dt = f(s), 0 ≤ s ≤ 1,

où f est une fonction connue.Cette équation s'écrit sous la forme:

(I − T )ϕ = f,

où

Tϕ(s) :=∮ 1

0

ϕ(t)t− s

dt, 0 ≤ s ≤ 1.

Rappelons que pour tout f ∈ H, cette équation admet une solution unique ϕ ∈ H, et l'opérateur intégral T estborné de H dans lui-même, de plus T ∗ = −T .

4

Soit (sn,j)nj=0 une grille sur [0, 1] telle que

0 ≤ sn,0 < sn,1 < . . . < sn,n ≤ 1.

Posonshn,i := sn,i − sn,i−1, i ∈ [[1, n ]], hn := (hn,1, hn,2, . . . , hn,n).

Considérons la suite (Πn)n≥1, des projections bornées et de rang �ni, telle que

Πnx :=n∑j=1

〈x, en,j〉 en,j ,

où

en,j :=φn,j√hn,j

, φn,j(s) :={

1 si s ∈]sn,j−1, sn,j [,0 sinon.

SoitJn := {sn,j , j ∈ [[0, n ]]} .

Nous dé�nissons le module de L2−intégrabilité de la fonction ψ ∈ H par rapport à hn comme suit:

ω2(ψ, Jn) := sup0≤δ≤‖hn‖∞

(∫ 1

0

|ψ(τ + δ)− ψ(τ)|2 dτ) 1

2

.

Toutes les fonctions sont prolongées par 0 en dehors de [0, 1]. Nous rappelons que

limn→∞

hn = 0 =⇒ limn→∞

ω2(ψ, Jn) = 0 pour tout ψ ∈ H,

et que, pour tout ψ ∈ H,

‖(I −Πn)ψ‖2 ≤ ω2(ψ, Jn).

Dans un premier temps considérons la projection de Galerkin

Tn = TGn := ΠnTΠn,

alors l'equation d'approximation de Galerkin suivante:

ϕGn − TnϕGn = Πnf,

admet une solution unique ϕGn , donnée par

ϕGn =n∑j=1

xn(j)en,j .

En résolvant le système linéaire ci-dessous, on retrouve les coe�cients xn(j):

(I −An)xn = bn,

où

An(k, j) :=1√

hn,jhn,k

∫ sn,k

sn,k−1

∮ n,sj

sn,j−1

dt

t− sds,

bn(k) :=1√hn,k

∫ sk

sk−1

f(s)ds.

5

Théorème 0.3 L'estimation suivante a lieu:

‖ϕGn − ϕ‖2 ≤ ω2(f, Jn) + ω2(Tϕ, Jn) + πω2(ϕ, Jn).

En deuxième lieu considérons la projection de Kulkarni

TKn := ΠnT + TΠn −ΠnTΠn.

La théorie a été développée pour le cas compact. Ici, nous proposons d'approcher notre opérateur borné noncompact T .

Soit ϕKn solution approchée de Kulkarni. On pose

un := ΠnϕKn .

Alors

un =n∑j=1

cn,jen,j .

En résolvant le système linéaire ci-dessous, on retrouve les coe�cients cn,j :

cn,i −n∑j=1

[〈Ten,j , en,i〉+

⟨T 2en,j , en,i

⟩−

n∑k=1

〈Ten,j , en,k〉〈Ten,k, en,i〉

]cn,j

= 〈f, en,i〉+ 〈Tf, en,i〉 −n∑k=1

〈f, en,k〉〈Ten,k, en,i〉 , i ∈ [[1, n ]].

Rappelons que

〈Ten,j , en,i〉 =1√

hn,jhn,i

∫ si

si−1

∮ sj

sj−1

dt

t− sds,

⟨T 2en,j , en,i

⟩=

1√hn,jhn,i

∫ si

si−1

∮ 1

0

1t− s

∮ sj

sj−1

dτ

τ − tdtds,

〈Tf, en,i〉 =1√hn,i

∫ si

si−1

∮ 1

0

f(t)t− s

dtds,

〈f, en,i〉 =1√hn,i

∫ si

si−1

f(s)ds.

La solution ϕKn est donnée parϕKn = un + (I −Πn)Tun + (I −Πn)f,

et donc

ϕKn (s) = un(s) +∮ 1

0

un(t)t− s

dt−n∑k=1

φn,k(s)hn,k

∫ sk

sk−1

∮ 1

0

un(τ)τ − t

dτdt+ f(s)−n∑k=1

φn,k(s)hn,k

∫ sk

sk−1

f(t)dt.

6

Théorème 0.4 L'estimation suivante est véri�ée:

‖ϕKn − ϕ‖2 ≤[2ω2(ϕ, Jn)

∥∥(I −Πn)T 2 (I −Πn)ϕ∥∥

2

] 12 .

Dans le quatrième chapitre, on présente la méthode de collocation en utilisant les polynomes de Tchebychev pourapprocher la solution de l'équation intégro-di�érentielle

ϕ′(x) +λ

π

∮ 1

−1

ϕ(t)t− x

dt = f(x), −1 < x < 1.

où f est une fonction connue et λ est une constante.Cette équation s'écrit:

Aϕ+ Tϕ = f,

où

(Aϕ)(x) := ϕ′(x), −1 < x < 1.

(Tϕ)(x) :=λ

π

∮ 1

−1

ϕ(t)t− x

dt, −1 < x < 1.

Nous allons introduire une approximation en utilisant les polynômes de Tchebychev de troisième espèce tn commesuit:

ϕn(x) = ω(x)n∑i=0

aiti(x),

où

ω(x) :=

√1 + x

1− x.

Nous rappelons que les polynômes de Tchebychev de troisième espèce tn sont dé�nis par:

tn(x) :=cos[(n+ 1

2 ) arccosx]cos( 1

2 arccosx),

et les polynômes de Tchebychev de quatrième espèce un sont dé�nis par:

un(x) :=sin[(n+ 1

2 ) arccosx]sin( 1

2 arccosx).

Considérons l'ensemble des zeros xj de un+1:

xj := − cos2j − 12n+ 3

π, j ∈ [[0, n ]].

7

Soit C0,λ([−1, 1],R) l'espace des fonctions ϕ sur [−1, 1] satisfaisant la condition de Hölder suivante: ∃M ≥ 0 telleque:

∀x1, x2 ∈ [−1, 1] , |ϕ(x1)− ϕ(x2)| ≤M |x1 − x2|λ ,

où 0 < λ ≤ 1.On pose

H :={ϕ ∈ L2([−1, 1],R) : ϕ′ ∈ L2([−1, 1],R), ϕ(−1) = 0

}.

L'opérateur T est borné de L2([−1, 1],R) dans lui-même et aussi de C0,λ([−1, 1],R) dans lui-même.Rappelons que

(A−1y)(s) =∫ s

−1

y(t)dt,

et que A−1 : L2([−1, 1],R)→ H est compact.Considérons les fonctions chapeau e0, e1, e2, . . . , en dans C0([−1, 1],R) telles que

ej(xk) = δj,k.

Soit la projection Pn de C0([−1, 1],R) sur l'espace des fonctions continues, dé�nie par:

Png(x) :=n∑j=0

g(xj)ej(x).

On pose

Dn := A−1PnT, D := A−1T.

Considérons l'équation de l'inconnue ϕn suivante:

ϕn +Dnϕn = A−1f.

Théorème 0.5 Soit f ∈ C0([−1, 1],R). Il existe une constante positive M , telle que

‖ϕn − ϕ‖∞ ≤ M‖Dnϕ−Dϕ‖∞.

pour n assez grand.

On trouve, le système:

Aϕn(xj) + Tϕn(xj) = f(xj), j ∈ [[0, n ]].

Et donc

n∑i=0

ai{ω′(xj)ti(xj) +ω(xj)1 + xj

[(i+12

)ui(xj)−12ti(xj)] + λui(xj)} = f(xj), j ∈ [[0, n ]].

8

Dans le cinquième chapitre nous présentons deux méthodes pour résoudre l'équation intégro-di�érentielle ànoyau de Cauchy

ϕ′(s) +∮ 1

−1

ϕ(t)t− s

dt = f(s), −1 ≤ s ≤ 1.

Cette équation s'écrit

ϕ+A−1Tϕ = A−1f,

où

Tϕ(s) :=∮ 1

−1

ϕ(t)t− s

dt, −1 ≤ s ≤ 1,

Aϕ(s) := ϕ′(s), −1 ≤ s ≤ 1.

SoientH := L2([−1, 1],R),

.D := {ϕ ∈ H : ϕ′ ∈ H, ϕ(−1) = 0} .

Soit (Ln)n≥0 la suite des polynômes de Legendre. On pose

ej :=

√2j + 1

2Lj ,

Soit (πn)n≥0 la projection orthogonale dé�nie par:

πnx :=n−1∑j=0

〈x, ej〉 ej .

En premier lieu, nous présentons le problème d'approximation

ϕn +A−1πnTϕn = A−1πnf.

Soit Hn sous-espace engendré par les n premiers polynômes de Legendre.Il est clair que ϕn ∈ D ∩Hn+1. Cela conduit au système linéaire suivant:

Anxn = bn,

où , pour i ∈ [[0, n− 1 ]] et j ∈ [[0, n ]],

An(i, j) :=

√2i+ 1

2

√2j + 1

2

[∫ 1

−1

L′j(s)Li(s)ds+∫ 1

−1

(∮ 1

−1

Lj(τ)τ − s

dτ)Li(s)ds],

An(n, j) := ej(−1),

bn(i) :=

√2i+ 1

2

∫ 1

−1

f(s)Li(s)ds,

bn(n) := 0.

Soient s > 0 et Hs([−1, 1],R) l'espace de Sobolev classique, muni de la norme ‖.‖s.

9

Théorème 0.6 Soit f ∈ Hs([−1, 1],R) pour un certain s > 0. Alors, il existe α > 0 telle que

‖ϕn − ϕ‖ ≤ α[n1−s‖Tϕ‖s−1 + n−s‖f‖s].

En deuxème lieu, nous présentons le problème d'approximation de Sloan:Trouver ϕSn ∈ D telle que

ϕSn +KπnϕSn = A−1f.

On trouve le système linéaire

(In +An)xn = bn,

où xn(j) :=⟨ϕSn , ej

⟩, et

An(i, j) :=

√2j + 1

2

√2i+ 1

2

∫ 1

−1

∫ s

−1

∮ 1

−1

Lj(τ)Li(s)τ − t

dτdtds,

bn(i) :=

√2i+ 1

2

∫ 1

−1

∫ s

−1

f(t)Li(s)dtds.

On pose

M := supn≥N

∥∥(I +Kπn)−1∥∥ .

Théorème 0.7 Soit f ∈ Hs([−1, 1],R) pour un certain s > 0. Il existe une constante positive β > 0, telle que

‖ϕSn − ϕ‖ ≤Mβ ‖K‖n−s‖ϕ‖s,

pour n assez grand.

Le dernier chapitre est consacré à une régularisation de l'équation intégrale

aϑ(x) +b

π

∮ 1

−1

ϑ(t)t− x

dt− µ∫ 1

−1

k(x, t)ϑ(t)dt = g(x), −1 < x < 1,

où a, b sont deux contantes, telles que a2 + b2 = 1. On suppose que g ∈ C0,α([−1, 1] ,C). Dans ce cas la solutions'écrit

ϑ(x) = ω(x)ϕ(x),

où la fonction ϕ véri�e la condition de Hölder, et ω est une fonction poids dé�nie comme suit:

ω(x) := (1− x)α(1 + x)β ,

où

α :=1

2πilog

a− iba+ ib

+ ν, β := − 12πi

loga− iba+ ib

+ ν′, −1 < α, β < 1.

10

ν et ν′ sont deux entiers liés à l'indice suivant:

κ := −(α+ β) = −(ν + ν′).

On pose

K0φ(x) := aω(x)φ(x) +b

π

∮ 1

−1

ω(t)φ(t)t− x

dt, φ ∈ C0,α([−1, 1] ,C), −1 < x < 1,

K1φ(x) :=∫ 1

−1

k(x, t)ω(t)φ(t)dt, φ ∈ C0,α([−1, 1] ,C), −1 < x < 1.

Kφ(x) := aω∗(x)φ(x)− b

π

∮ 1

−1

ω∗(t)φ(t)t− x

dt, φ ∈ C0,α([−1, 1] ,C), −1 < x < 1,

ω∗(x) :=1

ω(x),

h := g + µK1φ.

On trouve l'équation suivante:

ϕ− µKK1ϕ = Kg +b

πC,

que nous réécrivons

(T − λI)ϕ = f,

où

f :=−1µ

(Kg +b

πC), λ :=

1µ, T := KK1.

On suppose que k ∈ C1([−1, 1]2 ,C), g ∈ C1([−1, 1] ,C), alors l'opérateur T est compact de X := C0([−1, 1] ,C)dans X.

On poseψ := Tϕ = λϕ+ f.

On trouve

Pnψ =1λ

(PnTψ − PnTf).

Soit ψPn l'approximation de Kantorovich

ψPn =1λ

(PnTψPn − PnTf).

Théorème 0.8 L'opérateur inverse (I − 1λPnT )−1 existe et est uniformément borné pour n assez grand. De plus

il existe M0 > 0 tel que ∥∥ψ − ψPn ∥∥ ≤M0 ‖ψ − Pnψ‖ .

11

Notons

ϕ̃Pn :=1λ

(ψPn − f).

Théorème 0.9 ∥∥ϕ̃Pn − ϕ∥∥ ≤ C1 ‖T − PnT‖ et∥∥ϕ̃Pn − ϕ∥∥ ≤ M0

|λ|‖ψ − Pnψ‖ .

Notons

ψSn =1λ

(TPnψSn − Tf).

Théorème 0.10 Il existe N ∈ N tel que pour tout n ≥ N , l'opérateur (TPn − λI)−1existe, est uniformément

borné et ∥∥ψ − ψSn∥∥ ≤M1 ‖T‖ ‖Pnψ − ψ‖ ,

où

M1 := supn≥N

∥∥∥(TPn − λI)−1∥∥∥ .

Notons

ψGn =1λ

(PnTPnψGn − PnTf).

Théorème 0.11 Il existe N0 ∈ N tel que pour tout n ≥ N0, l'opérateur(I − 1

λPnT)−1

existe, est uniformément

borné et ∥∥ψ − ψGn ∥∥ ≤ γ ‖Pnψ − ψ‖ ,où

γ := supn≥N0

∥∥∥∥(I − 1λPnT )−1

∥∥∥∥ .Les di�érentes méthodes proposées dans cette thèse sont illustrées par un ensemble conséquent d'expériencesnumériques.

12

[for solving cauchy singular integral equations]

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