potential function in fredholm ــ volterra integral … · potential function in fredholm ــ...

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Umm Alــ Qura University

Faculty of Applied Sciences

Department of Mathematical Sciences

Potential Function in

Fredholm ــ Volterra Integral Equation

A Thesis Submitted in Partial Fulfillment of the Requirements of

Master's Degree

In

Applied Mathematics

( Integral Equations )

Prepared by Researcher

Faizah Mohamed Hamdi Alــ Saedy

Supervised by

Prof. Mohamed Abdella Ahmed Abdou

1427 AH2006 ــG

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References

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their origins in integral equation theory. Arch. Hist. Exact. Sci. Vol. 3 (1966) 1- 96.

[2] C.D.Green, Integral Equation Methods, NewYork, 1969.

[3] H.Hochstadt, Integral Equations, Awiley Inter Science Publication, NewYork,

1971.

[4] M.A.Golberg.ed. Solution Methods for Integral Equations, NewYork, 1979.

[5] F.G.Tricomi, Integral Equations, Dover, NewYork, 1985.

[6] T.A.Burton, Volterra Integral and Differential Equations, London, NewYork,

1983.

[7] R.P.Kanwal, Linear Integral Equations Theory and Technique, Boston, 1996.

[8] P.Schiavone, C.Constanda and A.Mioduchowski, Integral Methods in Science

and Engineering, Birkhauser Boston, 2002.

[9] N.I.Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, The

Netherland, 1953.

[10] Peter Linz, Analytic and Numerical Methods for Volterra Equations, SIAM,

Philadelphia, 1985.

[11] K.E.Atkinson, A Survey of Numerical Method for the Solution of Fredholm

Integral Equation of the Second Kind, Philadelphia, 1976.

[12] K.E.Atkinson, The Numerical Solution of Integral Equation of the Second Kind,

Cambridge University, Combridge, 1997.

[13] Christopher T.H.Baker, Treatment of Integral Equations by Numerical Methods ,

Academic Press, 1982.

[14] L.M.Delves and J.L.Mohamed, Computational Methods for Integral Equations,

NewYork, London, 1985.

[15] M.A.Golberg.ed, Numerical Solution for Integral Equations, NewYork, 1990.

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[16] M.A.Abdou, FredholmــVolterra integral equation of the first kind and contact

problem, J. Appl. Math. Comput. 125 (2002) 177193ــ.

[17] M.A.Abdou, FredholmــVolterra integral equation and generalized potential

kernel, J. Appl. Math. Comput. 131 (2002) 8194ــ.

[18] M.A.Abdou, On asymptotic methods for FredholmــVolterra integral equation of

the second kind in contact problem, J. Comp. Appl. Math. 154 (2003) 431446ــ.

[19] M.A.Abdou, FredholmــVolterra integral equation with singular kernel, J. Appl.

Math. Comput. 137 (2003) 231243ــ.

[20] M.A.Abdou, F.A.Salama, VolterraــFredholm integral equation of the first kind

and spectral relationships, Appl. Math. Comput. 153 (2004) 141153ــ.

[21] M.A.Abdou, O.L.Moustafa, FredholmــVolterra integral equation in contact

problem, J. Appl. Math. Comput. 138 (2003) 199215ــ.

[22] M.A.Abdou, A.A.Nasr, On the numerical treatment of the singular integral

eqution of the second kind, J. Appl. Math. Comput. 146 (2003) 373380ــ.

[23] M.A.Abdou, Fredholm integral equation with potential kernel and its structure

resolvent, Appl. Math. Comput . 107 (2000) 169 180 ـ .

[24] I.S.Gradshteyn and I.M.Ryzhik, Table of Integrals, Series and Products,

Academic Press, New York, 1980 .

[25] H.Bateman, A.Erdely, Higher Transcendental Functions, Vol.2, Nauka Moscow

1973 .

[26] S.M.Mkhitarian, M.A.Abdou, On different methods for solving the integral

eqution of the first kind with logarithmic kernel, Dokl. Acad. Nauk. Armenia 90

.10ــ1 (1990)

[27] E.V.Kovolenko, Some approximate methods of solving integral equations of

mixed problem, Appl. Math. Mech. 53 (1) (1989) 8592 ـ .

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[28] M.A.Abdou, N.Y.Ezzeldin, Krein's method with certain singular kernel for

solving the integral equation of the first kind, Period. Math. Hung. 28 (2) (1994)

.149ــ143

[29] M.A.Abdou, K.I.Mohamed and A.S.Ismal, Toeplitz matrix and product Nystrom

methods for solving the singular integral equation, Le Mathematicle, Vol. LVII

.37ــFasc 1.pp. 21ـ(2002)

[30] J.Frankel, A Galerkin solution to a regularized Cauchy singular integroــdiffere-

ntial equation, Quart. Appl. Math. 56 (1996) 409424ــ.

[31] Orsi, A.Palamara, Product integration for Volterra integral equation of the

second kind with weakly singular kernel. Math. Comp. Vol. 65 No.215 (1996)

.1212ــ1201

[32] N.K.Artiunian, Plane contact problems of the theory of creel. Appl. Math.Mech.

.923ــ901 (1959) 23

١٢٦

)مكة المكرمة(معة أم القرى جا

آـلـيـة الــعــــلوم الـتـطـبـيقـيـــة

قـســم الـعــلـوم الـريــاضـيـــــة

ـةــــي مــعــادلـــــــــد فــدالـــة جــهــ

فـردهـولـم ــ فـولـتـيـرا الـتـكـامـلـيــة

ث تكميلي مقدم لنيل درجة الماجستير ــحـب

فـــــــــي

الـتطبيـقـيـــــةاتـياضـريـال

)معادالت تكـامـلـيـــــــــة (

إعداد الباحثـــة

فايـــزة محمـد حـمـدي الـصـاعــــدي

تحت إشـراف

محمـد عبد الـاله أحمــد عبــده/ األســتاذ الـدآتــور

142٧ م٢٠٠٦ هـ ــ

٢

Contents

Introduction ………………………………………………………… i ـ v

Chapter 1 Basic Concepts

§ 1.1 Definitions and theorems . ………………………………………1

§ 1.2 Laplace transformation …………………………………………..12

§ 1.3 Classification of integral equations …………………………….. 16

§ 1.4 Fredholm theorems for continuous kernel ………………………..21

§ 1.5 Integral operator ………………………………………………….24

§ 1.6 Compact operator …………………………………………………28

Chapter 2 Volterra Integral Equation

§ 2.1 Existence and uniqueness solution of Volterra equation…………. 37

§ 2.2 The resolvent kernel method …………………………………….. 44

§ 2.3 Solution method using Laplace transformation …………………. 49

§ 2.4 Method of successive approximation …………………………… 51

§ 2.5 Volterra integral equation of the first kind ……………………… 53

Chapter 3 Fredholm ــ Volterra Integral Equation with Potential Kernel

§ 3.1 Introduction ……………………………………………….. 62 § 3.2 Existence and uniqueness solution of the integral equation……… 67

§ 3.3 Continuity and normality of integral operator …………………... 72

§ 3.4 The kernel of position …………………………………………… 75

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Chapter 4 Series Method

§ 4.1 Separation of variables method …………………………………. 86

§ 4.2 Discussion and special cases …………………………………… 94

Chapter 5 Applications for Potential Kernel

§ 5.1 Electrostatic potential…………………………………………… 99

§ 5.2 Torsion of an isotropic elastic plate …………………………… 103

§ 5.3 Mechanics and mixed problem ………………………………… 107

§ 5.4 Raditions and molecular condition …………………………… 107

§ 5.5 Discussion and results ………………………………………….. 109

Appendix ………………………………………………………………. 114

References................................................................................................ 115

Arabic summary ……………………………………………………. أ ــ ج