١
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.
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1971.
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[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
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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.
١١٩
[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.
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and spectral relationships, Appl. Math. Comput. 153 (2004) 141153ــ.
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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.
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second kind with weakly singular kernel. Math. Comp. Vol. 65 No.215 (1996)
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[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
٣
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
