Numerical Treatment Of The Fredholm Integral Equations Of .
An-Najah National UniversityFaculty of Graduate StudiesNumerical Treatment of The FredholmIntegral Equations of the Second KindByNjood Asad Abdulrahman RihanSupervised byProf. Naji QatananiThis Thesis is submitted in partial Fulfillment of the Requirementsfor the Degree of master of Science in Computational Mathematics,Faculty of Graduate Studies, An- Najah National University,Nablus, Palestine.2013
iiiDedicationI dedicate this thesis to my parents, my husband Jafar and mydaughter Shayma’a, withouttheir patience, understanding, supportand most of all love, this workwouldnot have been possible.
ivAcknowledgementI am heartily thankful to my supervisor, Prof. Dr. Naji Qatanani,whose encouragement, guidance and support from the initial to thefinal levelenabled me to develop and understanding the subject.My thanks and appreciation goes to my thesis committee membersDr.Yousef Zahaykah and Dr. Subhi Ruzieh for their encouragement,support,interest and valuable hints.I acknowledge An-Najah National University for supporting this work,and I wish to pay my great appreciation to all respected teachersandstaff in department of mathematics.Lastly, I offer my regards and blessings to all of those who supportedme in any respect during the completion of this thesis.
v ﺍﻹﻗﺭﺍﺭ ﺃﻨﺎ ﺍﻝﻤﻭﻗﻌﺔ ﺃﺩﻨﺎﻩ ﻤﻘﺩﻤﺔ ﺍﻝﺭﺴﺎﻝﺔ ﺍﻝﺘﻲ ﺘﺤﻤل ﺍﻝﻌﻨﻭﺍﻥ : Numerical Treatment of The Fredholm Integral Equations of the Second Kind ﺃﻗﺭ ﺒﺄﻥ ﻤﺎ ﺍﺸﺘﻤﻠﺕ ﻋﻠﻴﻪ ﻫﺫﻩ ﺍﻝﺭﺴﺎﻝﺔ ﺇﻨﻤﺎ ﻫﻭ ﻨﺘﺎﺝ ﺠﻬﺩﻱ ﺍﻝﺨﺎﺹ ، ﺒﺎﺴﺘﺜﻨﺎﺀ ﻤﺎ ﺘﻤﺕ ﺍﻹﺸﺎﺭﺓ ﺍﻝﻴﻪ ﺤﻴﺜﻤﺎ ﻭﺭﺩ ، ﻭﺃﻥ ﻫﺫﻩ ﺍﻝﺭﺴﺎﻝﺔ ﻜﻜل ، ﺃﻭ ﺃﻱ ﺠﺯﺀ ﻤﻨﻬﺎ ﻝﻡ ﻴﻘﺩﻡ ﻤﻥ ﻗﺒل ﻝﻨﻴل ﺃﻴﺔ ﺩﺭﺠﺔ ﻋﻠﻤﻴﺔ ﺃﻭ ﺒﺤﺙ ﻋﻠﻤﻲ ﺃﻭ ﺒﺤﺜﻲ ﻝﺩﻯ ﺃﻴﺔ ﻤﺅﺴﺴﺔ ﺘﻌﻠﻴﻤﻴﺔ ﺃﻭ ﺒﺤﺜﻴﺔ ﺃﺨﺭﻯ . Declaration The work provided in this thesis , unless otherwise referenced , is the researcher's own work , and has not been submitted elsewhere for any other degree or qualification. ﺇﺴﻡ ﺍﻝﻁﺎﻝﺒﺔ : ﺍﻝﺘﻭﻗﻴﻊ : ﺍﻝﺘﺎﺭﻴﺦ : Student's name Signature Date
viTable of onTable of ContentsList of FiguresList of TablesAbstractIntroductionChapter 1Mathematical Preliminaries1.1 Classification of integral equation1.1.1 Types of integral equations1.1.2. Linearity of integral equations1.1.3 Homogeneity of integral equations1.2 Kinds of kernels1.3 Review of spaces and operatorsChapter 2Analytical methods for solving Fredholm integral equationsof the second kind2.1 The existence and uniqueness2.2 Some analytical methods for solving Fredholm integralequations of the second kind2.2.1 The degenerate kernel methods2.2.2 Converting Fredholm integral equation to ODE2.2.3 The Adomain decomposition method2.2.4 The modified decomposition method2.2.5 The method of successive approximationsChapter 3Numerical methods for solving Fredholm integralequations of the second kind3.1 Degenerate kernel approximation methods3.1.1 The solution of the integral equation by the degeneratekernel method3.1.2 Taylor series approximation3.2 Projection methods3.2.1 Theoretical framework3.2.1.1 Lagrange polynomial interpolation3.2.1.2 Projection operators3.2.2 Collocation 3945495461626263676970707377
vii3.2.2.1 Piecewise linear interpolation3.2.3 Galerkin methods3.2.3.1 Bernstein polynomials3.2.3.2 Formulation of integral equation in matrix form3.2.4 The convergence of the projection methods3.3 Nyström methodChapter 4Numerical examples and results4.1 The numerical realization of equation (4.1) using thedegenerate kernel method4.2 The numerical realization of equation (4.1) using thecollocation method4.3 The numerical realization of equation (4.1) using theNyström method4.4 The error analysis of the Nyström methodConclusionReferencesAppendix?@ABC ا 818283848691959696103111117119120128 ب
viiiList of FiguresFigureTitlePage4.1The exact and numerical solution of applyingAlgorithm 1 for equation (4.1).1024.2The resulting error of applying algorithm 1 toequation (4.1)The exact and numerical solution of applyingAlgorithm 2 for equation (4.1).103The resulting error of applying algorithm 2 toequation (4.1)The exact and numerical solution of applyingAlgorithm 3 for equation (4.1).111The resulting error of applying algorithm 3 toequation (4.1)1164.34.44.54.6110116
ixList of TablesTableTitlePage4.1The exact and numerical solution of applyingAlgorithm 1 for equation (4.1) and the error.The exact and numerical solution of applyingAlgorithm 2 for equation (4.1) and the error.The exact and numerical solution of applyingAlgorithm 3 for equation (4.1) and the error.1024.24.3109115
xNumerical Treatment of The Fredholm Integral Equations of theSecond KindByNujood Asad Abdulrahman RihanSupervisorProf. Naji QatananiAbstractIn this thesis we focus on the mathematical and numerical aspects ofthe Fredholm integral equation of the second kinddue to their wide range ofphysical application such as heat conducting radiation, elasticity, potentialtheory and electrostatics. After the classification of these integral equationswe will investigate some analytical and numerical methods for solving theFredholm integral equation of the second kind. Such analytical methodsinclude: the degenerate kernel methods, converting Fredholm integralequation to ODE, the Adomain decomposition method, the modifieddecomposition method andthe method of successive approximations.The numerical methods that will be presented here are: Projection methodsincluding collocation method and Galerkin method, Degenerate kernelapproximation methods and Nyström methods.The mathematical framework of these numerical methods together withtheir convergence properties will be analyzed.Some numerical examples implementing these numerical methods havebeen obtained for solving a Fredholm integral equation of the second kind.The numerical results show a closed agreement with the exact solution.
1IntroductionThe subject of integral equations is one of the most importantmathematical tools in both pure and applied mathematics. Integralequations play a very important role in modern science such as numerousproblems in engineering and mechanics, for more details see [4] and [25].In fact, many physical problems are modeled in the form of Fredholmintegral equations, such problems as potential theory and Dirichletproblems which discussed in [4] and [37], electrostatics [34], mathematicalproblems of radiative equilibrium [23], the particle transport problems ofastrophysics and reactor theory [29], and radiative heat transfer problemswhich discussed in [40], [41], [42], and [49].Many initial and boundary value problems associated with ordinarydifferential equations (ODEs) and partial differential equations (PDEs) canbe solved more effectively by integral equations methods. Integralequations also form one of the most useful tools in many branches of pureanalysis, such as the theories of functional analysis and stochasticprocesses, see [27] and [32].Historical background of the integral equationAn integral equation is an equation in which an unknownfunction appears under one or more integral signs.
There is a close connection between differential and integralequations and some problems may be formulated either way. Themost basic type of integral equation is a Fredholm equation of thesecond kind λ λ where is a closed bounded set in , for some .G is a function called the kernel of the integral equation and isassumed to be absolutely integrable, and satisfy other propertiesthat are sufficient for the Fredholm Alternative Theorem, for moredetails see [4]. For , we have λ which is a non zero real orcomplex parameter and given, and we seek , this is thenonhomogeneos problem. For , equation (1) becomes aneigenvalue problem, and we seek both the eigenvalue λ and theeigenfunction .The integral equation (1) can be written abstractly as λ with is an integral operator on a Banach space to the sameBanach space X, e.g. or! "At the time in the early 1960’s, researchers were interested principally inone-dimensional case. It was for a kernel function that was at leastcontinuous; and then it was assumed that was several times
3continuously differentiable. This was the type of equation studied by IvarFredholm, and in his honor such equation is called Fredholm integralequation of the second kind. Today the work is with multi-dimensionalFredholm integral equations of the second kind in which the integraloperator is completely continuous and the integration region is commonly asurface in # , in addition, the kernel function is often singular.The Fredholm theory is still valid for such equations, and this theory iscritical for the convergence and stability analysis of associated numericalmethods. For more details see [4] and [14].There are many analytical methods which are developed forsolving Fredholm integral equations such methods as the degeneratekernel methods, converting Fredholm integral equation to ODE, tionmethod, the method of successive approximations and others. Formore details see [1], [14], [28], [30], [44] and [50].The numerical methods for solving Fredholm integral equationsmay be subdivided into the following classes: Degenerate kernelapproximation methods, Projection methods, Nyström methods. Formore details see [2], [5], [11], [13], [21], [36], [38] and [53]. All ofthese methods have iterative variants. There are other numericalmethods, but the above methods and their variants include the mostpopular general methods.
There are only a few books on the numerical solutions of integralequations as compared to the much larger number that have rtialdifferential equations. General books on the numerical solution ofintegral equations include, in historical order, [10], and [16], and[19]. More specialized treatments of numerical methods for integralequations are given in [4], [7], [31] and [33].
5Chapter 1Mathematical Preliminaries
6Chapter 1Mathematical PreliminariesDefinition 1.1An integral equation is an equation in which the unknown function appears under the integral sign. A standard integral equation is of the form & ' % ( ' " where ) and * are limits of integration, is a constant parameter,and is a function of two variables x and y called the kernel or thenucleus of the integral equation. The function that will be determinedappears under the integral sign, and sometimes outside the integral sign.The functions and are given. The limits of integration ) and * may be both variables, constants, or mixed, and they may be inone dimension or two or more.1.1 Classification of integral equations1.1.1 Types of integral equationsThere are four major types of integral equations, the first two are of mainclasses and the other two are related types of integral equations.
1. Fredholm integral equationsThe most standard form of Fredholm integral equations is given by theform , % with a closed bounded set in , for some .(i) If the function , , then " becomes simply % - " ". and this equation is called Fredholm integral equation of the second kind.(ii) If the function , , then " yields % "/ which is called Fredholm integral equation of the first kind.(iii) If h(x) is neither 0 nor 1 then (1.2) called Fredholm integral equationof the third kind2. Volterra integral equationsThe most standard form of Volterra integral equation is of the form
, ' % 1 "0 where the upper limit of integration is a variable and the unknown function appears linearly or nonlinearly under the integral sign.(i)If the function , , then equation "0 simply becomes' % 1 "2 and this equation is known as the Volterra integral equation of the secondkind.(ii)If the function , ' then equation "0 becomes % 1 "3 which is known as the Volterra integral equation of the first kind.(iii) If , is neither 0 nor 1 then (1.5) called Volterra integral equation ofthe third kind.3. Singular integral equationsWhen one or both limits of integration become infinite or when the kernelbecomes infinite at one or more points within the range of integration, theintegral equation is called singular. For example, the integral equation % 4 5 6 6 8 is a singular integral equation of the second kind. "7
9(i) Weakly singular integral equation: The kernel is of the form or where 9 isbounded 9 6 6: 9 ; 6 6 (thatis,severaltimescontinuouslydifferentiable) with 9 and α isa constant such that ? . For example, the equation 'A : ? "@ is a singular integral equation with a weakly singular kernel. For moredetails see [9] and [17].(ii) Singular integral equation: Here the kernel is of the form 9 where 9 is a differentiable function of with 9 , thenwhere the integral -E F GF is understood in the sense of Cauchythe integral equation is said to be a singular equation with Cauchy kernelD B ' C ' 8 CPrincipal Value (CPV) and the notation P.V. -Edenote this. ThusD B ' C ' 8 C GF is usually used to
9 H" I" E J D '8NKLM P NOA101Q9 9 GF GFS J 'RN J (iii) Strongly singular integral equations: if the kernel G(x, y) is of theform 9 !where 9 is a differentiable function of (x, y) with 9 , thenthe integral equation is said to be a strongly singular integral equation. Formore details see [22].4. Integro-differential equationsIn this type of equations, the unknown function appears as acombination of both ordinary derivative and under the integral sign. In theelectrical engineering problem, the current I (t) flowing in a closed circuitcontaining resistance, inductance and capacitance is governed by thefollowing integro-differential equation, V U UT AW T " where L is the inductance, R the resistance, C the capacitance, and W T theapplied voltage. Similar examples can be cited as follows X Z .A Y "
[ X '11 % A , (1.12)Equations (1.10) and (1.12) are of Volterra type integro-differentialequations, whereas equation (1.11) is Fredholm type integro-differentialequations.1.1.2. Linearity of integral equationsThere are two kinds of integral equations according to linearity and thisdepends on the unknown function under the integral sign.(i) Linear integral equationsThey are of the form & ' % ( ' " . where only linear operations are performed upon the unknown functioninside the integral sign, that is the exponent of the unknown inside theintegral sign is one, for example Z . . A" " / here the unknown function f appears in the linear form.(ii)Nonlinear integral equationsThey are of the form
& ' % 1( ' " 0 the unknown function f under the integral sign has exponent other thanone, or the equation contains nonlinear functions of , such as 4 \ , ] , ; , for example ' A " 2 1.1.3 Homogeneity of integral equationsIntegral equations of the second kind are classified as homogeneous ornon-homogeneous.(i)Homogeneous integral equationif the function g in the second kind of Volterra or Fredholm integralequations is identically zero, the equation is called homogeneous, forexample, " 3 and this kind of equations becomes an eigenvalue problem, and we seekboth the eigenvalue λ and the eigenfunction f, where by an eigenvalue (orcharacteristic value )we mean that the value of the constant λ, for whichthe homogeneous Fredholm equation has a solution identically zero on the non-zero solution eigenfunction, or characteristic function. which is not is called an
13(ii) Non-homogeneous integral equationif the function g in the second kind of Volterra or Fredholm integralequation is not equal zero, the equation is called non-homogeneous, forexample, Z Awhere is not equal zero. " 7 Notice that this property of classification holds for equations of thesecond kind only since . For more details see [4] and [50].1.2 Kinds of kernels1. Separable or degenerate kernelA kernel is called separable or degenerate if it can be expressedas the sum of a finite number of terms, each of which is the product of afunction of x only and a function of y only, (some authors say isdegenerate if it is of finite rank) that means, a F d b )c *c " " @ ceZThe functions )c and the functions *c are linearly independent.2. Symmetric (or Hermitian) kernelA complex-valued function is called symmetric if
1 f " where the asterisk denotes the complex conjugate. For a real kernel, thiscoincides with definition " " 3. Hilbert-Schmidt kernelIf the kernel G(x, y), for each sets of values of x, y in the square is such thatQQ 6 6! 11also for each value of x in isQ 6 6!1And for each value of y in Q is g g 6 6! g " 1has a finite value, then we call the kernel a regular kernel and thecorresponding integral equation is called a regular integral equation.4. Cauchy kernelIf the kernel is of the form 9 " . J
15where 9 is a differentiable function of (x, y) with 9 thenthe integral equation is said to be a singular equation with Cauchy kernel.5. Abel's kernelsIf the kernel is of the form 9 " / 6 6:where ? and the function 9 is assumed to be several timescontinuously differentiable such integral equations contain this kernel arecalled Abel integral equation.6. Hilbert kernelThe kernel is of the form where and hiT j k " 0 are real variables, is called the Hilbert kernel and is closelyconnected with the Cauchy kernel,
The integral equation (1) can be written abstractly as λ ˇ with is an integral operator on a Banach space ˆ to the same Banach space X, e.g. or ! " At the time in the early 1960’s, researchers were interested principally in one-dimensi
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