Integral Equations on Time Scales PDF
Preview Integral Equations on Time Scales
Atlantis Studies in Dynamical Systems Series Editors: Henk Broer · Boris Hasselblatt Svetlin G.Georgiev Integral Equations on Time Scales Atlantis Studies in Dynamical Systems Volume 5 Series editors Henk Broer, Groningen, The Netherlands Boris Hasselblatt, Medford, USA The“Atlantis StudiesinDynamical Systems”publishes monographs inthearea of dynamical systems, written by leading experts in the field and useful for both students and researchers. Books with a theoretical nature will be published alongside books emphasizing applications. More information about this series at http://www.atlantis-press.com Svetlin G. Georgiev Integral Equations on Time Scales Svetlin G.Georgiev Department ofDifferential Equations SofiaUniversity Sofia Bulgaria Atlantis Studies inDynamical Systems ISBN978-94-6239-227-4 ISBN978-94-6239-228-1 (eBook) DOI 10.2991/978-94-6239-228-1 LibraryofCongressControlNumber:2016951703 ©AtlantisPressandtheauthor(s)2016 Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorbyany means, electronic or mechanical, including photocopying, recording or any information storage and retrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher. Printedonacid-freepaper Preface Many problems arising in applied mathematics or mathematical physics, can be formulated in two ways namely as differential equations and as integral equations. In the differential equation approach, the boundary conditions have to be imposed externally, whereas in the case of integral equations, the boundary conditions are incorporated within the formulation, and this confers a valuable advantage to the latter method. Moreover, the integral equation approach leads quite naturally to thesolutionoftheproblemasaninfiniteseries,knownastheNeumannexpansion, the Adomian decomposition method, and the series solution method in which the successive terms arise from the application of an iterative procedure. The proof of the convergence of this series under appropriate conditions presents an inter- esting exercise in an elementary analysis. This book encompasses recent developments of integral equations on time scales. For many population models biological reasons suggest using their differ- ence analogues. For instance, North American big game populations have discrete birth pulses, not continuous births as is assumed by differential equations. Mathematical reasons also suggest using difference equations—they are easier to construct and solve in a computer spreadsheet. North American large mammal populations do not have continuous population growth, but rather discrete birth pulses, so the differential equation form of the logistic equation will not be con- venient. Age-structured models add complexity to a population model, but make themodelmorerealistic,inthatessentialfeaturesofthepopulationgrowthprocess are captured by the model. They are used difference equations to define the pop- ulation model because discrete age classes require difference equations for simple solutions. The discrete models can be investigated using integral equations in the case when the time scale is the set of the natural numbers. A powerful method introducedbyPoincaréforexaminingthemotionofdynamicalsystemsisthatofa Poincaré section. This method can be investigated using integral equations on the set of the natural numbers. The total charge on the capacitor can be investigated with an integral equation on the set of the harmonic numbers. This book contains elegant analytical and numerical methods. This book is intendedfortheuseinthefieldofintegralequationsanddynamiccalculusontime v vi Preface scales.Itisalsosuitableforgraduatecoursesintheabovefields.Thisbookcontains nine chapters.The chaptersin this book are pedagogically organized. Thisbook is specially designed for those who wish to understand integral equations on time scales without having extensive mathematical background. ThebasicdefinitionsofforwardandbackwardjumpoperatorsareduetoHilger. In Chap. 1 are given examples of jump operators on some time scales. The graininess function, which is the distance from a point to the closed point on the right,isintroducedinthischapter.Inthischapter,thedefinitionsfordeltaderivative and delta integral are given and some of their properties are deducted. The basic resultsinthischaptercanbefoundin[2].Chapter2introducestheclassificationof integral equations on time scales and necessary techniques to convert dynamic equationstointegralequationsontimescales.Chapter3dealswiththegeneralized Volterra integral equations and the relevant solution techniques. Chapter 4 is concerned with the generalized Volterra integro-differential equations and also solution techniques. Generalized Fredholm integral equations are investigated in Chap. 5. Chapter 6 is devoted on Hilbert–Schmidt theory of generalized integral equations with symmetric kernels. The Laplace transform method is introduced in Chap. 7. Chapter 8 deals with the series solution method. Nonlinear integral equations on time scales are introduced in Chap. 9. Theaim ofthisbookwastopresentaclearandwell-organizedtreatmentofthe concept behind the development of mathematics and solution techniques. The text material of this book is presented in highly readable, mathematically solid format. Many practical problems are illustrated displaying a wide variety of solution techniques. Nonlinear integral equations on time scales and some of their appli- cations in the theory of population models, biology, chemistry, and electrical engineeringwillbediscussedinaforthcomingbook“NonlinearIntegralEquations on Time Scales and Applications.” The author welcomes any suggestions for the improvement of the text. Paris, France Svetlin G. Georgiev June 2016 Contents 1 Elements of the Time Scale Calculus. .... .... .... .... ..... .... 1 1.1 Forward and Backward Jump Operators, Graininess Function .... 1 1.2 Differentiation... .... ..... .... .... .... .... .... ..... .... 8 1.3 Mean Value Theorems..... .... .... .... .... .... ..... .... 19 1.4 Integration . .... .... ..... .... .... .... .... .... ..... .... 22 1.5 The Exponential Function... .... .... .... .... .... ..... .... 39 1.5.1 Hilger’s Complex Plane .. .... .... .... .... ..... .... 39 1.5.2 Definition and Properties of the Exponential Function .... 49 1.5.3 Examples for Exponential Functions. .... .... ..... .... 63 1.6 Hyperbolic and Trigonometric Functions ... .... .... ..... .... 65 1.7 Dynamic Equations... ..... .... .... .... .... .... ..... .... 67 1.8 Advanced Practical Exercises.... .... .... .... .... ..... .... 74 2 Introductory Concepts of Integral Equations on Time Scales .. .... 77 2.1 Reducing Double Integrals to Single Integrals ... .... ..... .... 80 2.2 Converting IVP to Generalized Volterra Integral Equations.. .... 82 2.3 Converting Generalized Volterra Integral Equations to IVP.. .... 88 2.4 Converting BVP to Generalized Fredholm Integral Equation. .... 95 2.5 Converting Generalized Fredholm Integral Equation to BVP. .... 105 2.6 Solutions of Generalized Integral Equations and Generalized Integro-Differential Equations.... .... .... .... .... ..... .... 116 2.7 Advanced Practical Exercises.... .... .... .... .... ..... .... 121 3 Generalized Volterra Integral Equations.. .... .... .... ..... .... 131 3.1 Generalized Volterra Integral Equations of the Second Kind . .... 131 3.1.1 The Adomian Decomposition Method.... .... ..... .... 131 3.1.2 The Modified Decomposition Method.... .... ..... .... 138 3.1.3 The Noise Terms Phenomenon. .... .... .... ..... .... 144 3.1.4 Differential Equations Method.. .... .... .... ..... .... 146 3.1.5 The Successive Approximations Method.. .... ..... .... 158 vii viii Contents 3.2 Conversion of a Generalized Volterra Integral Equation of the First Kind to a Generalized Volterra Integral Equation of the Second Kind.... .... .... .... .... ..... .... 163 3.3 Existence and Uniqueness of Solutions .... .... .... ..... .... 166 3.3.1 Preliminary Results.. .... .... .... .... .... ..... .... 166 3.3.2 Existence of Solutions of Generalized Volterra Integral Equations of the Second Kind.. .... .... .... ..... .... 172 3.3.3 Uniqueness of Solutions of Generalized Volterra Integral Equations of Second Kind. .... .... .... .... ..... .... 175 3.3.4 Existence and Uniqueness of Solutions of Generalized Volterra Integral Equations of the First Kind .. ..... .... 177 3.4 Resolvent Kernels.... ..... .... .... .... .... .... ..... .... 178 3.5 Application to Linear Dynamic Equations .. .... .... ..... .... 184 3.6 Advanced Practical Exercises.... .... .... .... .... ..... .... 192 4 Generalized Volterra Integro-Differential Equations .... ..... .... 197 4.1 Generalized Volterra Integro-Differential Equations of the Second Kind... ..... .... .... .... .... .... ..... .... 197 4.1.1 The Adomian Decomposition Method.... .... ..... .... 197 4.1.2 Converting Generalized Volterra Integro-Differential Equations of the Second Kind to Initial Value Problems..... .... .... .... .... .... ..... .... 206 4.1.3 Converting Generalized Volterra Integro-Differential Equations of the Second Kind to Generalized Volterra Integral Equations.... .... .... .... ..... .... 212 4.2 Generalized Volterra Integro-Differential Equations of the First Kind. .... ..... .... .... .... .... .... ..... .... 219 4.3 Advanced Practical Exercises.... .... .... .... .... ..... .... 223 5 Generalized Fredholm Integral Equations. .... .... .... ..... .... 227 5.1 Generalized Fredholm Integral Equations of the Second Kind .... 227 5.1.1 The Adomian Decomposition Method.... .... ..... .... 227 5.1.2 The Modified Decomposition Method.... .... ..... .... 232 5.1.3 The Noise Terms Phenomenon. .... .... .... ..... .... 237 5.1.4 The Direct Computation Method.... .... .... ..... .... 241 5.1.5 The Successive Approximations Method.. .... ..... .... 246 5.2 Homogeneous Generalized Fredholm Integral Equations of the Second Kind... ..... .... .... .... .... .... ..... .... 251 5.3 Fredholm Alternative Theorem... .... .... .... .... ..... .... 258 Z Z b b 5.3.1 The Case When jKðx;YÞj2DXDY\1 .. ..... .... 258 a a 5.3.2 The General Case ... .... .... .... .... .... ..... .... 264 5.3.3 Fredholm’s Alternative Theorem.... .... .... ..... .... 274 5.4 The Schmidth Expansion Theorem and the Mercer Expansion Theorem .. ..... .... .... .... .... .... ..... .... 275 Contents ix 5.4.1 Operator-Theoretical Notations. .... .... .... ..... .... 275 5.4.2 The Schmidt Expansion Theorem... .... .... ..... .... 282 5.4.3 Application to Generalized Fredholm Integral Equation of the First Kind.... .... .... .... .... .... ..... .... 288 5.4.4 Positive Definite Kernels. Mercer’s Expansion Theorem . .... ..... .... .... .... .... .... ..... .... 289 5.5 Advanced Practical Exercises.... .... .... .... .... ..... .... 298 6 Hilbert-Schmidt Theory of Generalized Integral Equations with Symmetric Kernels .. ..... .... .... .... .... .... ..... .... 301 6.1 Schmidt’s Orthogonalization Process .. .... .... .... ..... .... 301 6.2 Approximations of Eigenvalues .. .... .... .... .... ..... .... 307 6.3 Inhomogeneous Generalized Integral Equations .. .... ..... .... 314 7 The Laplace Transform Method .... .... .... .... .... ..... .... 321 7.1 The Laplace Transform..... .... .... .... .... .... ..... .... 321 7.1.1 Definition and Examples.. .... .... .... .... ..... .... 321 7.1.2 Properties of the Laplace Transform. .... .... ..... .... 325 7.1.3 Convolution and Shifting Properties of Special Functions . .... .... .... .... .... ..... .... 333 7.2 Applications to Dynamic Equations ... .... .... .... ..... .... 347 7.3 Generalized Volterra Integral Equations of the Second Kind . .... 352 7.4 Generalized Volterra Integral Equations of the First Kind ... .... 357 7.5 Generalized Volterra Integro-Differential Equations of the Second Kind.... .... ..... .... .... .... .... .... ..... .... 360 7.6 Generalized Volterra Integro-Differential Equations of the First Kind. .... ..... .... .... .... .... .... ..... .... 366 7.7 Advanced Practical Exercises.... .... .... .... .... ..... .... 370 8 The Series Solution Method .... .... .... .... .... .... ..... .... 375 8.1 Generalized Volterra Integral Equations of the Second Kind . .... 375 8.2 Generalized Volterra Integral Equations of the First Kind ... .... 384 8.3 Generalized Volterra Integro-Differential Equations of the Second Kind.... .... ..... .... .... .... .... .... ..... .... 387 9 Non-linear Generalized Integral Equations.... .... .... ..... .... 395 9.1 Non-linear Generalized Volterra Integral Equations ... ..... .... 395 9.2 Non-linear Generalized Fredholm Integral Equations .. ..... .... 396 References.... .... .... .... ..... .... .... .... .... .... ..... .... 399 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 401
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