Integral equations and their applications

For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of integral equations governing the physical situation of the problems. It also contains elegant analytical and numerical methods, and an important topic of the variational principles. Primarily intended for senior undergraduate students and first year postgraduate students of engineering and science courses, students of mathematical and physical sciences will also find many sections of direct relevance. The book contains eight chapters, pedagogically organized. It is specially designed for those who wish to understand integral equations without having extensive mathematical background. Some knowledge of integral calculus, ordinary differential equations, partial differential equations, Laplace transforms, Fourier transforms, Hilbert transforms, analytic functions of complex variables and contour integrations are expected on the part of the reader. Cover......Page 1 Integral Equations and their Applications......Page 2 Copyright Page......Page 5 Contents......Page 6 Preface......Page 10 Acknowledgements......Page 14 1.1 Preliminary concept of the integral equation......Page 16 1.2 Historical background of the integral equation......Page 17 1.3 An illustration from mechanics......Page 19 1.4 Classification of integral equations......Page 20 1.5 Converting Volterra equation to ODE......Page 22 1.6 Converting IVP to Volterra equations......Page 23 1.7 Converting BVP to Fredholm integral equations......Page 24 1.8 Types of solution techniques......Page 28 1.9 Exercises......Page 29 References......Page 30 2.2 The method of successive approximations......Page 32 2.3 The method of Laplace transform......Page 36 2.4 The method of successive substitutions......Page 40 2.5 The Adomian decomposition method......Page 43 2.6 The series solution method......Page 46 2.7 Volterra equation of the first kind......Page 48 2.8 Integral equations of the Faltung type......Page 51 2.9 Volterra integral equation and linear differential equations......Page 55 2.10 Exercises......Page 58 References......Page 60 3.1 Introduction......Page 62 3.2 Various types of Fredholm integral equations......Page 63 3.3 The method of successive approximations: Neumann's series......Page 64 3.4 The method of successive substitutions......Page 68 3.5 The Adomian decomposition method......Page 70 3.6 The direct computational method......Page 73 3.7 Homogeneous Fredholm equations......Page 74 3.8 Exercises......Page 77 References......Page 78 4.1 Introduction......Page 80 4.2 The method of successive approximations......Page 81 4.3 Picard's method of successive approximations......Page 82 4.4 Existence theorem of Picard's method......Page 85 4.5 The Adomian decomposition method......Page 88 4.6 Exercises......Page 109 References......Page 111 5.1 Introduction......Page 112 5.2 Abel's problem......Page 113 5.3 The generalized Abel's integral equation of the first kind......Page 114 5.4 Abel's problem of the second kind integral equation......Page 115 5.5 The weakly-singular Volterra equation......Page 116 5.6 Equations with Cauchy's principal value of an integral and Hilbert's transformation......Page 119 5.7 Use of Hilbert transforms in signal processing......Page 129 5.8 The Fourier transform......Page 131 5.9 The Hilbert transform via Fourier transform......Page 133 5.10 The Hilbert transform via the ±π/2 phase shift......Page 134 5.11 Properties of the Hilbert transform......Page 136 5.13 Hermitian polynomials......Page 140 5.14 The finite Hilbert transform......Page 144 5.15 Sturm–Liouville problems......Page 149 5.16 Principles of variations......Page 157 5.17 Hamilton's principles......Page 161 5.18 Hamilton's equations......Page 166 5.19 Some practical problems......Page 171 5.20 Exercises......Page 176 References......Page 179 6.1 Introduction......Page 180 6.2 Volterra integro-differential equations......Page 181 6.3 Fredholm integro-differential equations......Page 192 6.4 The Laplace transform method......Page 199 References......Page 202 7.1 Development of Green’s function in one-dimension......Page 204 7.2 Green's function using the variation of parameters......Page 215 7.3 Green's function in two-dimensions......Page 222 7.4 Green's function in three-dimensions......Page 238 7.5 Numerical formulation......Page 259 7.6 Remarks on symmetric kernel and a process of orthogonalization......Page 264 7.7 Process of orthogonalization......Page 266 7.8 The problem of vibrating string: wave equation......Page 269 7.9 Vibrations of a heavy hanging cable......Page 271 7.10 The motion of a rotating cable......Page 276 7.11 Exercises......Page 279 References......Page 281 8.2 Ocean waves......Page 284 8.3 Nonlinear wave–wave interactions......Page 288 8.4 Picard's method of successive approximations......Page 289 8.5 Adomian decomposition method......Page 293 8.6 Fourth-order Runge–Kutta method......Page 297 8.7 Results and discussion......Page 299 8.8 Green's function method for waves......Page 303 8.9 Seismic response of dams......Page 314 8.10 Transverse oscillations of a bar......Page 321 8.11 Flow of heat in a metal bar......Page 324 8.12 Exercises......Page 330 References......Page 332 Appendix A: Miscellaneous results......Page 334 Appendix B: Table of Laplace transforms......Page 342 Appendix C: Specialized Laplace inverses......Page 356 Answers to some selected exercises......Page 360 G......Page 370 W......Page 371 For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of integral equations governing the physical situation of the problems. It also contains elegant analytical and numerical methods, and an important topic of the variational principles. Primarily intended for senior undergraduate students and first year postgraduate students of engineering and science courses, students of mathematical and physical sciences will also find many sections of direct relevance. The book contains eight chapters, pedagogically organized. This book is specially designed for those who wish to understand integral equations without having extensive mathematical background. Some knowledge of integral calculus, ordinary differential equations, partial differential equations, Laplace transforms, Fourier transforms, Hilbert transforms, analytic functions of complex variables and contour integrations are expected on the part of the reader. "While scientists and engineers can already choose from a number of books on integral equations, this book encompasses recent developments including some preliminary backgrounds of formulations of integral equations governing the physical situation of the problems. It also contains elegant analytical and numerical methods, and an important topic of the variational principles. Primarily intended for senior undergraduate students and first year postgraduate students of engineering and science courses, students of mathematical and physical sciences will also find many sections of divert relevance." "The book contains eight chapters, pedagogically organized. This book is specially designed for those who wish to understand integral equations without having extensive mathematical background. Some knowledge of integral calculus, ordinary differential equations, partial differential equations, Laplace transforms, Fourier transforms, Hilbert transforms, analytic functions of complex variables and contour integrations are expected on the part of the reader."--Jacket