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. 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.

Table of Contents:
1 IntroductionPreliminary concept of the integral equation; Historical background of the integral equation; An illustration from mechanics; Classification of integral equations; Converting Volterra equation to ODE; Converting IVP to Volterra equations; Converting BVP to Fredholm integral equations; Types of solution techniques; Exercises; References2 Volterra integral equationsIntroduction; The method of successive approximations; The method of Laplace transform; The method of successive substitutions; The Adomian decomposition method; The series solution method; Volterra equation of the first kind; Integral equations of the Faltung type; Volterra integral equation and linear differential equations; Exercises; References3 Fredholm integral equationsIntroduction; Various types of Fredholm integral equations; The method of successive approximations: Neumann's series; The method of successive substitutions; The Adomian decomposition method; The direct computational method; Homogeneous Fredholm equations; Exercises; References4 Nonlinear integral equationsIntroduction; The method of successive approximations; Picard's method of successive approximations; Existence theorem of Picard's method; The Adomian decomposition method; Exercises; References5 The singular integral equationIntroduction; Abel's problem; The generalized Abel's integral equation of the first kind; Abel's problem of the second kind integral equation; The weakly-singular Volterra equation; Equations with Cauchy's principal value of an integral and Hilbert's transformation; Use of Hilbert transforms in signal processing; The Fourier transform; The Hilbert transform via Fourier transform; The Hilbert transform via the eth/2 phase shift; Properties of the Hilbert transform; Analytic signal in time domain; Hermitian polynomials; The finite Hilbert transform; Sturm-Liouville problems; Principles of variations; Hamilton's principles; Hamilton's equations; Some practical problems; Exercises; References6 Integro-differential equationsIntroduction; Volterra integro-differential equations; Fredholm integro-differential equations; The Laplace transform method; Exercises; References7 Symmetric kernels and orthogonal systems of functionsDevelopment of Green's function in one-dimension; Green's function using the variation of parameters; Green's function in two-dimensions; Green's function in three-dimensions; Numerical formulation; Remarks on symmetric kernel and a process of orthogonalization; Process of orthogonalization; The problem of vibrating string: wave equation; Vibrations of a heavy hanging cable; The motion of a rotating cable; Exercises; References8 ApplicationsIntroduction; Ocean waves; Nonlinear wave-wave interactions; Picard's method of successive approximations; Adomian decomposition method; Fourth-order Runge-Kutta method; Results and discussion; Green's function method for waves; Seismic response of dams; Transverse oscillations of a bar; Flow of heat in a metal bar; Exercises; References