Integral Equations and Their Applications

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