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