Fourier Analysis And Allied Methods In Problems Of Scattering And Radiation Of Water Waves

Abstract

As has been discussed in the previous section, the growth of various methods (analytical methods like Green's function technique, application of Green's integral theorem, application of Fourier analysis—both Fourier transform as well as Fourier series, methods based on complex function theory such as Wiener-Hopf technique and Riemann-Hilbert problem, numerical methods like variational principle etc.) have taken place to solve this class of boundary value problems. Keeping in mind the limitation of these methods, in this thesis we have avoided the complexity appearing in certain methods, by reinvestigating some of the well-known problems by new methods, applied some of the mathematical generalisations to solve a more general class of problems and simplified some of the analysis in the existing methods. The thesis is based on the application of non-complex variable methods, except in Chapter 10 where we have used a direct complex variable method namely the Riemann-Hilbert problem to solve certain general integro-differential equations. Mainly, the work is based on suitable applications of the Fourier analysis, Green's integral theorem, the Green's function technique and the singular integral equations.