Fourier analysis and partial differential equations (Q2703811)

The book containing 8 chapters is an advanced introduction to the basic concepts of Fourier analysis and covers its recent applications to the solutions of some important nonlinear evolution equations.NEWLINENEWLINENEWLINEThe first and preliminary chapter presents the classical statements of boundary value problems for basic partial differential equations of mathematical physics together with some approaches to their solving such as the Fourier method of separation of variables and maximum principles. The two next chapters develop the elementary classical theory of Fourier series for piecewise continuous functions and introduce a class of generalized functions in particular, periodic distributions as a special type of those.NEWLINENEWLINENEWLINEThe conventional applications considered in Chapter 4 are concerned with linear evolution equations such as the heat, Schrödinger, and wave equations. Chapters 5 and 6 are devoted to studying the Cauchy problem for nonlinear evolution equations in the periodic setting. Mainly, here is analyzed the well-posedness of the generalized nonlinear Schrödinger equation and the Korteweg-de Vries equation. The two last chapters present the basic concepts of the theory of distributions and weighted Sobolev spaces with applications to some nonperiodic problems.