Methods of applied mathematics with a MATLAB overview (Q1422042)

The aim of this book is to provide an introduction to a number of methods of applied mathematics, especially those arising from Fourier analysis. Classical problems of mathematical physics nature are discussed throughout the book. Those problems represent the pretext or the basis for the elaboration of mathematical models. The book presents numerical schemes and analytical results as well. The structure of the book consists of nine chapters. The first chapter presents an overview of the topics that are about to be discussed in the book. The second chapter consists of an introduction to the theory and elementary applications of Fourier series. The applications include periodic solutions of ordinary differential equations, impedance methods for electric circuits and a discussion of the ``power spectrum'' notion derived from Parseval's theorem. In the third chapter the author discusses elementary boundary value problems as applications of the Fourier series results from chapter two. The fourth chapter is devoted to the study of higher dimensional, non-rectangular boundary value problems, emphasizing on the Sturm-Liouville expansions encountered in such models. The series solution and Bessel equations are also treated in this chapter. An introduction to functions of a complex variable is the central matter discussed in chapter five. Other topics treated in this chapter, within the frame of the complex functions are: the Cauchy integral, the residue theorem and some conformal mappings. Applications from fluid mechanics are also presented. The sixth chapter introduces the Laplace transform methods, with applications in the area of ordinary differential equations and transient circuit analysis. Chapter seven is devoted to the continuous time Fourier transforms. Topics like Fourier inversion and the special case of the inversion integral for Laplace transform, and a lot of applications to various fields are also treated. Discrete variable transforms are the subject of the eighth chapter. The finite discrete Fourier transform and the associated Fast Fourier Transform algorithm is discussed as well. The final chapter provides an introduction to some transform methods of a more specialized nature. Between others, topics like two-sided and Walsh transforms and integral transforms are also considered. Finally, I conclude with the idea that this book is an excellent material for students of applied mathematics and engineers, covering the theory of the Fourier analysis with interesting applications and numerical examples in MATLAB.