Introduction to Partial Differential Equations (eBook)

This exceptionally well-written and well-organized text is the outgrowth of a course given every year for 45 years at the Chalmers University of Technology, Goteborg, Sweden. The object of the course was to give students a basic knowledge of Fourier analysis and certain of its applications. The text is self-contained with respect to such analysis; however, in areas where the author relies on results from branches of mathematics outside the scope of this book, references to widely used books are given.Table of Contents:Chapter 1. Fourier series1.1 Basic concepts1.2 Fourier series and Fourier coefficients1.3 A minimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem1.4 Convergence of Fourier series1.5 The Parseval formula1.6 Determination of the sum of certain trigonometric seriesChapter 2. Orthogonal systems2.1 Integration of complex-valued functions of a real variable2.2 Orthogonal systems2.3 Complete orthogonal systems2.4 Integration of Fourier series2.5 The Gram-Schmidt orthogonalization process2.6 Sturm-Liouville problemsChapter 3. Orthogonal polynomials3.1 The Legendre polynomials3.2 Legendre series3.3 The Legendre differential equation. The generating function of the Legendre polynomials3.4 The Tchebycheff polynomials3.5 Tchebycheff series3.6 The Hermite polynomials. The Laguerre polynomialsChapter 4. Fourier transforms4.1 Infinite interval of integration4.2 The Fourier integral formula: a heuristic introduction4.3 Auxiliary theorems4.4 Proof of the Fourier integral formula. Fourier transforms4.5 The convention theorem. The Parseval formulaChapter 5. Laplace transforms5.1 Definition of the Laplace transform. Domain. Analyticity5.2 Inversion formula5.3 Further properties of Laplace transforms. The convolution theorem5.4 Applications to ordinary differential equationsChapter 6. Bessel functions6.1 The gamma function6.2 The Bessel differential equation. Bessel functions6.3 Some particular Bessel functions6.4 Recursion formulas for the Bessel functions6.5 Estimation of Bessel functions for large values of x. The zeros of the Bessel functions6.6 Bessel series6.7 The generating function of the Bessel functions of integral order6.8 Neumann functionsChapter 7. Partial differential equations of first order7.1 Introduction7.2 The differential equation of a family of surfaces7.3 Homogeneous differential equations7.4 Linear and quasilinear differential equationsChapter 8. Partial differential equations of second order8.1 Problems in physics leading to partial differential equations8.2 Definitions8.3 The wave equation8.4 The heat equation8.5 The Laplace equationAnswers to exercises; Bibliography; Conventions; Symbols; IndexWritten on an advanced level, the book is aimed at advanced undergraduates and graduate students with a background in calculus, linear algebra, ordinary differential equations, and complex analysis. Over 260 carefully chosen exercises, with answers, encompass both routing and more challenging problems to help students test their grasp of the material.