Introduction to Partial Differential Equations: From Fourier Series to Boundary-Value Problems
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Specificatii
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.
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