Book:Arne Broman/Introduction to Partial Differential Equations
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Arne Broman: Introduction to Partial Differential Equations from Fourier Series to Boundary-value Problems
Published $\text {1970}$, Dover Publications, Inc.
- ISBN 0-486-66158-X
Subject Matter
Contents
- Preface (Göteborg, March 1970)
- Chapter $1$ Fourier series
- $1.1$ Basic concepts
- $1.2$ Fourier series and Fourier coefficients
- $1.3$ A minimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem
- $1.4$ Convergence of the Fourier series
- $1.5$ The Parseval formula
- $1.6$ Determination of the sum of certain trigonometric series
- Chapter $2$ Orthogonal systems
- $2.1$ Integration of complex-valued functions of a real variable
- $2.2$ Orthogonal systems
- $2.3$ Complete orthogonal systems
- $2.4$ Integration of Fourier series
- $2.5$ The Gram-Schmidt orthogonalization process
- $2.6$ Sturm-Liouville problems
- Chapter $3$ Orthogonal polynomials
- $3.1$ The Legendre polynomials
- $3.2$ Legendre series
- $3.3$ The Legendre differential equation. The generating function of the Legendre polyomials
- $3.4$ The Tchebycheff polynomials
- $3.5$ Tchebycheff series
- $3.6$ The Hermite polynomials. The Laguerre polynomials
- Chapter $4$ Fourier transforms
- $4.1$ Infinite interval of integration
- $4.2$ The Fourier integral formula: a heuristic introduction
- $4.3$ Auxiliary theorems
- $4.4$ Proof of the Fourier integral formula. Fourier transforms
- $4.5$ The convolution theorem. The Parseval formula
- Chapter $5$ Laplace transforms
- $5.1$ Definition of the Laplace transform. Domain. Analyticity
- $5.2$ Inversion formula
- $5.3$ Further properties of Laplace transforms. The convolution theorem
- $5.4$ Applications to ordinary differential equations
- Chapter $6$ Bessel functions
- $6.1$ The gamma function
- $6.2$ The Bessel differential equation. Bessel functions
- $6.3$ Some particular Bessel functions
- $6.4$ Recursion formulas for the Bessel functions
- $6.5$ Estimation of Bessel functions for large values of $x$. The zeros of the Bessel functions
- $6.6$ Bessel series
- $6.7$ The generating function of the Bessel functions of integral order
- $6.8$ Neumann function
- Chapter $7$ Partial differential equations of first order
- $7.1$ Introduction
- $7.2$ The differential equation of a family of surfaces
- $7.3$ Homogeneous differential equations
- $7.4$ Linear and quasilinear differential equations
- Chapter $8$ Partial differential equations of second order
- $8.1$ Problems in physics leading to partial differential equations
- $8.2$ Definitions
- $8.3$ The wave equation
- $8.4$ The heat equation
- $8.5$ The Laplace equation
- Answers to exercises
- Bibliography
- Conventions
- Symbols
- Index
Source work progress
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