Book:Arne Broman/Introduction to Partial Differential Equations

Subject Matter

 * Partial Differential Equations

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
* : Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.2$ Definitions