Introduction to partial differential equations

From Fourier series to boundary-value problems

Arne Broman

190 pages, parution le 01/01/1989

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Résumé

Well-written advanced-level text covers basic Fourier analysis, applications. Fourier series, orthogonal systems, Laplace transforms, Bessel functions, much more. 266 exercises with solutions.

Fourier series
- Basic concepts
- Fourier series and Fourier coefficients
- A mimimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem
- Convergence of Fourier series
- The Parseval formula
- Determination of the sum of certain trigonemetric series
Orthogonal systems
- Integration of complex-valued functions of a real variable
- Orthogonal systems
- Complete orthogonal systems
- Integration of Fourier series
- The Gram-Schmidt orthogonalization process
- Sturm-Liouville problems
Orthogonal polynomials
- The Legendre polynomials
- Legendre series
- The Legendre differential equation. The generating function of the Legendre polynomials
- The Tchebycheff polynomials
- Tchebycheff series
- The Hermite polynomials. The Laguerre polynomials
Fourier transforms
- Infinite interval of integration
- The Fourier integral formula: a heuristic introduction
- Auxiliary theorems
- Proof of the Fourier integral formula. Fourier transforms
- The convention theorem. The Parseval formula
Laplace transforms
- Definition of the Laplace transform. Domain. Analyticity
- Inversion formula
- Further properties of Laplace transforms. The convolution theorem
- Applications to ordinary differential equations
Bessel functions
- The gamma function
- The Bessel differential equation. Bessel functions
- Some particular Bessel functions
- Recursion formulas for the Bessel functions
- Estimation of Bessel functions for large values of x. The zeros of the Bessel functions
- Bessel series
- The generating function of the Bessel functions of integral order
- Neumann functions
Partial differential equations of first order
- Introduction
- The differential equation of a family of surfaces
- Homogeneous differential equations
- Linear and quasilinear differential equations
Partial differential equations of second order
- Problems in physics leading to partial differential equations
- Definitions
- The wave equation
- The heat equation
- The Laplace equation
Answers to exercises
Bibliography
Conventions
Symbols
Index

Caractéristiques techniques

	PAPIER
Éditeur(s)	Dover
Auteur(s)	Arne Broman
Parution	01/01/1989
Nb. de pages	190
Format	14,3 x 21
Couverture	Broché
Poids	205g
Intérieur	Noir et Blanc
EAN13	9780486661582
ISBN13	978-0-486-66158-2