Partial Differential Equations

Analytical Methods and Applications
1. Auflage 2019
ISBN: 978-1-138-33983-5
Verlag: Chapman and Hall/CRC

Buch, Englisch, 396 Seiten, Format (B × H): 183 mm x 260 mm, Gewicht: 941 g

Reihe: Textbooks in Mathematics

Partial Differential Equations: Analytical Methods and Applications covers all the basic topics of a Partial Differential Equations (PDE) course for undergraduate students or a beginners’ course for graduate students. It provides qualitative physical explanation of mathematical results while maintaining the expected level of it rigor.

This text introduces and promotes practice of necessary problem-solving skills. The presentation is concise and friendly to the reader. The "teaching-by-examples" approach provides numerous carefully chosen examples that guide step-by-step learning of concepts and techniques. Fourier series, Sturm-Liouville problem, Fourier transform, and Laplace transform are included. The book’s level of presentation and structure is well suited for use in engineering, physics and applied mathematics courses.

Highlights:

- Offers a complete first course on PDEs

- The text’s flexible structure promotes varied syllabi for courses

- Written with a teach-by-example approach which offers numerous examples and applications

- Includes additional topics such as the Sturm-Liouville problem, Fourier and Laplace transforms, and special functions

- The text’s graphical material makes excellent use of modern software packages

- Features numerous examples and applications which are suitable for readers studying the subject remotely or independently

Zielgruppe

General

Autoren/Hrsg.

Henner, Victor

Belozerova, Tatyana

Nepomnyashchy, Alexander

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

Introduction

Basic definitions

Examples

First-order equations

Linear first-order equations

General solution

Initial condition

Quasilinear first-order equations

Characteristic curves

Examples

Second-order equations

Classification of second-order equations

Canonical forms

Hyperbolic equations

Elliptic equations

Parabolic equations

The Sturm-Liouville Problem

General consideration

Examples of Sturm-Liouville Problems

One-Dimensional Hyperbolic Equations

Wave Equation

Boundary and Initial Conditions

Longitudinal Vibrations of a Rod and Electrical Oscillations

Rod oscillations: Equations and boundary conditions

Electrical Oscillations in a Circuit

Traveling Waves: D'Alembert Method

Cauchy problem for nonhomogeneous wave equation

D'Alembert's formula

The Green's function

Well-posedness of the Cauchy problem

Finite intervals: The Fourier Method for Homogeneous Equations

The Fourier Method for Nonhomogeneous Equations

The Laplace Transform Method: simple cases

Equations with Nonhomogeneous Boundary Conditions

The Consistency Conditions and Generalized Solutions

Energy in the Harmonics

Dispersion of waves

Cauchy problem in an infinite region

Propagation of a wave train

One-Dimensional Parabolic Equations

Heat Conduction and Diffusion: Boundary Value Problems

Heat conduction

Diffusion equation

One-dimensional parabolic equations and initial and boundary conditions

The Fourier Method for Homogeneous Equations

Nonhomogeneous Equations

The Green's function and Duhamel's principle

The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions

Large time behavior of solutions

Ma

Über Autor(innen)

Victor Henner is a professor at the Department of Physics and Astronomy at the University of Louisville. He has Ph.Ds from the Novosibirsk Institute of Mathematics in Russia and Moscow State University. He co-wrote with Tatyana Belozerova Ordinary and Partial Differential Equations.

Tatyana Belozerova is a professor at Perm State University in Russia. Along with Ordinary and Partial Differential Equations, she co-wrote with Victor Henner Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions.

Alexander Nepomnyashchy is a mathematics professor at Northwestern University and hails from the Faculty of Mathematics at Technion-Israel Institute of Technology. His research interests include non-linear stability theory and pattern formation.