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Partial Differential Equations

Analytical Methods and Applications

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## Book Description

** 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

## Table of Contents

**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

Maximum principle

The heat equation in an infinite region

Elliptic equations

Elliptic differential equations and related physical problems

Harmonic functions

Boundary conditions

Example of an ill-posed problem

Well-posed boundary value problems

Maximum principle and its consequences

Laplace equation in polar coordinates

Laplace equation and interior BVP for circular domain

Laplace equation and exterior BVP for circular domain

Poisson equation: general notes and a simple case

Poisson Integral

Application of Bessel functions for the solution of Poisson equations in a circle

Three-dimensional Laplace equation for a cylinder

Three-dimensional Laplace equation for a ball

Axisymmetric case

Non-axisymmetric case

BVP for Laplace Equation in a Rectangular Domain

The Poisson Equation with Homogeneous Boundary Conditions

Green's function for Poisson equations

Homogeneous boundary conditions

Nonhomogeneous boundary conditions

Some other important equations

Helmholtz equation

Schrӧdinger equation

Two Dimensional Hyperbolic Equations

Derivation of the Equations of Motion

Boundary and Initial Conditions

Oscillations of a Rectangular Membrane

The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions

The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions

The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions

Small Transverse Oscillations of a Circular Membrane

The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions

Axisymmetric Oscillations of a Membrane

The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions

Forced Axisymmetric Oscillations

The Fourier Method for Equations with Nonhomogeneous Boundary Conditions

Two-Dimensional Parabolic Equations

Heat Conduction within a Finite Rectangular Domain

The Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange)

The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary conditions

Heat Conduction within a Circular Domain

The Fourier Method for the Homogeneous Heat Equation

The Fourier Method for the Nonhomogeneous Heat Equation

Heat conduction in an Infinite Medium

Heat Conduction in a Semi-Infinite Medium

Nonlinear equations

Burgers equation

Kink solution

Symmetries of the Burgers equation

General solution of the Cauchy problem.

Interaction of kinks

Korteweg-de Vries equation

Symmetry properties of the KdV equation

Cnoidal waves

Solitons

Bilinear formulation of the KdV equation

Hirota's method

Multisoliton solutions

Nonlinear Schrӧdinger equation

Symmetry properties of NSE

Solitary waves

Appendix A. Fourier Series, Fourier and Laplace Transforms

Appendix B. Bessel and Legendre Functions

**Appendix C. Sturm-Liouville problem and auxiliary functions for one and two dimensions**

Appendix D.

D1. The Sturm-Liouville problem for a circle

D2. The Sturm-Liouville problem for the rectangle

Appendix E.

**E1. The Laplace and Poisson equations for a rectangular domain with nonhomogeneous boundary conditions. **

**E2. The heat conduction equations with nonhomogeneous boundary conditions. **

## Author(s)

### Biography

**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.