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Über dieses Buch

This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise.
Within each section the author creates a narrative that answers the five questions: What is the scientific problem we are trying to understand?
How do we model that with PDE?What techniques can we use to analyze the PDE?How do those techniques apply to this equation?What information or insight did we obtain by developing and analyzing the PDE?The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
Continuous phenomena, such as wave propagation or fluid flow, are generally modeled with partial differential equations (PDE), which express relationships between rates of change with respect to multiple independent variables.
David Borthwick

Chapter 2. Preliminaries

Abstract
In this chapter we set the stage for the study of PDE with a review of some core background material.
David Borthwick

Chapter 3. Conservation Equations and Characteristics

Abstract
A conservation law for a physical system states that a certain quantity (e.g., mass, energy, or momentum) is independent of time. For continuous systems such as fluids or gases, these global quantities can be defined as integrals of density functions. The conservation law then translates into a local form, as a PDE for the density function. In this section we will study some first-order PDE that arise from conservation laws. We introduce a classic technique, called the method of characteristics, for analyzing these equations.
David Borthwick

Chapter 4. The Wave Equation

Abstract
In this chapter we will develop the wave equation as a model for the vibrating string problem, and derive d’Alembert’s explicit solution in one dimension using the method of characteristics introduced in Chap. 3.
David Borthwick

Chapter 5. Separation of Variables

Abstract
Some PDE can be split into pieces that involve distinct variables. For PDE that admit separation, it is natural to look for product solutions whose factors depend on the separate variables.
David Borthwick

Chapter 6. The Heat Equation

Abstract
In this chapter we will discuss the derivation of the heat equation and develop some basic properties, our first example of a PDE of parabolic type.
David Borthwick

Chapter 7. Function Spaces

Abstract
In this chapter we will introduce some basic concepts of functional analysis. These will give us the tools to address issues such as the construction of solutions by infinite series.
David Borthwick

Chapter 8. Fourier Series

Abstract
In this chapter we will analyze Fourier series in more detail, and show that the Fourier approach yields a general solution for the one-dimensional heat equation. The primary significance of this approach to PDE is the philosophy of spectral analysis that it inspired. The decomposition of functions with respect to the spectrum of a differential operator is a tool with enormous applications, both theoretical and practical.
David Borthwick

Chapter 9. Maximum Principles

Abstract
In this chapter we will consider another approach to issues of uniqueness and stability, based on maximum values. This method applies generally to elliptic equations, which describe equilibrium states, and to parabolic equations, which are generally used to model diffusion.
David Borthwick

Chapter 10. Weak Solutions

Abstract
As noted in Sect. 1.​2, it is possible to define weak solutions of a PDE that are not sufficiently differentiable to solve the equation literally. In this chapter we will discuss the mathematical formulation of this generalized notion of solution.
David Borthwick

Chapter 11. Variational Methods

Abstract
In this chapter we will develop a strategy that involves reformulating a PDE in terms of a minimization problem on a function space.
David Borthwick

Chapter 12. Distributions

Abstract
To define weak derivatives in Chap. 10, we measured the values of a solution using test functions. Taking this idea one step further leads to a generalization of the concept of a function, called a distribution.
David Borthwick

Chapter 13. The Fourier Transform

Abstract
In previous chapters we have made frequent use of Fourier series as an analytical tool. In this chapter we develop the continuous analog, an integral transform based on plane waves.
David Borthwick

Erratum to: Introduction to Partial Differential Equations

Without Abstract
David Borthwick

Backmatter

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