Partial Differential Equations: Analytical and Numerical Methods
Emphasizing techniques over theory, this book focuses its discourse mainly on physical and mathematical derivations of Laplace, wave, and heat equations.
TABLE OF CONTENTS Partial Differential Equations: Analytical and Numerical Methods
Chapter 2: Models in One Dimension
In this chapter, we present several different and interesting physical processes that can be modeled by ODEs or PDEs. For now we restrict ourselves to phenomena that can be described (at least approximately) as occurring in a single spatial dimension: heat flow or mechanical vibration in a long, thin bar, vibration of a string, diffusion of chemicals in a pipe, and so forth. In Chapters 5 7, we will learn methods for solving the resulting differential equations.
In Chapter 8, we consider similar experiments occurring in multiple spatial dimensions.
2.1 Heat flow in a bar; Fourier's law
We begin by considering the distribution of heat energy (or, equivalently, of temperature) in a long, thin bar. We assume that the cross-sections of the bar are uniform, and that the temperature varies only in the longitudinal direction. In particular, we assume that the bar is perfectly insulated, except possibly at the ends, so that no heat escapes through the side. By making several simplifying assumptions, we derive a linear PDE whose solution is the temperature in the bar as a function of spatial position and time.
We show, when we treat multiple space dimensions in Chapter 8, that if the initial temperature is constant in each cross-section, and if any heat source depends only on the longitudinal coordinate, then all subsequent temperature distributions depend only on the longitudinal coordinate. Therefore, in this regard, there is no modeling error in adopting a one-dimensional model. (There is modeling error associated with some of the other...
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