Heat Equation

1.1 Introduction

We wish to discuss the solution of elementary problems involving partial diﬀerential equations, the kinds of problems that arise in various ﬁelds of science and engineering. A partial diﬀerential equation (PDE) is a mathematical equation containing partial derivatives, for example, ∂u ∂u +3 = 0. ∂t ∂x (1.1.1)

We could begin our study by determining what functions u(x, t) satisfy (1.1.1). However, we prefer to start by investigating a physical problem. We do this for two reasons. First, our mathematical techniques probably will be of greater interest to you when it becomes clear that these methods analyze physical problems. Second, we will actually ﬁnd that physical considerations will motivate many of our mathematical developments. Many diverse subject areas in engineering and the physical sciences are dominated by the study of partial diﬀerential equations. No list could be all-inclusive. However, the following examples should give you a feeling for the type of areas that are highly dependent on the study of partial diﬀerential equations: acoustics, aerodynamics, elasticity, electrodynamics, ﬂuid dynamics, geophysics (seismic wave propagation), heat transfer, meteorology, oceanography, optics, petroleum engineering, plasma physics (ionized liquids and gases), and quantum mechanics. We will follow a certain philosophy of applied mathematics in which the analysis of a problem will have three stages: 1. Formulation 2. Solution 3. Interpretation We begin by formulating the equations of heat ﬂow describing the transfer of thermal energy. Heat energy is caused by the agitation of molecular matter. Two 1

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Chapter 1. Heat Equation

basic processes take place in order for thermal energy to move: conduction and convection. Conduction results from the collisions of neighboring molecules in which the kinetic energy of vibration of one molecule is transferred to its nearest neighbor. Thermal energy is thus spread by conduction even if the molecules themselves do not move their location appreciably. In addition, if a vibrating molecule moves from one region to another, it takes its thermal energy with it. This type of movement of thermal energy is called convection. In order to begin our study with relatively simple problems, we will study heat ﬂow only in cases in which the conduction of heat energy is much more signiﬁcant than its convection. We will thus think of heat ﬂow primarily in the case of solids, although heat transfer in ﬂuids (liquids and gases) is also primarily by conduction if the ﬂuid velocity is suﬃciently small.

1.2

Derivation of the Conduction of Heat in a One-Dimensional Rod

Thermal energy density. We begin by considering a rod of constant crosssectional area A oriented in the x-direction (from x = 0 to x = L) as illustrated in Fig. 1.2.1. We temporarily introduce the amount of thermal energy per unit volume as an unknown variable and call it the thermal energy density: e(x, t) ≡ thermal energy density. We assume that all thermal quantities are constant across a section; the rod is onedimensional. The simplest way this may be accomplished is to insulate perfectly the lateral surface area of the rod. Then no thermal energy can pass through the lateral surface. The dependence on x and t corresponds to a situation in which the rod is not uniformly heated; the thermal energy density varies from one cross section to another. z

φ(x,t) A φ(x + ∆x,t)

y x

x=0

x x + ∆x

x=L

Figure 1.2.1 One-dimensional rod with heat energy ﬂowing into and out of a thin slice.

Heat energy. We consider a thin slice of the rod contained between x and x+ ∆x as illustrated in Fig. 1.2.1. If the thermal energy density is constant throughout the volume, then the total energy in the slice is the product of the thermal energy

1.2 Conduction of Heat in One-Dimension

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density and the volume. In general, the energy density is not constant. However, if ∆x is...