24-12-2012, 02:57 PM
PARTIAL DIFFERENTIAL EQUATIONS MA 3132 LECTURE NOTES
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Introduction and Applications
This section is devoted to basic concepts in partial differential equations. We start the
chapter with definitions so that we are all clear when a term like linear partial differential
equation (PDE) or second order PDE is mentioned. After that we give a list of physical
problems that can be modelled as PDEs. An example of each class (parabolic, hyperbolic and
elliptic) will be derived in some detail. Several possible boundary conditions are discussed.
Applications
In this section we list several physical applications and the PDE used to model them. See,
for example, Fletcher (1988), Haltiner and Williams (1980), and Pedlosky (1986).
For the heat equation (parabolic, see definition 7 later).
Conduction of Heat in a Rod
Consider a rod of constant cross section A and length L (see Figure 1) oriented in the x
direction.
Let e(x, t) denote the thermal energy density or the amount of thermal energy per unit
volume. Suppose that the lateral surface of the rod is perfectly insulated. Then there is no
thermal energy loss through the lateral surface. The thermal energy may depend on x and t
if the bar is not uniformly heated. Consider a slice of thickness Δx between x and x+Δx.
A Vibrating String
Suppose we have a tightly stretched string of length L. We imagine that the ends are tied
down in some way (see next section). We describe the motion of the string as a result of
disturbing it from equilibrium at time t = 0, see Figure 4.
We assume that the slope of the string is small and thus the horizontal displacement can
be neglected. Consider a small segment of the string between x and x + Δx. The forces
acting on this segment are along the string (tension) and vertical (gravity). Let T(x, t) be
the tension at the point x at time t, if we assume the string is flexible then the tension is in
the direction tangent to the string, see Figure 5.
Diffusion in Three Dimensions
Diffusion problems lead to partial differential equations that are similar to those of heat
conduction. Suppose C(x, y, z, t) denotes the concentration of a substance, i.e. the mass
per unit volume, which is dissolving into a liquid or a gas. For example, pollution in a lake.
The amount of a substance (pollutant) in the given domain V with boundary Γ is given by The law of conservation of mass states that the time rate of change of mass in V is equal to
the rate at which mass flows into V minus the rate at which mass flows out of V plus the
rate at which mass is produced due to sources in V . Let’s assume that there are no internal
sources. Let q be the mass flux vector, then q · n gives the mass per unit area per unit time
crossing a surface element with outward unit normal vector n.