Mathematical Tools for Physics

Partial Differential Equations

If the subject of ordinary differential equations is large, this is enormous. I am going to examine only one corner
of it, and develop only one tool to handle it: Separation of Variables. Another major tool is the method of
characteristics and I’ll not go beyond mentioning the word. When I develop a technique to handle the heat
equation or the potential equation, don’t think that it stops there. The same set of tools will work on the
Schroedinger equation in quantum mechanics and on the wave equation in its many incarnations.

10.1 The Heat Equation
The flow of heat in one dimension is described by the heat conduction equation

P=−κA

∂T

∂x

(1)

wherePis the power in the form of heat energy flowing toward positivexthrough a wall andAis the area of the
wall.κis the wall’s thermal conductivity. Put this equation into words and it says that if a thin slab of material
has a temperature on one side different from that on the other, then heat energy will flow through the slab. If
the temperature difference is big or the wall is thin (∂T/∂xis big) then there’s a big flow. The minus sign says
that the energy flows from hot toward cold.
When more heat comes into a region than leaves it, the temperature there will rise. This is described by
the specific heat,C.

dQ=mCdT, or

dQ

dt

=mC

dT

dt

(2)

Again in words, the temperature rise in a chunk of material is proportional to the amount of heat added to it and
inversely proportional to its mass.

P(x,t)

x

A

P(x+ ∆x,t)

x+ ∆x

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