Fourier Series and Partial Differential Equations Lecture Notes

Chapter 1. Introduction 8
1.3.2 The heat equation
Also known as the diffusion equation, we will find T(x,t) such that
∂T
∂t = κ∂2 T
∂x 2,
(1.7)
where, for example, T(x,t) is a temperature at position x and time t, and κ is a positive
constant—the thermal diffusivity.
1.3.3 Laplace’s equation
In this case the problem is to find T(x,y) such that
∂2T ∂x2 + ∂2T = 0, (1.8)
∂y2 where, for example, T(x,y) may be a temperature and x and y are Cartesian coordinates
in the plane. In this case, Laplace’s equation models a two-dimensional system at steady
state in time: in three space-dimensions the temperature T(x,y,z,t) satisfies the heat
equation
∂T
∂t
∂2T = κ
∂x2 + ∂2T ∂y2 + ∂2T ∂z2
. (1.9)
Note that equation (1.9) reduces to (3.8) if T is independent of y and z. If the temperature
field is static, T is independent of time, t, and is a solution of Laplace’s equation in R 3 ,
∂ 2 T
∂x 2 + ∂2 T
∂y2 + ∂2T = 0, (1.10)
∂z2 and, in the special case in which T is also independent of z, of Laplace’s equation in R 2 ,
∂2T ∂x2 + ∂2T = 0. (1.11)
∂y2