
Chapter 5
Second-Order
Parabolic Equations with
Three or More Space Variables
5.1 Heat Equation
∂w
∂t
= a∆
3
w
5.1.1 Problems in Cartesian Coordinates
The three-dimensional sourceless heat equation in the rectangular Cartesian system of co-
ordinates has the form
∂w
∂t
= a
∂
2
w
∂x
2
+
∂
2
w
∂y
2
+
∂
2
w
∂z
2
.
It governs three-dimensional thermal phenomena in quiescent media or solids with constant
thermal diffusivity. A similar equation is used to study the corresponding three-dimensional
unsteady mass-exchange processes with constant diffusivity.
◮ Particular solutions:
w(x, y, z, t) = Ax
2
+By
2
+Cz
2
+2a(A+ B +C)t,
w(x, y, z, t) = A(x
2
+2at)(y
2
+2at)(z
2
+2at)+ B,
w(x, y, z, t) = A