
Chapter 3
Second-Order
Parabolic Equations
with One Space Variable
3.1 Constant Coefficient Equations
3.1.1 H eat Equation
∂w
∂t
= a
∂
2
w
∂x
2
This equation is often encountered in the theory of heat and mass transfer. It describes one-
dimensional unsteady thermal processes in quiescent media or solids with constant thermal
diffusivity. A similar equation is used in studying corresponding one-dimensional unsteady
mass-exchange processes w ith constant diffusivity.
◮ Particular solutions (A, B, and µ are arbitrary constants).
w(x) = Ax + B,
w(x, t) = A(x
2
+ 2at) + B,
w(x, t) = A(x
3
+ 6atx) + B,
w(x, t) = A(x
4
+ 12atx
2
+ 12a
2
t
2
) + B,
w(x, t) = A(x
5
+ 20atx
3
+ 60a
2
t
2
x)