
Chapter 4
Second-Order
Parabolic Equations
with Two Space Variables
4.1 Heat Equation
∂w
∂t
= a∆
2
w
4.1.1 B o u ndary Value Problems i n Cartesian Coordinates
In rectangular Cartesian coordinates, the two-dimensional sourceless heat equation has the
form
∂w
∂t
= a
∂
2
w
∂x
2
+
∂
2
w
∂y
2
.
It governs two-dimensional unsteady heat transfer processes in quiescent media or solid
bodies with constant thermal diffusivity a. A similar equation is used to study analogous
two-dimensional unsteady m ass transfer phenomena with constant diffusivity; in this case
the equation is called a diffusion equation.
◮ Particular solutions:
w(x, y) = Axy + C
1
x + C
2
y + C
3
,
w(x, y, t) = Ax
2
+