
8-88 Diff erential Equations
8.13.9 Laplace’s Equation or Potential Equation
or Two-dimensional Steady-state
Heat Flow Equation
The two-dimensional heat conduction equation is given by
()
22
22
22 2
, where is the Laplacian operator
xy
u
au
t
∂∂
∂∂
∂
==+
∂
——
In the case of steady-state heat fl ow
/0ut∂∂=
and equation reduces to
22
2
22
0
uu
u
xy
∂∂
=+=
∂∂
—
The solution u(x, y ) of the abo
ve equation can be obtained by the
method of separation of variab
les in a rectangular region both in the
Dirichlet problem as well as in Neumann’s problem. A rectangular thin
plate with its two faces insulated is considered so that the heat fl ow is
two-dimensional. The boundary conditions ...