ABSTRACT
In the previous chapter we solved the one-dimensional heat equation. Quite often we found that the transient solution died away, leaving a steady state. The partial differential equation that describes the steady state for two-dimensional heat conduction is Laplace’s equation
∂2u
∂x2 + ∂2u
∂y2 = 0. (12.0.1)
In general, this equation governs physical processes where equilibrium has been reached. It also serves as the prototype for a wider class of elliptic equations
a(x, t) ∂2u
∂x2 + b(x, t)
∂2u
∂x∂t + c(x, t)
∂2u
∂t2 = f
( x, t, u,
∂u
∂x , ∂u
∂t
) , (12.0.2)
where b2 < 4ac. Unlike the heat and wave equations, there are no initial conditions and the boundary conditions completely specify the solution. In this chapter we present some of the common techniques for solving this equation.
