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Chapter 5
2D Linear Second-Order Equations
In this chapter, we delve into the realm of linear partial differential equations in 2D domains.
Our focus lies on rectangular and disk-shape d domains, as they provide us with the opportu-
nity to employ the spatial variable separation technique. Within this framework, we explore
a range of partial differential equations, namely: 1) Laplace equations, 2) Poisson equations,
3) heat equations, and 4) wave equations.
Let Ω be an open bounded set in R
2
with a smooth or piecewise smooth boundary bnd(Ω).
We begin by considering the Laplace e quation, which takes the g en eral form ∆u(x; y)=0 for
all (x; y)2Ω. This equation is defined for all smooth function s u that satisfy a prescribed
boundary condition, typi cally of the form αu+β
@u
@n
=γ on the boundary bnd(Ω). Here, α; β,
and γ are constants.
The Laplace equation is a special case of the Poisson equation, which has the general form
¡∆u= f, where f is a function that is independent of u and depends only on the spatial
variables (x; y). In this equation, we seek a solution u that satisfies the prescribed boundary
conditions.
Moving on, we explore the steady state of a heat equation, which has the form u_
t=k∆u+ f. Here, u
t
represents the partial derivative of u with respect to time, k is a c o nstant
representing thermal diffusivity, ∆u is the Laplacian of u, and f represents an e xternal source
of heat.
Finally, we consider the wave equation or damped wave equation, given by
u
tt
+2ξu
t
=c
2
∆u + f. In this equation, u
tt
represents the second partial derivative of u
with respect to time, ξ is a non-negative constant known as the damping factor, c rep-
resents th e wave speed, ∆u is the Laplacian of u, and f represents an external forcing term.
Throughout this chapter, we aim to study the properties of these equations, investigate
their solutions, and u nderstand their physical interpretations in the context of 2D domains.
5.1 Overv iew of the e igenfunction expa nsion method
The eigenfunction expansion method for problems defined on bounded domains in the plane
R
2
is an ex tension of the approach used for 1D problems on bounded domains in R. To
demonstrate this method, let's consider the heat prob lem:
(
u
t
=k∆u + h on Ω; t > 0
αu + β
@u
@n
= 0 on bnd(Ω); t > 0
;
1