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Unformatted text preview: (x, y, z ) + ∇ · (κ(x, y, z )(∇u)) .
∂t The Dirichlet problem The reader will recall that the space of solutions to a homogeneous linear second
order ordinary diﬀerential equation, such as
d2 u
du
+ qu = 0
+p
dt2
dt
is twodimensional, a particular solution being determined by two constants. By
contrast, the space of solutions to Laplace’s partial diﬀerential equation
∂2u ∂2u
+ 2 =0
∂x2
∂y
118 (5.3) is inﬁnitedimensional. For example, the function u(x, y ) = x3 − 3xy 2 is a
solution to Laplace’s equation, because
∂u
= 3x2 − 3y 2 ,
∂x
∂u
= −6xy,
∂y
and hence ∂2u
= 6x,
∂x2 ∂2u
= −6x,
∂y 2 ∂2u ∂2u
+ 2 = 6x − 6x = 0.
∂x2
∂y Similarly,
u(x, y ) = 7, u(x, y ) = x4 − 6x2 y 2 + y 4 , and u(x, y ) = ex sin y are solutions to Laplace’s equation. A solution to Laplace’s equation is called
a harmonic function. It is not diﬃcult to construct inﬁnitely many linearly
independent harmonic functions of two variables.
Thus to pick out a particular solution to Lap...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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