Complex Analysis - Maths KU

φ = –2
y
D
φ
∇ 2 = 0 φ = 3
Figure 4.20 Figure for Example 1
x
4.5 Applications 225
A Method to Solve Dirichlet Problems We now present a
method for solving Dirichlet problems using Theorem 4.5. Let D be a domain
whose boundary consists of the curves C1, C2, ... Cn. Suppose that we wish
to find a function φ(x, y) that isharmonic in D and that takeson the values
k1, k2, ... kn on the boundary curves C1, C2, ... Cn, respectively. Our
method for solving such a problem consists of the following four steps.
Steps for Solving a Dirichlet Problem
1. Find an analytic function f(z) =u(x, y) +iv(x, y) that maps the
domain D in the z-plane onto a simpler domain D ′ in the w-plane
and that maps the boundary curves C1, C2, ... , Cn onto the curves
C ′ 1, C ′ 2, ... , C ′ n, respectively.
2. Transform the boundary conditions on C1, C2, ... Cn to boundary
conditions on C ′ 1, C ′ 2, ... , C ′ n.
3. Solve this new (and easier) Dirichlet problem in D ′ to obtain a harmonic
function Φ(u, v).
4. Substitute the real and imaginary parts u(x, y) and v(x, y) of f for the
variables u and v in Φ(u, v). By Theorem 4.5, the function φ(x, y) =
Φ(u(x, y), v(x, y)) is a solution to the Dirichlet problem in D.
We illustrate the general idea of these steps in Figure 4.19.
φ = k 2
C 2
y v
D
φ = k 1
w = f(z)
D′
x u
∇ φ 2 = 0 ∇ 2 Φ=
0
C 3
φ = k 3
Figure 4.19 Transforming a Dirichlet problem
C 1
Φ = k 2
C′ 2
C′ 3
Φ = k 3
C′ 1
Φ = k 1
EXAMPLE 1 Using Mappings to Solve a Dirichlet Problem
Let D be the domain in the z-plane bounded by the lines y = x and y = x +2
shown in color in Figure 4.20. Find a function φ(x, y) that isharmonic in D
and satisfies the boundary conditions φ(x, x +2)=−2 and φ(x, x) =3.
Solution We will solve this problem using the four steps given above.