Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 416

7.9.2 Transformation of Boundary-Value Problems
into Integral Equations

For boundary-value problems of Laplace equations in a simple, regular domains
we can obtain their solutions by using the Fourier method of expansion, separation
of variables, the integral transformation or the method of Green functions. These
methods do not work, however, for problems in domains that are not simple and
regular. We normally use either of the following two methods for those problems:

1. Find an analytical expression that contains the undetermined function and sat-
isfies the Laplace equation. The undetermined function is then determined by
imposing the boundary conditions.
2. Find the function set of functions satisfying the CDS. The solution is then deter-
mined by constructing harmonic functions from the set. This method belongs to
the direct category in mathematical equations of physics.

A typical example of the former is seeking solutions of boundary-value prob-
lems using the potential theory, i.e. by transforming boundary-value problems into
integral equations. If S is a )

are continuous on S , the single-layer and the double-layer potentials are harmonic
functions in both

Ляпунов surface and density functions τ ( P ) and μ ( P

Ω (the region outside S ). In particular,
the single-layer potential is continuous in both

Ω (the region inside S )and

Ω +

S ; its normal deriva-
tive also has the limit from both inside and outside of S . Therefore, we may use
the potentials as solutions of boundary value problems of Laplace equations. Their
density functions are undetermined functions and can be determined by imposing
the boundary conditions.
Here we discuss this method for solving boundary-value problems of Laplace
equations of the first and the second kind

Ω + S and

⎧
⎨ Δ u = 0 ,

Δ u = 0 ,

S =

or

∂ u

⎩

u | S = f ( M ) . f ( M ) .

∂ n

where the S is a Ляпунов surface and f ( M ) is a continuous function.

First Internal Boundary-Value Problems (Dirichlet Internal problems)

Assume that the double-layer potential with the undetermined density function τ ( P )

τ ( P ) cos ( PM , n )

u ( M )= d S (7.160a)

r PM

S

is a function that satisfies Δ u = 0, M ∈ Ω and u | S = f ( M ) , and is continuous in

Ω + S . By Theorem 7 in Section 7.8, we have, for M ∈ S