Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 418

Ω + S . By Theorem 8 in Section 7.8, we have, for M ∈ S

1
r PM

d S

∂ u

) ∂
∂ n − =

μ (

+

πμ (

)

P 2 M

∂ n

S

μ ( P ) cos ( PM , n )

= − d S + 2 πμ ( M ) .

r PM

S

Thus the Neumann internal problems are transformed into Fredholm integral equa-
tions of the second kind regarding

1
2

μ ( M )= K ( M , P ) μ ( P ) d S + f ( M ) , (7.162)

π

S

where K ( M , P ) is the same as in the Dirichlet problems.

Second External Boundary-Value Problems (Neumann External Problems)

By following a similar approach as that used in obtaining Eq. (7.162), we can trans-
form the Neumann external problems into Fredholm integral equations of the second
kind regarding

1
2

μ ( M )= − K ( M , P ) μ ( P ) d S + f ( M ) , (7.163)

π

S

S

)= ∂

u

Ω that is also the inner normal of

where f ( M , n is the outer normal for

∂ n

the closed surface S .

( , )

Remark 1 . In integral equations (7.160)-(7.163), the kernel K M P and its trans-

pose K ( P , M ) satisfy

K ( M , P )= − K ( P , M ) .

By using these relations, the four equations become

1
2

τ ( M )= K ( M , P ) τ ( P ) d S − f ( M ) , (7.160')

π

S

1
2

τ ( M )= − K ( M , P ) τ ( P ) d S + f ( M ) , (7.161')

π

S

1
2

μ ( M )= − K ( P , M ) μ ( P ) d S + f ( M ) , (7.162')

π

S

1
2

μ ( M )= K ( P , M ) μ ( P ) d S + f ( M ) . (7.163')

π

S