Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 415

is a finite region. For example, counterparts of Eq. (7.154) and (7.155) are, in the
three-dimensional case,

ϕ ( M )= λ K ( M , P ) ϕ ( P ) d Ω + f ( M ) , (7.156)

Ω

ϕ ( M )= λ K ( M , P ) ϕ ( P ) d Ω , (7.157)

Ω

respectively. Here Ω stands for a three-dimensional finite region, and M and P are

points in .

With known kernel K

Ω

( M , P ) and nonhomogeneous term g ( M ) , the integral equa-

tion

ψ ( M )= λ K ( M , P ) ψ ( P ) d Ω + g ( M ) , (7.158)

Ω

is called the transpose equation of Eq. (7.156) .
When the unknown function occurs only in the integrands, the equation is called
the Fredholm integral equation of the first kind .
Examples are

b

K ( x , ξ ) ϕ ( ξ ) d ξ = f ( x ) ,

a

K ( M , P ) ϕ ( P ) d Ω = f ( M ) . (7.159)

Ω

An integral equation that involves the unknown function outside of the integrands is
called a Fredholm integral equation of the second kind . Examples are Eqs. (7.154),
(7.156) and (7.158).
If the kernel is symmetric such that K

( , )= ( , )

M P K P M , the equation is called a

symmetric equation . If the kernel satisfies

H ( M , P )

K ( M , P )= , 0 < α < n ,

r PM

it is called a weakly-singular kernel . The corresponding equation is called an inte-
gral equation with a weakly-singular kernel .Here H is a continuous function.

r PM is the distance between P and M and n is the dimension of the integration do-
main.
An equivalent definition of a weakly-singular kernel is

( M , P )

C
r n − α

| ( , ) |≤

PM ,

< α < ,

K M P 0 n

where C is a constant. In three-dimensional space, the dimension of the integration
domain is two instead of three if the integrals involved are surface integrals.