Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 414

Theorem 8. Let S be a closed Ляпунов surface with n as its external normal. Sup-

pose that density function μ ( P ) is continuous on S .The u ( M ) is the single-layer

potential in Eq. (7.141). The normal derivative ∂

u

n has a discontinuity of the first ∂

kind at M ∈ S ,i.e.

∂

n + = ∂

u u ∂

n − = ∂

u u

n −

2 πμ ( M ) ,

n +

2 πμ ( M ) , M ∈ S , (7.153)

∂ ∂ ∂ ∂

∂ u ∂ u ∂ u

where and are the limits of

n as M tends to S along the normal from n − n + ∂ ∂ ∂

inside and outside of S , respectively.
For two-dimensional cases, the 2 π in Eq. (7.153) should be replaced by π .

7.9 Transformation of Boundary-Value Problems
of Laplace Equations to Integral Equations

This section starts with a brief discussion of integral equations. Potential theory is
thus applied to transform boundary-value problems of Laplace equations of the first
and the second kind into integral equations. Readers are referred to topics on integral
equations for methods of solving integral equations.

7.9.1 Integral Equations

An equation that contains an unknown function in the integrand is called an integral
equation . If the unknown function occurs in a linear form, the equation is called a
linear integral equation . For example, in the one-dimensional case, the equation

b

ϕ ( x )= λ K ( x , ξ ) ϕ ( ξ ) d ξ + f ( x ) (7.154)

a

is a linear integral equation. Here λ is a real-valued or complex-valued parameter.

is also a known function and is called the kernel of
integral equations . Two independent variables x and

f ( x ) is a given function. K ( x , ξ )

ξ vary in the region [ a , b ] . ϕ ( x )

is an unknown function.
If f

( x ) ≡ 0,wehave

b

ϕ ( x )= λ K ( x , ξ ) ϕ ( ξ ) d ξ , (7.155)

a

which is called the associated homogeneous equation of Eq. (7.154) . Equation
(7.154) is called a nonhomogeneous equation ,and f

( x ) is the nonhomogeneous

term .
In n -dimensional space, integrals in the integral equations are n -multiple, the
unknown function is a function of independent variables and the integration domain