4.9 Integral Equations

The integral equation under consideration in this context is the Fredholm integral equation of the second kind, which is commonly given in the form

                ∫
ϕ(s) = f(s) + λ    K (s,t)ϕ (t)dt.                     (4.179)
                 B
K(s,t) is the kernel of the integral equation. As the solution ϕ(s)  appears on both sides of Equation 4.179, it can be inserted into itself, thus yielding
        ∫ b       (          ∫              )
ϕ(s)=f(s) + λ   K (s,t)  f(s) + λ    K (s,t)ϕ (t)dt  dt =                 (4.180a)
         a                    B
        ∫                    ∫         ( ∫               )
=f(s) + λ   K (s,t)f(s)dt + λ2   K (s,t)     K (s,t)ϕ(t)dt  dt.     (4.180b)
         B                    B           B

Thus the function ϕ (s)  takes the form of a series

                                        ∞
                              2         ∑        i
ϕ(s) = ϕ0(s) + ϕ1(s)λ + ϕ2(s)λ  + ...=     ϕi(s)λ             (4.181)
                                        i=0
where
      ∫
ϕn =     K (s,t)ϕn −1(t)dt     ϕ0(s) = f(s).                (4.182)
       B
While this recursion relation is straightforward, an explicit expression for the ϕn (s)  depending only on ϕ0(s)=f(s) is favourable. It can be obtained by rewriting the recursion with a focus on the kernels instead of the functions to read.
     ∫
ϕn =    Kn (s,t)f(t)dt                          (4.183)
       B
Now the Kn(s,t) are given as
           ∫
Kn (s,t) =    Kn −1(s,t1)K (t1,t)dt    K1 (s,t) = K (s,t)         (4.184)
            B
which can be generalized to
            ∫

Kp+q (s,t) =    Kp (s,t1)Kq(t1,t)dt.                   (4.185)
             B
This representation is known as iterated kernels. The resulting series is known as a Neumann series or the resolvent series of the Equation 4.179 and takes the form
                                    2         ∞∑             i
R(s,t,λ) = K1 (s,t) + K2 (s,t)λ +  K3(s,t)λ  + ...=     Ki+1 (s,t)λ .       (4.186)
                                              i=0
Thus the expression of Equation 4.181 may also be expressed as
         ∫                             ∞  ∫
                                      ∑                i
ϕ(s)=f(s) + λ  B R(s,t,λ)f (t)dt = f (s ) + λ     B Ki+1(s,t)λ f(t)dt.     (4.187)
                                      i=0