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A method for the approximate solution of one-dimensional Fredholm integral equations of the second kind (cf. also [[Fredholm equation|Fredholm equation]]; [[Fredholm equation, numerical methods|Fredholm equation, numerical methods]]), based on replacing the kernel in a special way by a degenerate kernel, evaluating the resolvent of the degenerate equation and then improving the approximate solution through the use of a rapidly-convergent iterative algorithm.

Let the original integral equation be written as

$$ \tag{1 }
\phi ( x) - \lambda \int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  =  f ( x) .
$$

To construct the degenerate kernel, divide the square

$$ 
\{ a \leq  x \leq  b , a \leq  s \leq  b \}
$$

into  $  N $
strips

$$ 
\left \{ b- 
\frac{a}{N}
 i \leq  x \leq  b- 
\frac{a}{N}

( i + 1 ) , a \leq  s \leq  b \right \} ,\ \ 
i = 0 \dots N - 1 .
$$

In each strip, say the  $  i $-
th, the function  $  K ( x , s ) $
is approximated in the mean square, or uniformly, by functions

$$ 
K _ {i} ( x , s )  =  C _ {i} ( x) + P _ {i} ( x) Q _ {i} ( s) .
$$

In the simplest case,

$$ 
K _ {i} ( x , s )  =  K ( \xi _ {i} , s ) ,\ \ 
\xi _ {i} \in \left [ b- 
\frac{a}{N}
 i , b- 
\frac{a}{N}
 ( i + 1 ) \right ] .
$$

The function  $  K _ {i} ( x , s ) $
is now used to construct a [[Degenerate kernel|degenerate kernel]]:

$$ \tag{2 }
K _ {N} ( x , s )  =  \sum _ { i= 0} ^ { N- 1} [ \widehat{C}  _ {i} ( x) +
\widehat{P}  _ {i} ( x) Q _ {i} ( s)] ,
$$

$$ 
\widehat{P}  _ {i} ( x)  =  \left \{ 
\begin{array}{ll}
P _ {i} ( x) ,  &\ 
x \in \left [ b- 
\frac{a}{N}
 i , b- 
\frac{a}{N}
 ( i + 1 ) \right ] ,  \\
0,  & x \notin \left [ b- 
\frac{a}{N}
 i , b- 
\frac{a}{N}
 ( i + 1 ) \right ] ,  \\
\end{array}
\right.
$$

$$ 
\widehat{C}  _ {i} ( x)  =  \left \{ 
\begin{array}{ll}
C _ {i} ( x) ,  &\ 
x \in \left [ b- 
\frac{a}{N}
 i , b- 
\frac{a}{N}
 ( i + 1 ) \right ] ,  \\
0,  & x \notin \left [ b- 
\frac{a}{N}
 i , b- 
\frac{a}{N}
 ( i + 1 ) \right ] .  \\
\end{array}
\right.
$$

The solution of the equation with the degenerate kernel (2) approximates the solution of equation (1), generally, the better the larger the number  $  N $
of strips and the better the approximation of  $  K ( x , s ) $
in each strip is. The approximate solution  $  \phi _ {0} ( x) $
can be further improved by using the iterative algorithm

$$ \tag{3 }
\phi _ {k} ( x) - \lambda \int\limits _ { a } ^ { b }  K _ {N} ( x , s ) \phi _ {k} ( s)  d s =
$$

$$ 
= \ 
f ( x) + \lambda \int\limits _ { a } ^ { b }  [ K ( x , s ) -
K _ {N} ( x , s ) ] \phi _ {k-1} ( s)  d s .
$$

The iterations (3) converge in the mean square, or uniformly, to the solution of equation (1), provided the kernels  $  K _ {N} ( x , s ) $
approach  $  K ( x , s ) $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.N. Polozhii,   P.I. Chalenko,   "The strip method for solving integral equations"  ''Dop. Akad. Nauk UkrSSR'' :  4  (1962)  pp. 427–431  (In Ukrainian)  ((English abstract.))</TD></TR></table>

====Comments====
Excellent surveys on the numerical solution of Fredholm equations of the second kind can be found in [[#References|[a1]]] and [[#References|[a2]]]; the strip method is not mentioned there, although other degenerate kernel methods are discussed, cf. also [[Degenerate kernels, method of|Degenerate kernels, method of]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.E. Atkinson,   "A survey of numerical methods for the solution of Fredholm integral equations of the second kind" , SIAM  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.H. Baker,   "The numerical treatment of integral equations" , Clarendon Press  (1977)  pp. Chapt. 4</TD></TR></table>