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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.
 
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
 
Let the original integral equation be written as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\phi ( x) - \lambda \int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s = f ( x) .
 +
$$
  
 
To construct the degenerate kernel, divide the square
 
To construct the degenerate kernel, divide the square
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905102.png" /></td> </tr></table>
+
$$
 +
\{ a \leq  x \leq  b , a \leq  s \leq  b \}
 +
$$
 +
 
 +
into  $  N $
 +
strips
  
into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905103.png" /> strips
+
$$
 +
\left \{ b-
 +
\frac{a}{N}
 +
i \leq  x \leq  b-
 +
\frac{a}{N}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905104.png" /></td> </tr></table>
+
( i + 1 ) , a \leq  s \leq  b \right \} ,\ \
 +
i = 0 \dots N - 1 .
 +
$$
  
In each strip, say the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905105.png" />-th, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905106.png" /> is approximated in the mean square, or uniformly, by functions
+
In each strip, say the $  i $-
 +
th, the function $  K ( x , s ) $
 +
is approximated in the mean square, or uniformly, by functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905107.png" /></td> </tr></table>
+
$$
 +
K _ {i} ( x , s )  = C _ {i} ( x) + P _ {i} ( x) Q _ {i} ( s) .
 +
$$
  
 
In the simplest case,
 
In the simplest case,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905108.png" /></td> </tr></table>
+
$$
 +
K _ {i} ( x , s )  = K ( \xi _ {i} , s ) ,\ \
 +
\xi _ {i} \in \left [ b-
 +
\frac{a}{N}
 +
i , b-  
 +
\frac{a}{N}
 +
( i + 1 ) \right ] .
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s0905109.png" /> is now used to construct a [[Degenerate kernel|degenerate kernel]]:
+
The function $  K _ {i} ( x , s ) $
 +
is now used to construct a [[Degenerate kernel|degenerate kernel]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
K _ {N} ( x , s )  = \sum _ { i= } 0 ^ { N- }  1 [ \widehat{C}  _ {i} ( x) +
 +
\widehat{P}  _ {i} ( x) Q _ {i} ( s)] ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051011.png" /></td> </tr></table>
+
$$
 +
\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}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051012.png" /></td> </tr></table>
+
$$
  
The solution of the equation with the degenerate kernel (2) approximates the solution of equation (1), generally, the better the larger the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051013.png" /> of strips and the better the approximation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051014.png" /> in each strip is. The approximate solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051015.png" /> can be further improved by using the iterative algorithm
+
$$
 +
\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}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051017.png" /></td> </tr></table>
+
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
  
The iterations (3) converge in the mean square, or uniformly, to the solution of equation (1), provided the kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051018.png" /> approach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090510/s09051019.png" />.
+
$$ \tag{3 }
 +
\phi _ {k} ( x) - \lambda \int\limits _ { a } ^ { b }  K _ {N} ( x , s ) \phi _ {k} ( s)  d s =
 +
$$
 +
 
 +
$$
 +
= \
 +
f ( x) + \lambda \int\limits _ { a } ^ { b }  [ K ( x , s ) -
 +
K _ {N} ( x , s ) ] \phi _ {k-} 1 ( s)  d s .
 +
$$
 +
 
 +
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====
 
====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>
 
<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====
 
====Comments====

Revision as of 14:55, 7 June 2020


A method for the approximate solution of one-dimensional Fredholm integral equations of the second kind (cf. also Fredholm equation; 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:

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

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

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

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

[1] 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.))

Comments

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

References

[a1] K.E. Atkinson, "A survey of numerical methods for the solution of Fredholm integral equations of the second kind" , SIAM (1976)
[a2] C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4
How to Cite This Entry:
Strip method (integral equations). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strip_method_(integral_equations)&oldid=49451
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article