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Difference between revisions of "Bateman method"

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A method for approximating the integral operator of a one-dimensional integral Fredholm equation of the second kind; it is a particular case of the method of degenerate kernels (cf. [[Degenerate kernels, method of|Degenerate kernels, method of]]). In Bateman's method, the degenerate kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015370/b0153701.png" /> is constructed according to the rule:
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<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/b/b015/b015370/b0153702.png" /></td> </tr></table>
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<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/b/b015/b015370/b0153703.png" /></td> </tr></table>
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A method for approximating the integral operator of a one-dimensional integral Fredholm equation of the second kind; it is a particular case of the method of degenerate kernels (cf. [[Degenerate kernels, method of|Degenerate kernels, method of]]). In Bateman's method, the degenerate kernel  $  K _ {N} (x, s) $
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is constructed according to the rule:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015370/b0153704.png" />, are certain points on the integration segment of the integral equation considered. The method was proposed by H. Bateman [[#References|[1]]].
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$$
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K _ {N} (x, s) =
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$$
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$$
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= \
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\frac{\left |
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\begin{array}{cccc}
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0 &K (x,\
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s _ {1} )  &\dots  &K (x, s _ {N} )  \\
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K(x _ {1} , s)  &K (x _ {1} , s _ {1} )  &\dots  &K (x _ {1} , s _ {N} )  \\
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\dots  &\dots  &\dots  &\dots  \\
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K(x _ {N} , s)  &K (x _ {N} , s _ {1} )  &\dots  &K (x _ {N} , s _ {N} )  \\
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\end{array}
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\right | }{\left |
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\begin{array}{ccc}
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K(x _ {1} , s _ {1} )  &\dots  &K(x _ {1} , s _ {N} )  \\
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\dots  &\dots  &\dots  \\
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K(x _ {N} , s _ {1} )  &\dots  &K(x _ {N} , s _ {N} )  \\
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\end{array}
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\right | }
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,
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$$
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where  $  s _ {i} , x _ {i} , i = 1 \dots N $,  
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are certain points on the integration segment of the integral equation considered. The method was proposed by H. Bateman [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman,  ''Messeng. Math.'' , '''37'''  (1908)  pp. 179–187</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.V. Kantorovich,  V.I. Krylov,  "Approximate methods of higher analysis" , Noordhoff  (1958)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman,  ''Messeng. Math.'' , '''37'''  (1908)  pp. 179–187</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.V. Kantorovich,  V.I. Krylov,  "Approximate methods of higher analysis" , Noordhoff  (1958)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bateman,  ''Proc. Roy. Soc. A''  (1922)  pp. 441–449</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Bateman,  ''Proc. Roy. Soc. A''  (1922)  pp. 441–449</TD></TR></table>

Latest revision as of 10:33, 29 May 2020


A method for approximating the integral operator of a one-dimensional integral Fredholm equation of the second kind; it is a particular case of the method of degenerate kernels (cf. Degenerate kernels, method of). In Bateman's method, the degenerate kernel $ K _ {N} (x, s) $ is constructed according to the rule:

$$ K _ {N} (x, s) = $$

$$ = \ - \frac{\left | \begin{array}{cccc} 0 &K (x,\ s _ {1} ) &\dots &K (x, s _ {N} ) \\ K(x _ {1} , s) &K (x _ {1} , s _ {1} ) &\dots &K (x _ {1} , s _ {N} ) \\ \dots &\dots &\dots &\dots \\ K(x _ {N} , s) &K (x _ {N} , s _ {1} ) &\dots &K (x _ {N} , s _ {N} ) \\ \end{array} \right | }{\left | \begin{array}{ccc} K(x _ {1} , s _ {1} ) &\dots &K(x _ {1} , s _ {N} ) \\ \dots &\dots &\dots \\ K(x _ {N} , s _ {1} ) &\dots &K(x _ {N} , s _ {N} ) \\ \end{array} \right | } , $$

where $ s _ {i} , x _ {i} , i = 1 \dots N $, are certain points on the integration segment of the integral equation considered. The method was proposed by H. Bateman [1].

References

[1] H. Bateman, Messeng. Math. , 37 (1908) pp. 179–187
[2] L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)

Comments

References

[a1] H. Bateman, Proc. Roy. Soc. A (1922) pp. 441–449
How to Cite This Entry:
Bateman method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bateman_method&oldid=45996
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article