Bateman method
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 |
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