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)

References

[a1]	H. Bateman, Proc. Roy. Soc. A (1922) pp. 441–449