Difference between revisions of "Quadrature-sum method"
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) {{MR|0106537}} {{ZBL|0083.35301}} </TD></TR></table> |
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| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4 {{MR|0467215}} {{ZBL|0373.65060}} </TD></TR></table> |
Revision as of 17:00, 15 April 2012
A method for approximating an integral operator by constructing numerical methods for the solution of integral equations. The simplest version of a quadrature-sum method consists in replacing an integral operator, for instance of the form
 |
in an integral equation
 |
by an operator with finite-dimensional range, according to the rule
 | (1) |
The integral equation, in turn, is approximated by the linear algebraic equation
 |
On the right-hand side of the approximate equation (1) is a quadrature formula for the integral with respect to 
. Various generalizations of (1) are possible:
 | (2) |
where the 
are certain functions constructed from the kernel 
. The quadrature-sum method as generalized in the form (2) can be applied for the approximation of integral operators with singularities in the kernel and even of singular integral operators.
References
| [1] | L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) MR0106537 Zbl 0083.35301 |
Comments
References
| [a1] | C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4 MR0467215 Zbl 0373.65060 |
Quadrature-sum method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadrature-sum_method&oldid=16202