Namespaces
Variants
Actions

Mechanical quadrature, method of

From Encyclopedia of Mathematics
Revision as of 17:23, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

method of mechanical cubature

A method for solving integral equations, based on replacing an integral by a sum using quadrature (cubature) formulas. Consider the equation

(1)

where is a bounded open domain. Using a quadrature (cubature) process

one forms the system of linear equations

(2)

where , .

Let the absolute term and the kernel be continuous on and , respectively ( is the closure of ), and let (1) have a unique solution . Let as for any continuous function on . Then for sufficiently large the system (2) is uniquely solvable and

where and are positive constants and

as .

A mechanical quadrature method can be applied for the solution of non-linear integral equations [3] and eigen value problems for linear operators. The method converges even for a certain class of equations with discontinuous kernels [4].

References

[1] V.I. Krylov, V.V. Bobkov, P.I. Monastyrnyi, "Numerical methods" , 2 , Moscow (1977) (In Russian)
[2] I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)
[3] M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[4] G.M. Vainikko, "On the convergence of the method of mechanical quadratures for integral equations with discontinuous kernels" Sib. Math. J. , 12 : 1 (1971) pp. 29–38 Sibirsk. Mat. Zh. , 12 : 1 (1971) pp. 40–53
[5] S.G. [S.G. Mikhlin] Michlin, S. Prössdorf, "Singular integral operators" , Springer (1986) (Translated from German)


Comments

References

[a1] H. Brunner, P.J. van der Houwen, "The numerical solution of Volterra equations" , North-Holland (1986)
[a2] C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4
[a3] H. Engels, "Numerical quadrature and cubature" , Acad. Press (1980)
[a4] K.E. Atkinson, "A survey of numerical methods for the solution of Fredholm integral equations of the second kind" , SIAM (1976)
How to Cite This Entry:
Mechanical quadrature, method of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mechanical_quadrature,_method_of&oldid=29822
This article was adapted from an original article by G.M. Vainikko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article