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''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

\begin{equation}\label{1}
x(t) =  \int\limits_\Omega K(t,s) x(s) \, ds + y(t),
\end{equation}

where $ \Omega \subset \mathbf{R}^n $ is a bounded open domain. Using a quadrature (cubature) process

\[ \int\limits_\Omega z(s) \, ds = \sum_{j=1}^{n} \alpha_{jn} z(s_{jn}) + \phi_n(z) \]

one forms the system of linear equations

\begin{equation}\label{2}
x_{in} = \sum_{j=1}^{n}\alpha_{jn} K(s_{in}, s_{jn})x_{jn} + y(s_{in}), \quad i = 1, \dots, n,
\end{equation}

where $ x_{in} \approx x(s_{in}) $, $ i = 1, \dots, n $.

Let the absolute term $ y $ and the kernel $ K $ be continuous on $ \overline\Omega $ and $ \overline\Omega\times \overline\Omega $, respectively ($ \overline\Omega $ is the closure of $ \Omega $), and let \eqref{1} have a unique solution $ x(t) $. Let $ \phi_n(z) \to 0 $ as $ n \to \infty $ for any continuous function $ z(t) $ on $ \overline\Omega $. Then for sufficiently large $ n $ the system \eqref{2} is uniquely solvable and

\[ c_1 \epsilon_n \le \max_{1\le i \le n}|x_{in} - x(s_{in})| \le c_2 \epsilon_n, \quad n \ge n_0, \]

where $ c_1 $ and $ c_2 $ are positive constants and

\[ \epsilon_n = \max_{1\le i\le n} |\phi_n (K(s_{in}, s)x(s))| \to 0 \]

as $ n\to \infty $.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Krylov,   V.V. Bobkov,   P.I. Monastyrnyi,   "Numerical methods" , '''2''' , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.S. Berezin,   N.P. Zhidkov,   "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Krasnosel'skii,   G.M. Vainikko,   P.P. Zabreiko,   et al.,   "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.G. [S.G. Mikhlin] Michlin,   S. Prössdorf,   "Singular integral operators" , Springer  (1986)  (Translated from German)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Brunner,   P.J. van der Houwen,   "The numerical solution of Volterra equations" , North-Holland  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.T.H. Baker,   "The numerical treatment of integral equations" , Clarendon Press  (1977)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Engels,   "Numerical quadrature and cubature" , Acad. Press  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K.E. Atkinson,   "A survey of numerical methods for the solution of Fredholm integral equations of the second kind" , SIAM  (1976)</TD></TR></table>