Mechanical quadrature, method of

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

\[ x(t) = \int\limits_\Omega K(t,s) x(s) \, ds + y(t), \]

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

\[ x_{in} = \sum_{j=1}^{n}\alpha_{jn} K(s_{in}, s_{jn})x_{jn} + y(s_{in}), \quad i = 1, \dots, n, \]

where $ x_{in} \approx x(x_{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 (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 (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 [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)

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)