View source for 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632902.png" /> is a bounded open domain. Using a quadrature (cubature) process

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632903.png" /></td> </tr></table>

one forms the system of linear equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632904.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632906.png" />.

Let the absolute term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632907.png" /> and the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632908.png" /> be continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m0632909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329010.png" />, respectively (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329011.png" /> is the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329012.png" />), and let (1) have a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329013.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329014.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329015.png" /> for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329017.png" />. Then for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329018.png" /> the system (2) is uniquely solvable and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329019.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329021.png" /> are positive constants and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329022.png" /></td> </tr></table>

as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329023.png" />.

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>