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''method of mechanical cubature''
 
''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
 
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>
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\begin{equation}\label{1}
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x(t) = \int\limits_\Omega K(t,s) x(s) \, ds + y(t),
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\end{equation}
  
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
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where $ \Omega \subset \mathbf{R}^n $ 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>
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\[ \int\limits_\Omega z(s) \, ds = \sum_{j=1}^{n} \alpha_{jn} z(s_{jn}) + \phi_n(z) \]
  
 
one forms the system of linear equations
 
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>
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\begin{equation}\label{2}
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x_{in} = \sum_{j=1}^{n}\alpha_{jn} K(s_{in}, s_{jn})x_{jn} + y(s_{in}), \quad i = 1, \dots, n,
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\end{equation}
  
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" />.
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where $ x_{in} \approx x(s_{in}) $, $ i = 1, \dots, n $.
  
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
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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
  
<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>
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\[ c_1 \epsilon_n \le \max_{1\le i \le n}|x_{in} - x(s_{in})| \le c_2 \epsilon_n, \quad n \ge n_0, \]
  
where <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
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where $ c_1 $ and $ c_2 $ 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>
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\[ \epsilon_n = \max_{1\le i\le n} |\phi_n (K(s_{in}, s)x(s))| \to 0 \]
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063290/m06329023.png" />.
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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]]].
 
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]]].

Latest revision as of 06:14, 2 June 2013


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 [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=17875
This article was adapted from an original article by G.M. Vainikko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article