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''computational model''
 
''computational model''
  
A typical problem used as a model for investigating and developing numerical methods for some class of problems. For example, in the theory of [[Quadrature|quadrature]] the problem of calculating integrals of functions satisfying a condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643501.png" /> is considered. The processing of methods for the solution of the [[Cauchy problem|Cauchy problem]] for systems of ordinary differential equations historically was done by investigating the properties of the methods on models from a sequence of increasing complexity (with integration interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643502.png" />):
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A typical problem used as a model for investigating and developing numerical methods for some class of problems. For example, in the theory of [[Quadrature|quadrature]] the problem of calculating integrals of functions satisfying a condition $  | f ^ { ( n) } | \leq  A $
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is considered. The processing of methods for the solution of the [[Cauchy problem|Cauchy problem]] for systems of ordinary differential equations historically was done by investigating the properties of the methods on models from a sequence of increasing complexity (with integration interval $  [ 0 , X ] $):
  
1) the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643503.png" />;
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1) the equation $  y  ^  \prime  = 0 $;
  
2) the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643505.png" /> of order 1 (models 1) and 2) correspond to the problem of integration on small time intervals of systems with smooth solutions);
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2) the equation $  y  ^  \prime  = m y $,  
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$  | m | X $
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of order 1 (models 1) and 2) correspond to the problem of integration on small time intervals of systems with smooth solutions);
  
3a) the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643507.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643508.png" />; this model corresponds to the problem of integration on large time intervals of systems with stable solutions;
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3a) the equation $  y  ^  \prime  = m y $,  
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$  m < 0 $,  
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$  | m | X \gg 1 $;  
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this model corresponds to the problem of integration on large time intervals of systems with stable solutions;
  
3b) the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m0643509.png" />; a model of an equation with singularities in the derivatives of solutions;
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3b) the equation $  y  ^  \prime  = x  ^  \lambda  $;  
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a model of an equation with singularities in the derivatives of solutions;
  
4) the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m06435010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m06435011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m06435012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m06435013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064350/m06435014.png" /> of order 1; a model of so-called stiff differential systems (cf. [[Stiff differential system|Stiff differential system]]), in which one component varies relatively slowly and the other rapidly.
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4) the system $  y _ {1}  ^  \prime  = m _ {1} y _ {1} $,  
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$  y _ {2}  ^  \prime  = m _ {2} y _ {2} $,  
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0 > m _ {1} > m _ {2} $,  
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$  | m _ {2} | X \gg | m _ {1} | X $,  
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$  | m _ {1} | X $
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of order 1; a model of so-called stiff differential systems (cf. [[Stiff differential system|Stiff differential system]]), in which one component varies relatively slowly and the other rapidly.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) {{MR|0362811}} {{ZBL|0524.65001}} </TD></TR></table>

Latest revision as of 08:01, 6 June 2020


computational model

A typical problem used as a model for investigating and developing numerical methods for some class of problems. For example, in the theory of quadrature the problem of calculating integrals of functions satisfying a condition $ | f ^ { ( n) } | \leq A $ is considered. The processing of methods for the solution of the Cauchy problem for systems of ordinary differential equations historically was done by investigating the properties of the methods on models from a sequence of increasing complexity (with integration interval $ [ 0 , X ] $):

1) the equation $ y ^ \prime = 0 $;

2) the equation $ y ^ \prime = m y $, $ | m | X $ of order 1 (models 1) and 2) correspond to the problem of integration on small time intervals of systems with smooth solutions);

3a) the equation $ y ^ \prime = m y $, $ m < 0 $, $ | m | X \gg 1 $; this model corresponds to the problem of integration on large time intervals of systems with stable solutions;

3b) the equation $ y ^ \prime = x ^ \lambda $; a model of an equation with singularities in the derivatives of solutions;

4) the system $ y _ {1} ^ \prime = m _ {1} y _ {1} $, $ y _ {2} ^ \prime = m _ {2} y _ {2} $, $ 0 > m _ {1} > m _ {2} $, $ | m _ {2} | X \gg | m _ {1} | X $, $ | m _ {1} | X $ of order 1; a model of so-called stiff differential systems (cf. Stiff differential system), in which one component varies relatively slowly and the other rapidly.

References

[1] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) MR0362811 Zbl 0524.65001
How to Cite This Entry:
Model for calculations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Model_for_calculations&oldid=14160
This article was adapted from an original article by N.S. Bakhvalov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article