# Model for calculations

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=47867
This article was adapted from an original article by N.S. Bakhvalov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article