Difference between revisions of "Domain of influence"
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− | ''of a point | + | {{TEX|done}} |
+ | ''of a point $M$ (of a set $A$ of points $M$)'' | ||
− | The set | + | The set $B(M)$ (respectively, $B(A)$) of all points at which the solution of a differential equation or of a set of differential equations changes as a result of a perturbation of it at the point $M$ (or at the set $A$). In the simplest cases of linear partial differential equations the domain of influence is independent of the solution; for most non-linear problems the domain of influence depends both on the solution itself and on the nature of the perturbations. In such a case infinitely small perturbations are considered. For hyperbolic equations, the domain of influence of the point $M$ is often the union of the characteristic conoid (cf. [[Characteristic manifold|Characteristic manifold]]) passing through the point $M$ and its interior; for parabolic and elliptic equations, the domain of influence of the point $M$ is usually the domain of definition of the solution. |
Latest revision as of 21:43, 14 April 2014
of a point $M$ (of a set $A$ of points $M$)
The set $B(M)$ (respectively, $B(A)$) of all points at which the solution of a differential equation or of a set of differential equations changes as a result of a perturbation of it at the point $M$ (or at the set $A$). In the simplest cases of linear partial differential equations the domain of influence is independent of the solution; for most non-linear problems the domain of influence depends both on the solution itself and on the nature of the perturbations. In such a case infinitely small perturbations are considered. For hyperbolic equations, the domain of influence of the point $M$ is often the union of the characteristic conoid (cf. Characteristic manifold) passing through the point $M$ and its interior; for parabolic and elliptic equations, the domain of influence of the point $M$ is usually the domain of definition of the solution.
Comments
This notion is usually met in the context of initial value problems (also called Cauchy problems, cf. Cauchy problem) for first-order, or general hyperbolic, partial differential equations (cf. Differential equation, partial, of the first order; Hyperbolic partial differential equation). For a more precise discussion of the latter case see [a1], Chapt. 6 Par. 7.
A related notion is that of the domain of dependence (cf. Cauchy problem).
References
[a1] | R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) |
Domain of influence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Domain_of_influence&oldid=15452