View source for Domain of influence

''of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338101.png" /> (of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338102.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338103.png" />)''

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338104.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338105.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338106.png" /> (or at the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338107.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338108.png" /> is often the union of the characteristic conoid (cf. [[Characteristic manifold|Characteristic manifold]]) passing through the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d0338109.png" /> and its interior; for parabolic and elliptic equations, the domain of influence of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033810/d03381010.png" /> 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|Cauchy problem]]) for first-order, or general hyperbolic, partial differential equations (cf. [[Differential equation, partial, of the first order|Differential equation, partial, of the first order]]; [[Hyperbolic partial differential equation|Hyperbolic partial differential equation]]). For a more precise discussion of the latter case see [[#References|[a1]]], Chapt. 6 Par. 7.

A related notion is that of the domain of dependence (cf. [[Cauchy problem|Cauchy problem]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR></table>