Domain of influence

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)