# 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.