Knowlpedia

Definition

Principal part of a differential operator

The sum of the highest-order derivative terms in a linear differential operator.

If a has order mm,

L=∑∣α∣≤maα(x)∂α,L=\sum_{|\alpha|\le m}a_\alpha(x)\partial^\alpha,

its principal part is

Lm=∑∣α∣=maα(x)∂α.L_m=\sum_{|\alpha|=m}a_\alpha(x)\partial^\alpha.

The remaining terms have lower differential order.

Example and conventions

For L=∂x2+2∂x∂y+3∂y2+b(x,y)∂x+c(x,y)L=\partial_x^2+2\partial_x\partial_y+3\partial_y^2+b(x,y)\partial_x+c(x,y), the first three terms form the principal part. In an evolution equation, authors may use a parabolic order that counts one time derivative like two space derivatives; that convention must be declared.

An asymptotic argument may also call its leading approximation a “principal operator.” Leading size in a small parameter and highest differential order are different criteria and need not select the same terms.

References