nLab distributional solution of a PDE

Idea

Given a linear differential operator $P$, then a generalized solution or weak solution of the corresponding homogenous differential equation is a distribution $u$ (hence a “generalized function” via this prop) satisfying

$$P u = 0 \,,$$

where the differential operator on the left now acts via derivatives of distributions. This means that for all $b \in \mathcal{D}$ we have

$$u(P^\ast(b)) = 0 \,,$$

where $P^\ast$ is the formally adjoint differential operator.

This is such that in the case that $u = u_f$ happens to be a non-singular distribution given by an ordinary smooth function $f$, then $u_f$ is a generalized solution precisely if $f$ is an ordinary solution:

$$\begin{aligned} P u_f = 0 \;\; & \Leftrightarrow\;\; u_f(P^\ast b) = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D} \\ & \Leftrightarrow\;\; \int f P^\ast b \, dvol = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D} \\ & \Leftrightarrow\;\; \int (P f) b \, dvol = 0 \phantom{AAA}\text{for all}\, b \in \mathcal{D} \\ & \Leftrightarrow\;\; P f = 0 \end{aligned}$$

Similarly there are generalized solutions for the inhomogeneous equation, and in fact now the inhomogeneity may itself be a distribution. In particular for delta distribution-inhomogeneity

$$P u = \delta$$

the generalized solutions $u$ are the fundamental solutions or Green's functions of $P$.

Related concepts

• fundamental solution

• Green function

• formal adjoint differential operator

• Green hyperbolic differential equation

References

• Ram Kanwal, section 10.2 of Generalized Functions: Theory and Applications, Springer 2004