Introduction to linear partial differential equations

The purpose of this article is to describe some elementary properties of second-order linear partial differential equations (p.d.e.) with constant coefficients, in other words, in the case of two variables, equations of the form : $a\frac{{\partial }^{2}u}{\partial x^2}+b\frac{{\partial }^{2}u}{\partial x\partial y}+c\frac{{\partial }^{2}u}{\partial y^2}+\alpha \frac{\partial u}{\partial x}+\beta \frac{\partial u}{\partial y}+\mathit{\gamma u}=F\left(x,\mathrm{\hspace{0.17em}}y\right)$

where a, b, c, α, β, γ denote six given real numbers (a, b, c being not all zero), F a continuous function of two real variables defined on an open U of the plane and u an unknown function, assumed to be of class C ² .

A priori, two types of problems can be distinguished:

• those in which the time variable t does not intervene, and which therefore depend only on the spatial variables x, y, z; these are called stationary problems;

• those in which, in addition to the spatial variables x, y and z, the time variable t is involved; these are called evolution problems.

• Most often, we're looking for solutions that satisfy boundary conditions, meaning that the considered solution u, a priori defined on the open U of the plane, satisfies certain conditions on the boundary of U. There are two types of boundary conditions: Dirichlet and Neumann.

The Dirichlet conditions require the solution u to be continuous on the adherence of U, i.e. on U and its boundary, and then to be equal to a given function on the boundary of U.

The Neumann conditions require the solution u to be continuous on the adherence of U, i.e. on U and its boundary, and to admit at any point on the boundary of U a derivative u/ N along the normal vector N directed outwards from the boundary of U (assumed sufficiently regular) equal to a given function.

In an evolution problem, we also look for solutions that satisfy certain...