 Linear partial differential equation

A linear partial differential equation is a partial differential equation in which the unknown function and its partial derivatives occur linearly, with coefficients independent of the unknown. Its homogeneous solutions obey the superposition principle.

For example, the heat equation permits adding independently decaying Fourier modes. For an equation $Lu=f$ with a fixed nonzero PDE source term, adding two solutions gives $L(u+v)=2f$, so the sum generally solves a different problem. Likewise, adding solutions adds their initial conditions and boundary conditions.