DefinitionThe second is Laplaces equation, which is a second-order partial differential equation: ∇2ϕ=0, where the laplacian is ∇2. It models steady-state heat conduction, electrostatics, and incompressible fluid flow. Harmonic functions- solutions to Laplaces equation- occur in the study of various phenomena in physics and engineering. For example, in two dimensions (x and y), Laplaces equation is ∂ϕ∂x2+∂ϕ∂y2=0. A solution is ϕ(x,y)=x2−y2.