As examples of elliptic partial differential equations , we consider the Laplace equation , Poisson equation , and Helmholtz equation . Recall that the Laplacian of the function u(x,y) is . With this notation, we can write the Laplace, Poisson, and Helmholtz equations in the following forms: It is often the case that the boundary values for the function u(x,y) are known at all points on the sides of a rectangular region R in the plane. In this case, each of these equations can be solved by the numerical technique known as the finite-difference method.