DIRICHLET PROBLEM

in mathematics, the problem of formulating and solving certain partial differential equations that arise in studies of the flow of heat, electricity, and fluids. Initially, the problem was that of determining the equilibrium temperature distribution on a disk from measurements taken along the boundary. The temperature at points inside the disk must satisfy a partial differential equation called the Laplace equation corresponding to the physical condition that the total heat energy contained in the disk shall be a minimum. A slight variation of this problem occurs when there are points inside the disk at which heat is added (sources) or removed (sinks) as long as the temperature still remains constant at each point (stationary flow), in which case Poisson's equation is satisfied. The Dirichlet problem can also be solved for any simply connected regioni.e., containing no holesif the temperature varies gradually along the boundary. In the related Neumann problem, heat is supplied and removed across the boundary in such a way as to maintain a stationary temperature distribution. In Robin's problem, heat is merely allowed to be lost through radiation across the boundary at a rate proportional to the temperature drop across it, resulting in the eventual stabilization of the temperature distribution. Aside from heat flow, there are other phenomena that result in similar mathematical formulations, as in electrical charge distribution and steady fluid flow. These are special cases of the more general boundary-value problems of the class of second-order partial differential equations called elliptic equations.