Cauchy boundary condition

In mathematics, a Cauchy (French: [koʃi]) boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. It is named after the prolific 19th-century French mathematical analyst Augustin Louis Cauchy.

Cauchy boundary conditions are simple and common in second-order ordinary differential equations,

For partial differential equations, Cauchy boundary conditions specify both the function and the normal derivative on the boundary. To make things simple and concrete, consider a second-order differential equation in the plane

Notice the difference between a Cauchy boundary condition and a Robin boundary condition. In the former, we specify both the function and the normal derivative. In the latter, we specify a weighted average of the two.

We would like boundary conditions to ensure that exactly one (unique) solution exists, but for second-order partial differential equations, it is not as simple to guarantee existence and uniqueness as it is for ordinary differential equations. Cauchy data are most immediately relevant for hyperbolic problems (for example, the wave equation) on open domains (for example, the half plane).[1]