Concept

# Cauchy boundary condition

Summary
In mathematics, a Cauchy (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 as 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. Second-order ordinary differential equations Cauchy boundary conditions are simple and common in second-order ordinary differential equations, :y''(s) = f\big(y(s), y'(s), s\big), where, in order to ensure that a unique solution y(s) exists, one may specify the value of the function y and the value of the derivative y' at a given point s=a, i.e., :y(a) = \alpha, and
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