Cauchy boundary

In mathematics, a Cauchy (French: [koʃi]) boundary conditions 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 conditions. 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,

where, in order to ensure that a unique solution ${\displaystyle y(s)}$ exists, one may specify the value of the function ${\displaystyle y}$ and the value of the derivative ${\displaystyle y'}$ at a given point ${\displaystyle s=a}$, i.e.,

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