Cauchy boundary condition

Boundary-value problem in differential equations 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 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 ( y ( s ) , y ′ ( s ) , s ) , {\displaystyle y''(s)=f{\big (}y(s),y'(s),s{\big )},} where, in order to ensure that a unique solution y ( s ) {\displaystyle y(s)} exists, one may specify the value of the function y {\displaystyle y} and the value of the derivative y ′ {\displaystyle y'} at a given point s = a {\displaystyle s=a} , i.e., y ( a ) = α , {\displaystyle y(a)=\alpha ,} and y ′ ( a ) = β , {\displaystyle y'(a)=\beta ,} whe