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Boundary value problems are similar to initial value problems.A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
Cauchy boundary conditions are simple and common in second-order ordinary differential equations, ″ = ((), ′ (),), where, in order to ensure that a unique solution () exists, one may specify the value of the function and the value of the derivative ′ at a given point =, i.e.,
Explicit formulas for eigenvalues and eigenvectors of the second derivative with different boundary conditions are provided both for the continuous and discrete cases. In the discrete case, the standard central difference approximation of the second derivative is used on a uniform grid.
Let be a domain (an open and connected set) in .Let be the Laplace operator, let be a bounded function on the boundary, and consider the problem: {() =, = (),It can be shown that if a solution exists, then () is the expected value of () at the (random) first exit point from for a canonical Brownian motion starting at .
Spectral theory for second order ordinary differential equations on a compact interval was developed by Jacques Charles François Sturm and Joseph Liouville in the nineteenth century and is now known as Sturm–Liouville theory. In modern language, it is an application of the spectral theorem for compact operators due to David Hilbert.
Points satisfying the barrier condition are called regular points of the boundary for the Laplacian. These are precisely the points at which one is guaranteed to obtain the desired boundary values: as x → y , u ( x ) → φ ( y ) {\displaystyle x\rightarrow y,u(x)\rightarrow \varphi (y)} .
For a system of the form (,, ′) =, some sources also require that the Jacobian matrix (,,) be non-singular in order to call this an implicit ODE [system]; an implicit ODE system satisfying this Jacobian non-singularity condition can be transformed into an explicit ODE system.
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.