Definition:Mutually Consistent Boundary Conditions

This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.

Definition

Let $\map {\mathbf y} x$, $\map {\boldsymbol \psi} {\mathbf y}$ be an $n$-dimensional vectors.

Consider the system of differential equations:

$(1): \quad \mathbf y = \map {\mathbf f} {x, \mathbf y, \mathbf y'}$

Let derivatives of $\mathbf y$ satisfy:

$\bigintlimits {\mathbf y'} {x \mathop = x_1} {} = \bigintlimits {\map {\boldsymbol \psi^{\paren 1} } {\mathbf y} } {x \mathop = x_1} {}$

$\bigintlimits {\mathbf y'} {x \mathop = x_2} {} = \bigintlimits {\map {\boldsymbol \psi^{\paren 2} } {\mathbf y} } {x \mathop = x_2} {}$

If every solution of $(1)$ satisfying conditions at $x = x_1$ automatically satisfies conditions at $x = x_2$ (or vice versa), then these boundary conditions are called mutually consistent.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 6.31$: Consistent Boundary Conditions. General Definition of a Field