Definition:Mutually Consistent Boundary Conditions
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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