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Cauchy problem

From Biocrawler, the free encyclopedia.

Consider a smooth hypersurface $S\in\mathbb{R}^n$ having a continuous, non-tangential direction field described by unitary vectors $\hat{\mathbf{d}(\mathbf{x})},\mathbf{x}\in S$ , i.e.

$\hat{\mathbf{d}(\mathbf{x})}\cdot\hat{\mathbf{n}(\mathbf{x})}\neq0, \forall\mathbf{x}\in S$

where $\hat{\mathbf{n}(\mathbf{x})}$ is the unitary vector perpendicular to $S$ .

The general Cauchy problem, consists on finding the solution $u$ of a $κ -$ order differential equation that also satisfies the conditions:

$u(\mathbf{x})=f_0(\mathbf{x}), \mathbf{x}\in S$

$\frac{\part^m u(\mathbf{x})}{\part d^m}=f_m(\mathbf{x}), m=1,\ldots,\kappa-1$

where $f m$ are given functions defined on surface $S$ (Cauchy Surface).

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Cauchy problem

From Biocrawler, the free encyclopedia.

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