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Discrete Laplacian

`L = del2(U)`

`L = del2(U,h)`

`L = del2(U,h1,...,hN)`

returns
a discrete approximation of Laplace’s differential
operator applied to `L`

= del2(`U`

)`U`

using the default
spacing, `h = 1`

, between all points.

If the input `U`

is a matrix, the interior
points of `L`

are found by taking the difference
between a point in `U`

and the average of its four
neighbors:

$${L}_{ij}=\left[\frac{\left({u}_{i+1,j}+{u}_{i-1,j}+{u}_{i,j+1}+{u}_{i,j-1}\right)}{4}-{u}_{i,j}\right]\text{\hspace{0.17em}}.$$

Then, `del2`

calculates the values on the edges
of `L`

by linearly extrapolating the second differences
from the interior. This formula is extended for multidimensional `U`

.

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