Integrating Conditionals/piecewise cont. functions

• To: mathgroup at smc.vnet.net
• Subject: [mg26221] Integrating Conditionals/piecewise cont. functions
• From: "Richard Lindenberg" <georgepburdell at hotmail.com>
• Date: Sun, 3 Dec 2000 18:05:54 -0500 (EST)
• Organization: University of Illinois at Urbana-Champaign
• Sender: owner-wri-mathgroup at wolfram.com

```I am trying to do some simplified finite elements that simply requires the
integration of some roof functions (i.e. piecewise continous functions that
are 0 everywhere except where they ramp up and then down in a specified
domain). I was hoping that I could do everything symbolically, but
Mathemtica refuses to integrate every conditional I give it. I can get it to
do the numerically, but it comes with baggage. This is what I have right
now:

These are roof functions differentiated

Piecewise continuous
b'[x_, n_] :=
Which[x < n, 0, n <= x <= n + 1, 1, n + 1 < x < n + 2, -1, x >= n + 2, 0]

Another way to look at it
v'[x_, n_] := 0 /; x < n
v'[x_, n_] := 1 /; n <= x <= n + 1
v'[x_, n_] := -1 /; n + 1 < x < n + 2
v'[x_, n_] := 0 /; x >= n + 2

I want to simply do this...

Integrate[v'[x,m] v'[x,n]]  for m=n=1...5 (or something)

(If you copy the below item into Mathematica you will see exactly, I
think...)

Essentially I am trying to make a matrix full of the these functions. The
numerical integrate seems to work, but keeps spitting up after trying to
integrate integrands of zero. I suppose if I could turn this off that would
be good. The other stuff in the table function just makes it a tridiagonal.

\!\(Table[
Switch[i - j, \(-1\),
N[\[Integral]\_0\%5\((\(v'\)[x, i]\ \(v'\)[x, j])\)
\[DifferentialD]x],
0, N[\[Integral]\_0\%5\((\(v'\)[x, i]\ \(v'\)[x,
j])\) \[DifferentialD]x], 1,
N[\[Integral]\_0\%5\((\(v'\)[x, i]\ \(v'\)[x,
j])\) \[DifferentialD]x], _, 0], {i, 5}, {j, 5}]\)

Appreciate any help/ideas...

Thanks,
Rich Lindenberg
UIUC

```

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