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Topic: meaningful solution to the differential eqn
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Alexei Boulbitch

Posts: 483
Registered: 2/28/08
meaningful solution to the differential eqn
Posted: Mar 6, 2013 5:54 AM
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I have been trying to get a meaningful solution to the differential eqn. for the voltage and current in an inductor:
V[t] == -L i'[t]. For smooth voltage waveforms, the current function is well-behaved and a reasonable solution. But when V[t] is a piecewise function, either my own, or one of the built-in ones, such as SquareWave, the solutions obtained are unreasonable. They may satisfy the differential equation, but they increase in magnitude without limit. I tried integrating the SquareWave[t] Mathematica function, and I could see why DSolve was probably having a problem. The square wave has values +1 or -1 on each half-cycle. Integrate[] sees these constants in the Piecewise statement and simple-mindedly replaces them with +t and -t. So as t gets larger or smaller, the integral(s) yield ramp functions that alternate in sign during every half-cycle of the square wave. What Integrate should yield is a positive-going ramp on the + half-cycle, and a decreasing positive-valued ramp on the - half-cycle of the square wave. The correct function should be the sum of SquareWave[t]*dt over
t. The
integrals created over each (continuous, differentiable) half-cycle should then be added to create a new function. I can't figure out how to implement Integrate on piecewise functions like this. Does anyone know how?

P.S. I couldn't find a way to search MathGroup postings from the group's pages, so I may be posting an already solved problem. If so, please let me know how I can use the group more efficiently.

Yes somebody knows. This:

Plot[SquareWave[x], {x, 0, 5}]

is the plot of a SquareWave. Evaluate it. And this:

f[z_] := Integrate[SquareWave[x], {x, 0, z}];

is a definition of the integral of the square wave. One may now plot it. Evaluate this:

Plot[f[z], {z, 0, 5}]

It behaves exactly as you described it should.

Alexei BOULBITCH, Dr., habil.
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