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meaningful solution to the differential eqn
Posted:
Mar 6, 2013 5:54 AM


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 wellbehaved and a reasonable solution. But when V[t] is a piecewise function, either my own, or one of the builtin 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 halfcycle. Integrate[] sees these constants in the Piecewise statement and simplemindedly 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 halfcycle of the square wave. What Integrate should yield is a positivegoing ramp on the + halfcycle, and a decreasing positivevalued ramp on the  halfcycle of the square wave. The correct function should be the sum of SquareWave[t]*dt over t. The integrals created over each (continuous, differentiable) halfcycle 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. IEE S.A. ZAE Weiergewan, 11, rue Edmond Reuter, L5326 Contern, LUXEMBOURG
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