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Integrating piecewise functions.
*To*: mathgroup at smc.vnet.net
*Subject*: [mg120024] Integrating piecewise functions.
*From*: mrkhrdr at gmail.com
*Date*: Tue, 5 Mar 2013 04:20:05 -0500 (EST)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*Delivered-to*: l-mathgroup@wolfram.com
*Delivered-to*: mathgroup-newout@smc.vnet.net
*Delivered-to*: mathgroup-newsend@smc.vnet.net
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.
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