meaningful solution to the differential eqn

*To*: mathgroup at smc.vnet.net*Subject*: [mg120044] meaningful solution to the differential eqn*From*: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>*Date*: Wed, 6 Mar 2013 05:56:12 -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. 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, L-5326 Contern, LUXEMBOURG Office phone : +352-2454-2566 Office fax: +352-2454-3566 mobile phone: +49 151 52 40 66 44 e-mail: alexei.boulbitch at iee.lu