Timeline for Acausality in solving time-domain inhomogeneous differential equations with Fourier transforms?
Current License: CC BY-SA 3.0
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| Oct 5, 2012 at 6:48 | history | edited | Qmechanic♦ | CC BY-SA 3.0 |
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| Oct 5, 2012 at 6:37 | history | edited | Qmechanic♦ | CC BY-SA 3.0 |
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| Oct 5, 2012 at 0:12 | vote | accept | Deep Blue | ||
| Oct 5, 2012 at 7:21 | |||||
| Oct 5, 2012 at 0:09 | comment | added | Deep Blue | I'll see if I can get that result this evening. | |
| Oct 4, 2012 at 23:39 | comment | added | Qmechanic♦ | Right, after you impose the appropriate initial condition, then your FT formula for $Q(t)$ must be equivalent to the manifestly causal formula in my answer. | |
| Oct 4, 2012 at 23:20 | comment | added | Deep Blue | I agree, but does that solution not solve the original equation? Since the equation is causal, I expect the solution to maintain that property. I remember from school that when solving wave equations in classical EM, a delta function pops out during integration and enforces this causality (this raises an interesting question of what would happen when it is solved on a finite interval where the wavenumbers are discrete, but that's for another time) - but I can't see a way that such a causality-ex-machina would show up in any general problem. | |
| Oct 4, 2012 at 23:08 | comment | added | Qmechanic♦ | The very first step: to use an object (the FT) that depends on the far future, cf. section I. | |
| Oct 4, 2012 at 23:01 | comment | added | Deep Blue | I agree that the problem may be solved like you have wrote, but what step goes wrong in the FT approach that seems to break causality? If the solution I outlined above it valid, what would happen if I try to evaluate it for some input function? I suspect that if I switch the order of integration, the first (homogeneous) solution term you wrote may pop out from the transform solution, but I've yet to check it out. Also, as I said in, I'm interested in this difficulty in a more general situation, and only used the RC circuit to illustrate the problem. | |
| Oct 4, 2012 at 20:50 | history | edited | Qmechanic♦ | CC BY-SA 3.0 |
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| Oct 4, 2012 at 20:41 | history | edited | Qmechanic♦ | CC BY-SA 3.0 |
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| Oct 4, 2012 at 20:06 | history | answered | Qmechanic♦ | CC BY-SA 3.0 |