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I have a linear ODE derived from electrical engineering of the form:

$$\omega\cos(\omega t) - W\left(\frac{e^{\omega\cos(\omega t)}}{C}\right) = \frac{Ai'(t)}{i(t)}$$

Where A, C are constants, and W is the Lambert W function.

Wolfram Alpha returns the solution in the form of an integral:

$$i(t) = c_1\exp\int_{1}^{t}\frac{\omega\cos(\omega \zeta) - W\left(\frac{e^{\omega\cos(\omega \zeta)}}{C}\right)}{A}d\zeta$$

However when I insert a third constant term, B, on the RHS as:

$$\omega\cos(\omega t) - W\left(\frac{e^{\omega\cos(\omega t)}}{C}\right) = \frac{Ai'(t)}{i(t) + B}$$

The system seems unable to assist with a solution.

However it seems that when C is small with respect to $\omega$ that

$$W\left(\frac{e^{\omega\cos(\omega t)}}{C}\right)$$

this term seems to behave more-or-less as a periodic sinusoid with an added constant/"DC offset."

I'm hoping for an approximation to the expression involving the W function when constant C is small with respect to omega that could allow for a more practical explicit solution to the previous ODE in that regime.

MattyZ
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    Is anything known about $\omega$ and $C$ separately? For instance, is it possible that $C$ is fixed and $\omega$ is arbitrarily large? If you're only considering the case when $\omega$ is fixed and $C$ is small, there is a simple approximation $$i(t) = C_0 (-C \ln C)^{t/A} - B.$$ – Maxim Sep 03 '19 at 00:09

1 Answers1

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That still has a solution with the $B$ added, just follow the same steps that got you the first solution. This is a separable equation, so we get

$$\frac{\omega \cos (\omega t) - W\left(\frac{e^{\omega \cos (\omega t)}}{C} \right)}{A} = \frac{i'}{i+B}$$

$$\implies \int_0^t \frac{\omega \cos (\omega \zeta) - W\left(\frac{e^{\omega \cos (\omega \zeta)}}{C} \right)}{A} d\zeta + K = \log|i + B|$$

$$i(t) = K\exp\left( \int_0^t \frac{\omega \cos (\omega \zeta) - W\left(\frac{e^{\omega \cos (\omega \zeta)}}{C} \right)}{A} d\zeta \right) - B$$

Ninad Munshi
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  • Thanks. It would still be nice if there was a way to simplify the solution or original ODE such that the integral is simplified or eliminated as it's not entirely obvious (to me at least) how this solution behaves with respect to variation of the parameter omega in the domain I mentioned. – MattyZ Sep 02 '19 at 20:04
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    @Bitrex Letting $C\ll \omega$ wouldn't solve your problem either because then the Lambert W term becomes infinite – Ninad Munshi Sep 02 '19 at 20:08
  • This approximation for W in $0 < x < e$ may work for my needs https://math.stackexchange.com/questions/26682/approximation-for-lambert-w-function-near-zero?noredirect=1&lq=1 – MattyZ Sep 02 '19 at 21:18
  • @Bitrex how can that be if you say $C \ll \omega$ ? Or are your considerations different from that? – Ninad Munshi Sep 02 '19 at 21:21
  • Not precisely as written but when $z = \frac{e^{\omega\cos(\omega t)}}{C}$, $\log(1 + z)$ times some constant scale factor looks to be a decent approximation to the $W(\frac{e^{\omega\cos(\omega t)}}{C})$ when $C≪ω$ (I cannot seem to @ you currently, sorry about that) – MattyZ Sep 02 '19 at 22:12
  • @Bitrex I don't think you're understanding. That approximation does not apply! It's only valid for small x, but if $C \ll \omega$, then $x=\frac{e^{\omega \cos \omega \zeta}}{C} \to \infty$ – Ninad Munshi Sep 02 '19 at 22:25
  • for these two example particular choices of C and omega does the "fit" of these plots break down, somewhere? e.g. https://imgur.com/a/fe2fb5I – MattyZ Sep 02 '19 at 23:06
  • @Bitrex have a look at these plots. You can see where the approximation only holds for small $z$. On the other hand, there is an asymptotic expression for large $z$, $W(z) \approx \log z + O(\log \log z)$. It seems like for large $\omega$, the $z$ actually swings between very large and very small values because of the cosine. So no one approximation should work. Choose unsymmetric value for $\omega$ and $C$. Also, physically, do those two parameters have the dimension? Otherwise how are we comparing them? – Ninad Munshi Sep 02 '19 at 23:18
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    @Bitrex and here is another plot that disproves your approximation working, which I got by choosing unsymmetric values for the parameters. I don't think an approximation is possible because of the swing given by the cosine. – Ninad Munshi Sep 02 '19 at 23:27
  • looks like it works "okay" for some specific values but probably no way to find a general method to fit them. It's alright, I'm not married to the idea. The W function under the integral isn't much fun, though. – MattyZ Sep 02 '19 at 23:37
  • The equation describes how current flows through a certain circuit involving a capacitor and diode so omega is units of angular frequency and C is capacitance. 1 Farad (C = 1) is a very large unrealistic value of capacitance, might be more like omega = 1e6 and C = 1e-9 – MattyZ Sep 02 '19 at 23:40