Timeline for Using Fourier Series to acquire Nonlinear ODE Periodic Solutions
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Jul 5, 2017 at 18:56 | comment | added | Bazinga | Alright, then I was thinking exactly the same thing. Therefore, it is more of a geometric derivation of qp and I totally can understand it. Thank you. | |
| Jul 5, 2017 at 18:54 | comment | added | bbgodfrey | @Mitscaype Look at the second figure, and you will realize why the distance from the origin to the turning point is a quarter period. | |
| Jul 5, 2017 at 18:52 | comment | added | Bazinga | I see, now it seems more reasonable. And then you connect the turning point with the quarter period of the solution. Is that a formula? I found some stuff in wiki about the quarter period elliptic function but it does not mention the turning point as a perquisite to find qp. Again, thanks for taking the time to explain. | |
| Jul 5, 2017 at 14:30 | comment | added | bbgodfrey | @Mitscaype Cases[f[[1]], ArcSin[z_] -> z, Infinity, 1] extracts the argument of ArcSin from the left side of f. That argument must lie between -1 and 1 for the ArcSin to have a real value. Thus turning points occur when the argument of ArcSin has one of those two values. I chose 1 for convenience. | |
| Jul 5, 2017 at 11:37 | comment | added | Bazinga | Hello and excuse this late comment of mine. I was going through the steps again, since I would like to implement what we did here to another problem and I noticed that as you say the turning point is given by: Solve[First@Cases[f[[1]], ArcSin[z_] -> z, Infinity, 1] == 1, x][[1, 1]] . Can you please explain why f[[1]], after you solved it for ArcSin[z_]==z, has to be equal to 1 for it to be a turning point? I am actually not familiar with elliptic integrals and since the problem was about Fourier I did not think I would need it. But thats the beauty of it in math, isn't it? :) | |
| May 14, 2017 at 22:43 | history | edited | bbgodfrey | CC BY-SA 3.0 | added definition of qp |
| May 12, 2017 at 21:11 | comment | added | bbgodfrey | @Mitscaype I cannot answer your question with certainty without seeing your code. With respect to your earlier question, the frequency and first three coefficients of the Fourier solution corresponding to c -> 2 above are {w -> 0.996162, a[1] -> 1.282,a[3] -> -0.000326235, a[5] -> 5.9772*10^-6 }. This very fast convergence is consistent with numerical comparisons I also have made. I shall add these results to the answer in an hour or so. | |
| May 12, 2017 at 19:21 | comment | added | Bazinga | Interestingly enough, Mathematica gives an error of maximum stepsize when I tried to solve the ODE, for the same ICs with NDSolve. Why would this happen? The solution does exist in an implicit form so I am guessing some problem with the domain of $x(t)$? | |
| May 12, 2017 at 12:56 | comment | added | Bazinga | Thank you very much. :) | |
| May 12, 2017 at 12:45 | comment | added | bbgodfrey | @Mitscaype The solutions provided in my answer all are periodic, and all periods (or frequencies) are allowed. Because the equation is nonlinear, the waveform depends on the frequency. I shall look at the Fourier amplitudes question later today. | |
| May 12, 2017 at 5:20 | comment | added | Bazinga | So if I understand it correctly, the fact that you could compute a period means that there are indeed periodic solutions to the problem, right? I really would like the procedure to acquire the Fourier $A_n$ along with $\omega$ and then plot this solution to the one acquired by DSolve. Again, many thanks for your time and help. Anytime tomorrow is fine :) | |
| May 12, 2017 at 5:10 | comment | added | bbgodfrey | @Mitscaype The Elliptic functions are part of the solution obtained by DSolve. They are not uncommon in ODEs that can be reduced to first integrals of the motion. I found the period by computing the point at which x'[t] vanished. Given the results in my answer, it would not be difficult to obtain the corresponding Fourier coefficients, but I felt that you would not need them, given the complete answer. I can provide answers to other questions, if any, tomorrow. It is late here now. | |
| May 12, 2017 at 5:04 | comment | added | Bazinga | Thank you for taking the time to provide an answer. Nevertheless, I have some questions. At the part where you use {DSolve} you have some Elliptic functions involved. How did this come up? Also, the way I understand it, is that you did not took at all the Fourier approximation of $x(t)$, but you found period in a different way, is that correct? | |
| May 11, 2017 at 20:50 | history | edited | bbgodfrey | CC BY-SA 3.0 | added second plot and corresponding explanation |
| May 11, 2017 at 20:42 | history | edited | bbgodfrey | CC BY-SA 3.0 | added second plot and corresponding explanation |
| May 11, 2017 at 20:28 | history | answered | bbgodfrey | CC BY-SA 3.0 |