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I'm having a hard time finding some references on series solutions for "nonlinear" ODE's, the most I could find was a small excerpt on Wikipedia.

https://en.wikipedia.org/wiki/Power_series_solution_of_differential_equations

Most books just say something along the lines of ... and the method is applicable to nonlinear ODE's. But none I've seen go into detail let alone an example. Can anyone suggest me a good book or reference (in particular for 2nd order nonlinear ODEs)?

Thanks

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  • $\begingroup$ Are you looking for more than a local existence and uniqueness theorem? If not, the statement and proof of the Cauchy-Kovalevski theorem for PDE's applies directly to a system of first order nonlinear ODE's. Any 2nd order nonlinear ODE can be "prolonged" into a system of first order nonlinear ODE's. $\endgroup$
    – Deane Yang
    Commented Sep 29, 2010 at 3:07
  • $\begingroup$ Thanks, I'm actually not concerned too much about the theory at the moment. I want to see how the recurrence relations turn out. I know these will be nonlinear but would like to get "comfortable" with them. If that makes any sense. I am staring at an example in a paper where the author applies series methods. Turns out he gets a cubic recurrence relation and intstead of substituting a power series with all powers of x, he substitutes a power series with odd values only, I was wondering why this is. $\endgroup$
    – AUK1939
    Commented Sep 29, 2010 at 3:26
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    $\begingroup$ In case you need a package to play with, you may want to try FriCAS, and look at the routines seriesSolve (to obtain the first few coefficients), guessRec (to guess a recurrence) and guessADE (to guess a differential equation). Please email [email protected] or myself for more info. $\endgroup$
    – Martin Rubey
    Commented Sep 29, 2010 at 5:08

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Nonlinear differential equations is hard to find good references on-partly due to the difficulty of the subject and partly due to the highly specialized nature of most of the research problems connected with them. But a lot of these problems are really problems of numerical approximation-so I think you'll have greater luck if you begin searching the literature on THAT,math.

A very good book to start with that has a lot of great material on this is Atkinson and Kan's Theoretical Numerical Analysis. Not only is it terrifically written and comprehensive with lots of examples,it's one of the most scholarly texts I've ever seen with complete and opiniated references. I think you'll find this book's references will give you a great deal of direction for further study on nonlinear solution of ODE's.

An older book that has a lot of nice material on power series and other numerical methods for ODE's is Einar Hille's Lectures On Ordinary Differential Equations. Why most of Hille's texts-which are all wonderful-are out of print mystifies me.

That should help you get started,especially the Atkinson/Han book. Good hunting!

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  • $\begingroup$ Thanks for the reference, its an excellent book ... i've been looking for a book like this for a long time. $\endgroup$
    – AUK1939
    Commented Oct 25, 2010 at 2:30
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I recommend the book of Bender and Orszag "Advanced mathematical methods for scientists and engineers". The first chapter is a whirlwind review of "exact methods for ODE" which includes Frobenius series and other standard tricks, and has a section on techniques specific to nonlinear ODE. The rest of the book describes perturbation theory and other approximation techniques which are more broadly applicable in practice and in my opinion often more "physically" enlightening (that is, often providing useful reasons "why" solutions to an equation behave the way they do).

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  • $\begingroup$ As much as I dislike engineering maths books, this one was quite helpful, even has a short section on non linear recurrences. $\endgroup$
    – AUK1939
    Commented Oct 25, 2010 at 2:29
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I have found another nonlinear example at sosmath.com here but it appears to contain mistakes.

Also you can play with the examples of power series solutions in Maple using dsolve with the series option.

Some further examples could probably be found in connection with the study of the Painleve property and the Painleve test of integrability for (systems of) ODEs, see e.g. the book Integrability and Nonintegrability of of dynamical systems (full text available at this URL) by Alain Goriely and references therein.

Another useful keyword to search for is formal solution instead of the power series solution, although the convergence will probably not be discussed in the context of formal solutions.

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