Hello Maxima.
I am a new member of the mailing list, so please forgive me if I do something stupid.
I have become very interested in the Homotopy Analysis Method, which is a powerful tool for analyzing nonlinear systems. The basic principle is to use homotopy to continuously transform a guess solution into the correct one.
In the case of differential equations, one begins by constructing a power series in the homotopy parameter q with an unknown function of x as the coefficient of q^k. One then repeatedly takes the homotopy derivative of both sides, thus building a series of recurrence relations from which the unknown functions of x can be built. The solution to the problem is the homotopy series when q=1.
Here is my problem: I wish to automate the homotopy derivative, whose properties can be derived from the ordinary derivative. Here is the definition of the homotopy derivative: Let y be a homotopy series.
D[m](y)=1/m!d^m(y)/dq^m | q=0
The main property here is that the nth homotopy derivative of a homotopy series is the nth function of x in that series.
There are, theoretically, infinitely many of these functions. I've tried constructing the series using Maxima's sum function. The problem is that an arbitrary power series like that is meaningless to maxima, and the homotopy derivative, as I've defined it, just returns zero every time.
To start things off, how could I represent an undetermined power series in Maxima (do I necessarily have to truncate the series first? I don't have to when doing calculations by hand, so why should I?)
-- thank you for your patients,
Casey