I got the Maths of the thesis. For the application part and better understanding of what is to be done I should first get my hands on friCAS I suppose. Any other important thing I can look out?
On Wed, Feb 26, 2014 at 2:21 AM, Waldek Hebisch <[email protected]>wrote: > Varun Chopra wrote: > > Fricas-1.2.2 downloaded. > > Can you send the pdf on my mail. The link is directing but pdf is not > > opening. > > and how should I proceed now? Can there be any use of solving > differential > > equation model in simulink using MATLAB in this project? > > What should I read and learn ? > > > > I have sent you pdf in the mail, I hope you can read it. The project > is about _exact_ (symbolic) computations. We start from differential > operator L which with coefficient beeing rational functions of x. > The coefficient are known exactly (in simplest case rational functions > have rational coefficients which can be represented exactly in > the computer). Fundamental thing is that equaltion Lf = 0 > has n-dimensional solution space if we allow sufficiently many > f as solutions. One could try to find solutions among rational > functions, but many equations have no rational solutions. > Larger class is elementary functions (build from rational > functions, exp and log by composition) or Liouvillian > functions (beside rational functions, exp and log one can > use integration to express them). There are differential > equation which have no Liouvillian solutions, so we need larger > class of solutions. A lot of equations can be solved by > power series. However, even power series are not enough. > For example f' - 2*x*f = 0 has x^(1/2) as solutions, > which is not a normal power series at 0 (it needs fractional > exponents). f' + (1/x^2)*f = 0 has exp(1/x) as a solutions > and exp(1/x) has no power series expansion at 0. Sometimes > solutions have logarithmic singularity. When > fractional (possibly irrational) power are allowed then > there are n linearly independent solutions of form > > f = e*\sum log(x)^i*p_i > > where e = exp(r) where r is rational function of 1/k for some > integer k, and p_i are power series in fractional powers > (Puiseux series). This looks similar to traditional theorem > about existence of solutions to differential equations, but > here coeffients of L may have pole at 0, so most traditonal > results do not apply (they need smooth coeffiecients), and > we say nothing about convergence of p_i. Our power series > are formal power series, so we do only operations on coefficients. > > Generalized exponents capture e and leading term of power > series. They tell you how solutions behave close to x = 0. > And they contain important information about possible > factorizations of L. Generalized exponents may be found > by factoring L over ring of Puiseux series (that is we allow > as factors operators having series as coefficients). And > factors with series coefficients help find normal factors. > > BTW. All of the above is in van Hoeij's thesis. > > To implement van Hoeij's algorithms one needs to operate > with series. FriCAS has apropriate domains to do this. > Series are represented in lazy way, namely there is some > number of already computed coefficients and a rule to > compute next term. So on demand we can compute as many > coefficients as needed. Basic operations like addition, > inverse, multiplication are already done. van Hoeij's > algorithms will need some special operations on series. > > OK, I think it is enough for today. > > -- > Waldek Hebisch > [email protected] > > -- > You received this message because you are subscribed to a topic in the > Google Groups "FriCAS - computer algebra system" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/fricas-devel/JYqZq8rAHlM/unsubscribe. > To unsubscribe from this group and all its topics, send an email to > [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/fricas-devel. > For more options, visit https://groups.google.com/groups/opt_out. > -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/fricas-devel. For more options, visit https://groups.google.com/groups/opt_out.
