Thanks for your reply. So do I start formally working on my proposal (Solving first order ODE's using Lie Groups), that is creating a wiki page for it. And if I do, who is most likely to be my mentor? . Because I've seen quite a few good proposals not accepted last time, because of a need of a mentor.
On Fri, Mar 8, 2013 at 1:01 PM, Aaron Meurer <[email protected]> wrote: > On Thu, Mar 7, 2013 at 2:58 AM, Manoj Kumar > <[email protected]> wrote: > > Hello again. > > > > Sorry to bring this post up again, but I thought it would be good to keep > > all my ideas on a single thread. > > > > I've been reading about lie groups and their application in solving > > differential equations and I believe I've got my concepts right about > > solving a linear first order differential equation. Maple, (according to > the > > papers cited in the SoC ideas pages), has two separate papers dedicated > to > > solving differential equations of the first and second order using lie > > groups. I would like to like to code a clone for solving first order > > differential equations using lie groups in sympy for my SoC proposal. > > > > Theory: > > There are many differential equations that cannot be solved using the > > present dsolve in ode.py. Suppose there is a differential equation of the > > form > > (dr / ds) = f(r), since the slope is independent of s, moving in the s > > direction would merge one solution into another. The principal aim of > using > > lie groups is to reduce any given differential equation to the above > > mentioned form. > > > > The coordinates r and s are called canonical coordinates because any lie > > group transformation which leaves the differential equation invariant > would > > transform it into the form (r, s + alpha) where r remains invariant.If we > > are able to find such coordinates, then the equation becomes separable. > > > > The problem is finding these canonical co-ordinates. This can be broken > into > > four steps(This is for first order differential equations) > > > > 1. Find η and ξ , the infinitesimals using this partial diffential > equation, > > which > > are basically the differentials of the transformed co-ordinates with > respect > > to > > the transformation parameter. > > > > ηx - ξy h2 + (ηy - ξx )h - (ξhx + ηhy ) = 0 > > > > This should be the toughest part of the project. As you can see solving a > > linear first order ODE, leads to solving a much more complicated PDE. > > However in this PDE, some assumptions could be made for the > infinitesimals. > > > > Maple has 6 separate algorithms just for finding the infinitesimals as > > described in the paper. Once this is done. > > > > 2. rx ξ + ry η = 0, > > sx ξ + sy η = 1, > > > > Solving these two equations would give, r and s , this can be done by. > dx / > > ξ = dy / η = r(x,y) and dx / ξ = ds > > > > 3. After this substituting r and s in > > ds / dr = sx + h(x,y)sy / rx + h(x, y)ry and simplifying in terms of r > and s > > would give a separable form. > > > > 4. And after solving the differential equation in terms of r and s, > > substituting back r and s in terms of x and y. > > > > I haven't focussed on the implementation part yet, but I think it should > be > > structured similar to dsolve, or should it be implemented in dsolve. > > > > I've focussed on only first order differential equations here, even Maple > > also has support only upto the second order.I feel this project would be > > challenging for me and also a addition to the ODE solver of sympy. > > > > @Aaron, Stefan, and Smichr Please do give your feedback. > > I haven't formally studied Lie groups and their applications to > solving ODEs, but from my understanding, they are indeed a powerful > method for doing so. This would definitely be a welcome project for > GSoC. > > Aaron Meurer > > > > > > > -- > > You received this message because you are subscribed to the Google Groups > > "sympy" 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/sympy?hl=en. > > For more options, visit https://groups.google.com/groups/opt_out. > > > > > > -- > You received this message because you are subscribed to a topic in the > Google Groups "sympy" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/sympy/C4btttnGJss/unsubscribe?hl=en. > 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/sympy?hl=en. > For more options, visit https://groups.google.com/groups/opt_out. > > > -- Regards, Manoj Kumar, Mech Undergrad. BPGC Blog <http://manojbits.wordpress.com> -- You received this message because you are subscribed to the Google Groups "sympy" 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/sympy?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
