I am kind of stuck in my proposal. For solving 2nd order DE's I need their symmetries. In every source on the ideas page as well as other papers I read much is written about solving the ODE after the symmetries have been extracted. I am not able to find a concise algorithm which helps me in getting my infinitesimals from ansatz.
For example: If I assume that the infinitesimals are bivariate in nature, what are the sufficient conditions that would make the guess true and then how to extract the infinitesimals ? I contacted @Manoj Kumar who wrote the code for 1st order DE's. He told me to get in touch with @Raoul for help. Can anyone guide me here ? On Wednesday, March 4, 2015 at 1:23:05 PM UTC+5:30, Mihir Wadwekar wrote: > > I have made a draft at: > > > https://github.com/sympy/sympy/wiki/GSoC-2015-Application-Mihir-Wadwekar:-Lie-Group-Methods-for-Second-Order-Differential-Equations > > <https://github.com/sympy/sympy/wiki/GSoC-2015-Application-Mihir-Wadwekar:-Lie-Group-Methods-for-Second-Order-Differential-Equations.> > > Will keep on adding implementation details of each method. I have tried to > put theory in brief and have avoided proofs to keep it short and simple. Is > it enough or is a deeper explanation required? Also do you see anything > wrong with the process flow ? > Do let me know any concerns. > > Thanks. > > > On Monday, March 2, 2015 at 6:07:31 AM UTC+5:30, Aaron Meurer wrote: >> >> The scope seems good. >> >> I believe there are other issues. Search the issue tracker. Some may >> be out of scope (like some issues with the systems solver, unless the >> Lie Group methods can be used to solve systems). >> >> Aaron Meurer >> >> On Sun, Mar 1, 2015 at 3:25 PM, Mihir Wadwekar <m.mi...@gmail.com> >> wrote: >> > Looking at Maple is exactly what I am doing. Both their online help and >> the >> > papers listed in the ideas page have helped me immensely in forming a >> way to >> > implement lie group methods. Will add a draft soon to the wiki page >> > discussing exactly how I plan to go ahead. Does the scope of the >> project >> > seem right ? I am not sure whether I would be able to do any other >> major >> > different topic along with lie groups for this summer. Maybe there are >> some >> > other smaller issues with dsolve that can be synced along with this >> proposal >> > ? >> > >> > >> > On Monday, March 2, 2015 at 12:37:47 AM UTC+5:30, Aaron Meurer wrote: >> >> >> >> I don't know enough about Lie Group methods to comment on how they >> >> should be implemented, but looking at how Maple does it (and reading >> >> the papers referenced in their docs) is definitely a good place to >> >> start. The current ODE module is heavily influenced by Maple (at >> >> least the parts I wrote are). >> >> >> >> Aaron Meurer >> >> >> >> On Fri, Feb 27, 2015 at 3:45 PM, Mihir Wadwekar <m.mi...@gmail.com> >> wrote: >> >> > Hi, >> >> > >> >> > I am Mihir Wadwekar, a 3rd year undergrad pursuing computer science >> at >> >> > IIIT >> >> > Hyderabad. I have been tinkering with Sympy for some time now and >> have >> >> > committed 4 patches in it. I do understand how the codebase works >> and >> >> > would >> >> > like to work upon the ODE module as part of my GSOC 2015 project. >> >> > >> >> > I am planning to extend the solving of 1st-order differential >> equations >> >> > using lie groups to also include 2nd order differential equations. >> >> > >> >> > Currently for first-order differential equation, infinitesimals are >> >> > generated which fit the linearized symmetry condition. Canonical >> >> > co-ordinates are derived from these infinitesimals by solving >> relatively >> >> > simpler PDE's. These coordinates on substitution make the equation a >> >> > problem >> >> > of quadrature. After solving it, the original variables are >> substituted >> >> > back >> >> > to get the solution. >> >> > >> >> > Finding a pair of infinitesimals is the most difficult part of it. >> There >> >> > is >> >> > no easy way to generate them yet and as a result various intelligent >> >> > guesses >> >> > are made and tried out till one fits the symmetry condition and >> makes >> >> > the >> >> > ODE invariant. Similar procedure can be used for 2nd order >> differential >> >> > equations. However their symmetry condition and the type of guesses >> for >> >> > the >> >> > infinitesimals is different. Also the infinitesimals here have an >> >> > additional >> >> > variable( dy/dx ). >> >> > >> >> > The 2nd order part can be built upon the previous structure. There >> is a >> >> > function 'infinitesimals' which generates infinitesimals by calling >> all >> >> > heuristics functions. This function can be changed to recognize the >> >> > order of >> >> > the ODE and call the heuristic functions of that ODE. Some >> heuristics >> >> > are >> >> > common for both ODE's, but still would require some changes due to >> the >> >> > presence of an additional variable in 2nd order equations. Other new >> >> > guesses >> >> > would be written from scratch. A brief summary of the guesses for >> 2nd >> >> > order >> >> > is provided here. >> >> > >> >> > After generating infinitesimals for 2nd order there are multiple >> ways to >> >> > approach the remaining part. Solving through canonical coordinates >> as >> >> > done >> >> > in 1st order eqautions is one of the way. Since it is already >> >> > implemented >> >> > for 1st order, I plan to go forward with this approach. However >> >> > canonical >> >> > variables may sometimes cause computation failures due to inverse >> >> > transformations and hence I also want to implement the method of >> first >> >> > integrals. >> >> > >> >> > A lot of depth can be added here. More intelligent guesses for 1st >> and >> >> > 2nd >> >> > order equations can also be implemented. Lie groups is a vast field >> and >> >> > could itself be a proper module. Your views on the complexity would >> >> > really >> >> > help as there is a lot of material to read. >> >> > >> >> > It would be a great boost for ODE solving module if 2nd order >> >> > differential >> >> > equations can also be solved through lie groups. Using lie groups is >> a >> >> > more >> >> > generic method and does not require any special classification of >> ODE. >> >> > This >> >> > can form a base for higher-degree ODE's. >> >> > >> >> > I do have a vision for implementing this is depth but would like to >> know >> >> > your initial thoughts on this before I go ahead. >> >> > >> >> > Thanks. >> >> > >> >> > -- >> >> > 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 sympy+un...@googlegroups.com. >> >> > To post to this group, send email to sy...@googlegroups.com. >> >> > Visit this group at http://groups.google.com/group/sympy. >> >> > To view this discussion on the web visit >> >> > >> >> > >> https://groups.google.com/d/msgid/sympy/d618eb30-1fd2-401e-9956-c2a629e22510%40googlegroups.com. >> >> >> >> > For more options, visit https://groups.google.com/d/optout. >> > >> > -- >> > 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 sympy+un...@googlegroups.com. >> > To post to this group, send email to sy...@googlegroups.com. >> > Visit this group at http://groups.google.com/group/sympy. >> > To view this discussion on the web visit >> > >> https://groups.google.com/d/msgid/sympy/33c69676-185e-490d-acff-2b5bec1dbc70%40googlegroups.com. >> >> >> > >> > For more options, visit https://groups.google.com/d/optout. >> > -- 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 sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/50aa856e-ec9e-4c63-9ea4-f14e8573223a%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
