Hello:
I am PhD student of the Kiev Institute of Mathematics, National
Academy of Science, Ukraine. I have an honors master's degree in
mathematical physics and computer sciences.
Currently in applied math department I study the theory of Lie groups
and it's application to the differential equations. The most recent
works however concerned with supersymmetry of Schrödinger-Pauli
equation of arbitrary dimension. The obtained results widely extend
the number of exactly solvable Quantum Mechanic problems as well as
generalize the approach of finding such problems.
I am interested in participating in Sympy 2011 GsoC project. I have
required mathematical skills as well as good knowledge of different
CAS (Mathematica, Maple, Sage) and experience to write extensions for
them. The proposed ideas were interesting for me and I would be glad
to help with most of them, especially with ones concerned with
integration, series, ordinary and partial differential equation, Lie
groups, supersymmetry and quantum mechanics.
I would also like to propose my own ideas which I found helpful in my
study.
1. Implement functions to be able to collect similar members of
expression containing functions and to gather coefficients of linearly
independent members.
e.g. expression
x*(a+b*y+sin(y))+y*(c+3*x+x^2)+x*sin(y+3)+x*cos(y)+f(x,y,t)
should be translated to
a*x+(b+c)*y+3*x*y+x^2*y+(1+cos(3))*x*sin(y)+(1+sin(3))*x*cos(y)
+f(x,y,t)
and functions and coefficients should be gathered
[a, b+c, 3, 1, 1+cos(3), 1+sin(3), 1]
[x, y, x*y, x^2*y, x*sin(y), x*cos(y), f(x,y,t)]
where x, y are variables and a, b, c, t - unknown constants
That is, to do the same things that collect and coeffs functions do
with polynomials.
2. Implement case wise methods for expressions depending on
parameters.
e.g. Solve ODE containing constant parameters
diff(f(r),r)=a*f(r)+b*r+c
has two significantly different solutions
f(r)=C1*exp(a*r)-b*r/a-(b+a*c)/a^2 if a != 0
f(r)=b*r^2/2+c*r+C1 if a = 0
where r - independent variable, f - dependent variable, a, b, c -
arbitrary constants
This problems become extremely painful, when we have bunch of more
complex equations and expressions. Unfortunately there are no
solutions in commercial CAS, such as Maple, Mathematica and MatLab as
well as in open source such as Maxima and Sage. Actually Symbolic
Tools Box (former MuPad) in MatLab tried to solve the second problem,
but the solution does not work well. Mentioned problems are connected
with each other and with improvement of pattern matching problem in
Sympy. I have complete solution for Maple and porting it to
Mathematica. I think it won't be a problem to implement this algorithm
in Sympy.
Also I want to concentrate attention on some small problems, solution
of which can improve usability of Sympy
e.g. It is possible to simplify usage of dsolve function in case when
dependent and independent variables are obvious.
The equation
eq=diff(y,x)+a*y+b*x+c
has y as dependent variable and x as independent, so it is better to
write
dsolve(eq)
The solution is complete (as patch for Sage, but it can be easily
adopt to Sympy).
I think that it can be interesting to implement this algorithms in
Sympy in terms of the GSoC project.
If you are interested in my proposal please contact me and I will send
you precise plan and if not, I will be glad to participate in one of
the proposed projects.
Best regards,
Yuri Karadzhov
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