I think that a project like this could tie in quite well with a separate
Lie group/lie algebra module, especially if we have something that would
store a lot of information about some of the most common Lie groups
(SL(2,C), SU(2), SO(3), etc).
On Sunday, 24 February 2013 11:38:41 UTC, Manoj Kumar wrote:
>
> Hello Everyone,
>
> I've been trying to solve bugs related to the ODE Solver .I would like my
> proposal to be in this area since I've got a general idea of the source
> code. I found the following features missing in ode.py. I could possibly
> integrate these ideas into a single project called "Improving ODE solver in
> sympy" for my SOC proposal.
>
> Most of my references are from Differential Equations with Applications
> abd Historical Notes from George F. Simmons
>
> 1 . Solving a set of homogenous equations with constant coefficients:
>
> Right now sympy has no support for solving a system of differential
> equations. I would like to implement a method by which a two system diff.
> equation can be solved when there are constant coefficients. If a function
> of t is also present in the system of differential equations, then try
> using variation of parameters to solve the system.The actual session would
> be something like this. I thought of trying to implement it in dsolve, but
> wouldn't it make things a bit messy? So maybe use another function called
> dsolve_system(?)
>
> >>> from sympy import *
> >>> from sympy.abc import x, y, t
> >>> f1 = Derivative(x, t)
> >>> f2 = Derivative(y, t)
> >>> eq1 = f1 - 3*x - (4*y)
> >>> eq2 = f2 - x + y
> >>> dsolve_system(eq1, eq2)
> Output: [x = 2 * C1 * exp(t) + C2 * (1 + 2*t) * exp(t) , y = C1 * exp(t) +
> C2 * t * exp(t)]
> >>> eq1 = f1 + 3*sin(x) - (4*y)
> >>> dsolve_system(eq1, eq2)
> Output: ValueError: dsolve_system can work only when there are constant
> coefficients.
>
> I have studied methods to implement this only when there are constant
> coefficients, if there are methods to do this when the coefficients are
> functions of x and y do let me know i'll read them up.
>
> 2.Power series solution of homogenous second order linear equations:
> The user can input the point about which the power series solution can be
> found. If not the point can be assumed to be zero.
>
> Implementing it in dsolve might be a pain, because the user would want the
> equation upto n terms and sometimes the nth term.
> So, define a SeriesSolve class, evaluate a general term and return when a
> method called term is called (The simplest way I could think of, feel free
> to criticize me)
> Something like this, maybe
>
> >>> from sympy.solvers.ode import SeriesSolve
> Lets look at various cases.
> >>> f = Function('f')(x)
> >>> d1 = Derivative(f, x)
> >>> d2 = Derivative(f, x, x)
> >>> eq = (1 - x**2)*d2 - (2*x)d1 + 6 * f
> >>> series = SeriesSolve(eq)
> >>> series.term(0)
> Output: [A0 , A1]
> >>> series.term(2)
> Output: [-3 *A0 , - 2 * A1 / 3)
> >>> series = SeriesSolve(eq, point = 2)
> >>> series.term(0)
> Output : [A0 , A1]
> >>> series.term(2)
> Output: [A0 , -2*A1/3]
> >>> series.equation(n)
> #Can give the first n terms of both the series.
>
> And if the given point is singular check if it is regular, If regular
> attempt to find the Frobenius series solution.
> >>> eq = (2*x**2)*d2 + x * (2*x + 1)*d1 - f
> >>> series = SeriesSolve(eq)
> >>> series.term(1)
> Output: [(-2 / 5)*A0 , - A1]
> >>> series.equation(n)
> Same goes, first n terms of both the series
>
> If the roots of the indicial equation are equal, return a ValueError. If
> the second series cannot be found, just return the first series.
>
> If point is not regular, return ValueError
> >>> eq = (2*x**3)*d2 + x * (2*x + 1)*d1 - f
> Output: ValueError: Point is irregular singular, power series solution
> cannot be found.
>
> And if the equation is of the form of a Gauss HyperGeometric one, apply
> the short-cut , transform it into a GHE , and get the general solution.
>
> If possible try and implement initial value conditions. Since till now
> there seems to be no support for values like f'(0) I suppose the easiest
> way would be
>
> >>> equation = SeriesSolve(eq, point = x, (point1 , value of function) ,
> (point2, value of derivative of function))
>
> * This would be a bit tough in the case where there is a three recursion
> formula, where I'm not sure but I shall verify this as soon as possible.
>
> * In cases like Frobenius Series Solution, sometimes there might be only
> one series, then simply ignore the second condition?
>
> 3.Implementing Laplace transform to solve equations of second degree with
> initial conditions.
>
> Laplace transforms could be used in a simpler way to solve the above
> mentioned equations.
>
> This could be again implemented in dsolve, but since initial conditions
> aren't specified, this could again be a big pain.
>
> >>> from sympy.solvers.ode import laplace_eq
> >>> laplace_eq(d2 - 4*d1 + 4*f , (0 , 0) , (0, 3))
> Output:
> 3 * x * exp(2*x)
>
> This could be extended to nth order, but supplying initial conditions and
> finding laplace inverse would then become difficult
>
> These are the ideas I had done as part of my coursework last semester.
> Please do tell me , if they are trivial(which I hope not) , places where I
> could improve, maybe new methods that I could try and read up to fit in a
> period of ten weeks , so that the possibilty of me getting selected could
> increase.
>
> Regards,
> Manoj Kumar,
> Mech Undergrad.
> BPGC
> Blog <http://manojbits.wordpress.com>
>
>
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