GSoC – Mid term report

Hi,  I passed the midterm evaluations of GSoC, this is what I have accomplished till now.

1] Built the basic infrastructure of the pde module, and added hints that could solve general Partial Differential Equations with constant coefficients

2] Added a function, infinitesimals , that would try to find out the required infinitesimals of any given first order ODE. The following heuristics have been implemented.

a] abaco1_simple (Assumes one of the infinitesimals to be zero, and the other to be a function to x or y)
b] abaco1_product (Assumes one of the infinitesimals to be zero, and the other to be a product of a function of x and y)
c] abaco2_similar (Assumes both infintesimals to be a function of x or y)
d] abaco2_unique_unknown (This is when, one infinitesimals is a function of x and the other to be a function of y)
e] abaco2_unique_general (This is a more general case of the above mentioned hint)
f] linear (Infinitesimals are of the form ax + by + c
g] bivariate (Infinitesimals are bivariate, more general form of the above mnetioned hint)
h] function_sum (When the infinitesimals are the sum of two functions)
i] chi (Finds a polynomial \chi , which helps in calculating the infinitesimals directly)

3] Added a function checkinfsol , which helps in checking if the infinitesimals are the actual solutions to the PDE.

As far as this week went, I couldn’t do much but I managed to add a hint which helps in solving Linear PDE’s with variable coefficients. The general form of such a PDE is

a(x, y)\frac{\partial u}{\partial x} + b(x, y)\frac{\partial u}{\partial y} + c(x, y)u(x, y) = d(x, y) , where a(x, y), b(x, y), c(x, y) and d(x, y) are arbitrary functions in x and y.  This can be done using the method of characteristics. However a simpler way, according to a paper that I skimmed through is, to convert the PDE into an ODE of one variable. The change of coordinates is \xi is the constant in the solution of the differential equation \frac{dy}{dx} = \frac{b}{a} and \eta = x (I don’t know why surely though) is selected such that the Jacobian doesn’t become zero. This is the Pull request,

TODO’s for this week

1. Get the heuristics PR merged.
2. Try integrating the PDE hint with the ODE hint, (I can foresee a few problems here)

I guess that’s it. Cheers to a new life.


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