I need to approximate the solutions of a differential equation using rational functions.
A) The solution is of the form f(x,y,z) = N / D where D needs to be a polynomial of higher order than N so it goes to zero as x and y increases.
B) I am assuming that f(x,y,z)=f(-x,y,z)=f(x,-y,z) for the solution, so i need to test different rational functions and they all must be even functions (i guess).
C) The maximum order for the polynomials should be N1 = N2 = 4, for the numerator and denominator respectively, but polynomials with lower orders should be considered as well.
I know that if both numerator and denominator polynomials have the same parity, then the resulting function will be even.
But is there a way of sistematically generating all polynomial divisions that will suffice the above conditions?
Thanks a lot.
PS: After i have a set of rational functions that agree with the above conditions, i will use those functions to create an analytical solution aproximation of the original equation (diffusion equation) using optimization algorithms to find the optimal polynomial coefficients.