- #1
Leonardo Machado
- 57
- 2
- TL;DR Summary
- I am not being able to determine the behavior of my solutions for improper boundaries if the behavior of the solution is expected to diverge.
Hello,
I am trying to compute some non-linear equations with pseudospectral/collocation methods. Basically I am expanding the solution as
$$
y(x)=\sum_{n=0}^{N-1} a_n T_n(x),
$$
Being the basis an Chebyshev polynomial with the mapping x in [0,inf].
Then we put this into a general differential equation
$$
Ly(x)=f(x,y),
$$
which leads to
$$
\sum_{n=0}^{N-1} L T_n(x) a_n = f(x,y).
$$
This function is evaluated at the collocation points associated with the Chebyshev polynomials as usual, leading to N-1 non-linear equations. Also, there is also equations for the boundaries, i. e.,
$$
\sum_{n=0}^{N-1}a_n T_n(0)=A,
$$
$$
\sum_{n=0}^{N-1}a_n T_n(inf)=B.
$$
My problem is here. How can I treat a boundary condition which leads to infinity somehow? For example$$
\sum_{n=0}^{N-1}a_n \frac{dT_n(inf)}{dx}=1,
$$
or even,
$$
\sum_{n=0}^{N-1}a_n T_n(inf)=inf.
$$
I am trying to compute some non-linear equations with pseudospectral/collocation methods. Basically I am expanding the solution as
$$
y(x)=\sum_{n=0}^{N-1} a_n T_n(x),
$$
Being the basis an Chebyshev polynomial with the mapping x in [0,inf].
Then we put this into a general differential equation
$$
Ly(x)=f(x,y),
$$
which leads to
$$
\sum_{n=0}^{N-1} L T_n(x) a_n = f(x,y).
$$
This function is evaluated at the collocation points associated with the Chebyshev polynomials as usual, leading to N-1 non-linear equations. Also, there is also equations for the boundaries, i. e.,
$$
\sum_{n=0}^{N-1}a_n T_n(0)=A,
$$
$$
\sum_{n=0}^{N-1}a_n T_n(inf)=B.
$$
My problem is here. How can I treat a boundary condition which leads to infinity somehow? For example$$
\sum_{n=0}^{N-1}a_n \frac{dT_n(inf)}{dx}=1,
$$
or even,
$$
\sum_{n=0}^{N-1}a_n T_n(inf)=inf.
$$