- #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.

$$