I am trying to numerically solve the initial value problem (see later discussion for ICs)
$$ x \frac{\partial f}{\partial t} = \frac{\partial}{\partial x} (1-x^2) \frac{\partial f}{\partial x} – f$$
for $$ f= f(x,t), \quad x \in X = [-1, 1] \quad \text{and} \quad t \in T = [0,1] $$
where $$ f(1, \ t), \ f(-1, \ t) \text{ are finite} \ \forall \ t \in T $$
and f is continuous for all x, t. Please note that my question only concerns solution using iterative methods, not a solution by eigenfunction expansion.
I attempted to solve this problem given an initial condition $$ \breve{f}(x, t = 0)$$
with a simple BTCS method. This leads to reverse diffusion for x < 0, hence the solution gets lost in oscillations.
Question 1 – Is there any way to solve this problem this problem directly? How can I check the solution even exists?
I was recommended to solve reformulated problem, so that there are two piecewise initial conditions:
$$ \breve{f}_0(x_1, t = 0), \ x_1 \in [0, 1] \quad \breve{f}_1(x_0, t = 1), \ x_0 \in [-1, 0] $$
and march backwards in time for $x \in [-1, 0]$. I am however unsure how to deal with boundary conditions at $x = 0$. The solution is continuous for all $x, t$ which leads to the following requirement at the boundary
$$ \lim_{|x| \rightarrow 0} \frac{\frac{\partial ^ 2 f}{\partial x^2} – f}{x} \text { is finite } \forall \ t \in T $$
I cannot solve the both parts of X interval simultaneously however, which seems to prevent me from imposing this BC.
Question 2 – How can I impose the boundary condition at $x = 0$ to ensure the final solution is continuous? Are there any restrictions on $\breve{f}_1(x_0, t = 1) \ \text{given} \ \breve{f}_0(x_1, t = 0)$ ?
Any other suggestions on solution are appreciated.
Thank you
Best Answer
This kind of problem has been discussed in J. Math. Phys. vol 22, page 954 and vol 24, page 1932.