I have the following second order partial differential equation (PDE) on $[0,T] \times \mathbb{R},~ T >0 $:
\begin{equation}
\left(1 + \frac{1}{(1 + b f)^2}\right) \frac{\partial f}{\partial t}
-\left(\frac{1}{(1 + b f)^2}\right) \frac{\partial f}{\partial x}
= \frac{\partial^2 f}{\partial x^2} .
\end{equation}
supplied with following initial and boundary conditions:
\begin{equation}
f(x,t=0)=\begin{cases}
1, & x\leq 0\\
0, & x > 0, ~ \text{the Heaviside unit step function}.
\end{cases}
\end{equation}
\begin{equation}
\frac{\partial f}{\partial x} =0, ~\mbox{as}~|x| \rightarrow \infty,
\end{equation}
Here $b$ is a non-negative constant.
My questions:
- Does there exists any analytic solution to this PDE (e.g., some approximation solution).
- Does there exists any finite difference scheme or any numerical scheme to solve this PDE.
P.S. I have some idea how to solve non-linear PDEs with constant coefficients for time derivative. Buy I have no guess how to start for stated PDE.
It will be great help, if someone suggest some idea. Thanks!
Best Answer
A little more details as requested (in private email)
Sorry I made a mistake above, of course you want $b \ll 1$. If that is the case you can use whats called "regular perturbation theory" (as opposed to singular perturbation theory which is what you would need if for example the $f_{xx}$ term was multiplied by a small parameter).
Make the following assumption $f(x,t) = f_0(x,t) + b f_1(x,t) + O(b^2)$
Now plug that in and collect terms corresponding to different powers of $b$. You get the following
$$2 \partial_t f_0 - \partial_x f_0 = \partial_x^2 f_0$$
And $f_0$ has the same BCs as you had for $f$ above. This problem is much easier to solve. Then and should give you an approximation when $b \ll 1$. You can even derive a correction by considering the equation you would get for the $f_1$ term. This equation will be forced (i.e. be non-homogenous) by the solution of $f_0$.
Have you looked at any finite difference schemes for parabolic PDEs? Unless you actually try to implement some scheme and run into specific problems, I don't know what you're expceting people on here to tell you about numericas for this thing.