IBVP with transformed boundary conditions

Question

I am trying to use the following result

Theorem: A pde of the form $$\frac{\partial w}{\partial t} = F\{x, \frac{\partial w}{\partial x},\frac{\partial^2 w}{\partial^2 x}\}$$ has an Additive separable solution $$w\left(x, t\right) = M t + N + \phi(x),$$ where $M$,$N$ are arbitrary constants, and the function $\phi(x)$, is determined by the ordinary differential equation $$F\{x, \frac{d \phi}{dx}, \frac{d^2 \phi}{d^2 x}\} = M.$$

The problem, I am studying has an initial condition and two Dirichlet boundary conditions, but I have had to perform a few change of variables to get the PDE, I am studying to look like the one in the above result. As a consequence of this the initial and the boundary conditions in my problem have transformed to the form \begin{eqnarray} \qquad w\left(f(x) + g(t), 0\right) &=& \delta\left(f(x) + g(t) - x_0\right),\\ w\left(f(x) + g(t) = 0, h(t)\right) &=& w\left(\pi / 2, h(t)\right) = 0 \qquad (t > 0) \end{eqnarray} where $f, g, h$ are some common but non-linear functions and $\delta$ is the dirac delta function, instead of having a form like $w\left(x, 0\right)$, $w\left(0, t\right)$ and $w\left(C, t\right)$, which I think is what is required to get the solution.

I am wondering what techniques or options are available for me to be able to use such initial and the boundary conditions. I have been searching to find such examples of problems, but I have not had much luck with this.

Any help will be greatly appreciated.

Edit

The pde I am trying to apply this theorem to, is \begin{equation} \frac{\partial w(x, t)}{\partial t} = - \frac{2 b \kappa x}{\left(x^2+1\right)^2} \frac{\partial w(x, t)}{\partial x} + \frac{b \kappa}{\left(x^2+1\right)^2} \frac{\partial ^2 w(x, t)}{\partial x^2}% \end{equation} and the initial condition is $$ w\left(x, 0\right) = \delta\left(x - x_0\right) $$ and the boundary conditions are $w = 0$ along the curves $$ \{(\pi/2, t): t> 0\} \text{ and } \{ (x,t) : \tan(x) + a (2\kappa t)^{\frac{1}{2\kappa}} = 0, t > 0\}, $$ and $a, b, \kappa$ are constants from my model. The ode corresponding to this problem is \begin{equation} \frac{d^2 \phi}{d^2 x} - 2 x \frac{d \phi}{dx} = \frac{M}{b \kappa}\left(x^2+1\right)^2 \end{equation} which has the general solution \begin{equation} \phi\left(x\right) = \frac{11 M x^2}{8 b \kappa}\, _2F_2\left(1,1;\frac{3}{2},2;x^2\right) - \frac{M x^4}{8 b \kappa} - \frac{7 M x^2}{8 b \kappa} + C_1 \int e^{x^2} \,dx + C_2, \end{equation} where $C_1$ and $C_2$ are constants of integration and we have a generalized hypergeometric function in the first term of the general ode solution.

Answer 1

You can't use your theorem. Your boundaries are not straight. Furthermore, your boundary conditions are independent of $t$, which requires $M = 0$, and this puts strong requirements on the initial data.

Basically:

1. Your theorem shows that there exists some family of solutions to a PDE of the form as written. Emphatically, however, you need to understand that this is a very special solution and most (in fact almost all) solutions to the PDE cannot be described in that form.

2. If you freely prescribe initial and boundary data, chances are the solution that satisfies your initial and boundary constraints will not intersect the special family the theorem mentions.

3. For linear equations, this is pretty much why separation of variables are usually done multiplicative and not additive. This way Dirichlet boundary conditions of type $w|_{\partial\Omega} = 0$ or Neumann boundary conditions $\partial_\nu w|_{\partial\Omega} = 0$ are preserved once it holds for the initial data.

Additionally:

4. I am pretty sure your "change of variables" doesn't do what you think it does. Your ansatz would only work to "straighten out" the domain if the domain were constant width in time. It is not. This is to say nothing of you also introducing $t$ dependence in to the coefficients of the equation.