A Modified Heat Equation: Any Hope for Analytical Approximation of Solution I am interested in solving the PDE 
$$\frac{\partial}{\partial t} f(x,t) = c(x,t) \frac{\partial^2}{\partial x^2} f(x,t)$$
The goal is to solve this in $f$, given the boundary conditions $\frac{\partial}{\partial x} f(\pm \infty, t) = 0$ and the initial condition $f(x,0)=g(x)$ for some known $g$.
If $c(x,t)$ was constant, this would be a standard heat equation, with the solution being $f(x,t) = [g \star k_{ct}](x)$, where $k_{\alpha}(x)$ is the heat kernel $\frac{1}{\sqrt{4 \pi \alpha}}e^{-\frac{x^2}{4\alpha}}$ and $\star$ is the convolution operator.
However, since $c$ varies in both $x$ and $t$, I am not sure if an analytical solution exists. If not, then what is a reasonable analytical approximation to the solution $f$?
For example, one idea is to forget the dependency of $c$ on $x,t$... solve the heat equation, and only after that replace c with $c(x,t)$ in the solution form. So the approximate solution would be $\hat{f}(x,t) = \Big([g \star k_{ct}](x) \Big)_{|\,c=c(x,t)}$. Is this a reasonable idea? If not, are there more reasonable ideas for approximating $f$?
 A: We have
$$\frac{\partial}{\partial t} f(x,t) = c(x,t) \frac{\partial^2}{\partial x^2} f(x,t)$$
subject to initial boundary conditions on $f$, e.g., $f(x,0)=g(x)$.  Write $c(x,t) = c_0 + \epsilon c_1(x,t)$.  Then assume that the solution $f$ behaves as $f(x,t) = f_0(x,t)+\epsilon f_1(x,t)$, where $\epsilon^2=0$.  Equating coefficients of $\epsilon$, we get two equations:
$$\frac{\partial}{\partial t} f_0(x,t) - c_0 \frac{\partial^2}{\partial x^2} f_0(x,t) = 0 \quad f_0(x,0)=g(x)$$
$$\frac{\partial}{\partial t} f_1(x,t) - c_0 \frac{\partial^2}{\partial x^2} f_1(x,t) = c_1(x,t) \frac{\partial^2}{\partial x^2} f_0(x,t)  \quad f_1(x,0)=0$$
The first equation is the typical homogeoneous heat equation with a constant diffusivity.  We incorporate the initial boundary condition here.
The second equation is an inhomogeoneous heat equation with a constant diffusivity.  Note that the RHS is known.  Also we impose zero initial conditions here.
The idea is that you split up the diffusivity term into a constant term and a smaller, non-constant term.  The value of $\epsilon$ can be what you need it to be.
