What is the analytical solution for the following diffusion partial differential equation (initial value problem)? $$\frac{\partial f}{\partial t} = (ax^2+b)\frac{\partial f}{\partial x}+\frac{\partial^2 f}{\partial x^2},$$ where $a$ and $b$ are real number constants.
We can separate the variables or take the Fourier transform $\tilde f(x)$ of $f$ in the time domain $t$, and turn the above into an ordinary differential equation eigenvalue problem in $x$: $$k\tilde f= (ax^2+b)\frac{d\tilde f}{d x}+\frac{d^2 \tilde f}{d x^2}.$$ where $k$ can be views as an eigenvalue for the differential operator on the left hand side. Now we can further transform this into the Sturm-Liouville form.
However, I can not immediate recognize a transformation that can turn the above into a known form that admits an analytic solution. Can someone help?
Using Maple we get the solution is: $$f(x,t)=F1(x)\cdot F2(t)$$ where $F1$ and $F2$ are functions such that $$F1_{xx}=c_1\cdot F1-(a\cdot x^2-b)F1_x \quad\text{ and } $$$$F2_t=c_1\cdot F2, $$ where $c_1$ is an arbitrary constant. The ODE for $F1$ has an "explicit" solution in terms of the Heun Triconfluent function (very ugly! Maple is not making some simplifications because it is assuming a and b are complex numbers) and the second ODE is just $F2(t)= c_2 \exp(c1\cdot t)$.
$F1(x) = c_3\cdot HeunT\left(-3^{2/3}\cdot a^2\cdot c_1/(a^2)^{4/3}, -3\cdot \sqrt{a^2}/a, a\cdot b\cdot 3^{1/3}/(a^2)^{2/3}, (1/3)\cdot 3^{2/3}\cdot (a^2)^{1/6}\cdot x\right)\cdot \exp\left(-(1/6)\cdot x\cdot (a\cdot x^2+3\cdot b)\cdot ((a^2)^{1/6}\cdot a+(a^2)^{2/3})/(a^2)^{2/3}\right)+c_4\cdot HeunT\left(-3^{2/3}\cdot a^2\cdot c_1/(a^2)^{4/3}, 3\cdot \sqrt{a^2}/a, a\cdot b\cdot 3^{1/3}/(a^2)^{2/3}, -(1/3)\cdot 3^{2/3}\cdot (a^2)^{1/6}\cdot x\right)\cdot \exp\left((1/6)\cdot x\cdot (a\cdot x^2+3\cdot b)\cdot ((a^2)^{1/6}\cdot a-(a^2)^{2/3})/(a^2)^{2/3}\right)$