Solving PDE analytically

I am solving a partial differential equation numerically (by Explicit Scheme method), but also I have to check the consistency and stability of the method by comparing it with the analytical result. Could you help me find its analytical solution?

Consider the following PDE defined on $\Omega \in R^n$ $( x \in [0, L] \quad \text{and} \quad t \in [0,t_{\text{max}}])$: $$(E) \left\{ \begin{array}{lcc} \dfrac{\partial\theta }{\partial t}+rx\dfrac{\partial\theta }{\partial x}-\sigma^2\dfrac{x^2}{2}\dfrac{\partial ^2\theta }{\partial x^2}+r\theta=0, & 0<x<L, & 0<t<t_{\text{max}}\cr \theta(0,t)=0, & & \forall t \in [0,t_{\text{max}}] \cr \dfrac{\partial \theta }{\partial x}(L,t)=0,& & \forall t \in [0,t_{\text{max}}] \end{array} \right.$$

$r$ and $\sigma^2$ are constants.

Can anyone help me to find its Analytical solution please?

• Show something of what have you tried. – Test123 Nov 26 '17 at 9:54
• What is $r$ ? ... – JJacquelin Nov 26 '17 at 10:02
• You could change variables $x=e^{y}$ so that the equation becomes constant-coefficient, then separate variables $\theta(t,y)=e^{-rt}T(t)Y(y)$. – user254433 Nov 26 '17 at 10:08
• r and $\sigma^2$ are give constants – Balde Nov 26 '17 at 10:09
• if I change variable $x=e^y$, how to perform the partial derivatives ? – Balde Nov 26 '17 at 10:21