When solving a parabolic PDE with boundary conditions (e.g. the reflection method of the barrier option), we can change the parabolic PDE into a heat equation by applying some substitution. For example, changing: $$\left\{\begin{array}{cc} \frac{\partial u}{\partial t}+\frac{1}{2} \sigma^{2} \frac{\partial^{2} u}{\partial x^{2}}+\left(r-q-\frac{\sigma^{2}}{2}\right) \frac{\partial u}{\partial x}-r u=0 \\ & \left(x \in \mathbf{R}_{+}, 0 \leqslant t \leqslant T\right) \\ u(x, T)=\left(e^{x}-K_{B}\right)^{+}, & (0<x<\infty) \\ u(0, t)=0, & (0 \leqslant t \leqslant T) \end{array}\right.$$
into: $$ \left\{\begin{array}{l} \frac{\partial W}{\partial t}+\frac{1}{2} \sigma^{2} \frac{\partial^{2} W}{\partial x^{2}}=0 \\ W(x, T)=e^{-\alpha x}\left(e^{x}-K_{B}\right)^{+} \\ W(0, t)=0 \end{array}\right.$$
And then apply an odd extension to change it into a Cauchy problem.
My question is, in the two-dimensional condition, for example: $$ \begin{cases} \begin{aligned} &\frac{\partial V}{\partial t}-\frac{1}{2} \delta^{2} \frac{\partial^{2} V}{\partial x^{2}}-\delta \sigma \gamma \frac{\partial^{2} V}{\partial x \partial y}-\frac{1}{2} \sigma^{2} \frac{\partial^{2} V}{\partial y^{2}} -a(t) \frac{\partial V}{\partial x}-b(t) \frac{\partial V}{\partial y}+cV=0 \quad (x \in \mathcal{R},y \in \mathcal{R}^+,t\in [0,T]) \\ &V(x,y,0)=e^y(e^x-\bar{I}) \quad (0 < y< +\infty)\\ &V(x,0,t)=0 \quad (0 \leqslant t \leqslant T) \end{aligned} \end{cases}$$ I can change the following into a two-dimensional heat equation: $$ \begin{cases} \begin{aligned} &\frac{\partial W}{\partial t}-\frac{1}{2} \delta^{2} \frac{\partial^{2} W}{\partial x^{2}}-\delta \sigma \gamma \frac{\partial^{2} W}{\partial x \partial y}-\frac{1}{2} \sigma^{2} \frac{\partial^{2} W}{\partial y^{2}}=0 \\ &W(x,y,0)=e^{(1-b)y}(e^x-\bar{I}) \quad (0 < y< +\infty)\\ &W(x,0,t)=0 \quad (0 \leqslant t \leqslant T) \end{aligned} \end{cases}$$ Can I also apply an odd extension to this heat equation (but the coefficients of the second-order partial derivatives are not the same) and change it into a Cauchy problem? Does the odd extension only work in two-dimensional heat equations with the same second-order partial differential coefficients? If not, how to solve the following two-dimensional parabolic pde with a Dirichlet boundary condition?
I would really appreciate it if someone could help me, as most references on extension only deal with one-dimensional equations. Thank you!