Let $\mathcal{L} = \partial/\partial t + \frac{1}{2}\sigma^2x^2 \partial^2/\partial x^2 + rx \partial/\partial x - r $ be the Black-Scholes operator.
In the paper he mentioned about the "method of image solution" (Sec. 2.2), which transforms the PDE \begin{align*} \mathcal{L}P &= 0 \\ P(t,B) &= 0, \; 0 \leq t < T \\ P(T,x) &= f(x), \; 0 \leq x < B \end{align*} to another PDE with extended boundary condition that has the same solution as above \begin{align*} \mathcal{L}P &= 0 \\ P(t,B) &= 0, \; 0 \leq t < T \\ P(T,x) &= f(x)1_{\{x < B\}} - \left(\dfrac{B}{x} \right)^{\frac{2r}{\sigma^2} - 1} f\left(\dfrac{B^2}{x} \right)1_{\{x > B\}} , \; 0 \leq x < \infty \end{align*}
He takes the reference from a paper that I have no access to, and I can't figure out why one can think of this transformation.