I have come across the following Heat equation IBVP but I am not quite sure how to solve it \begin{equation} \frac{\partial v\left(x, t\right)}{\partial t} = \alpha \frac{\partial^2 v\left(x, t\right)}{\partial x^2}, \quad t \in \left(0, T\right], x \in \Omega. \label{heatEqn} \end{equation} with the following initial and boundary conditions \begin{equation} v\left(x, 0\right) = \delta\left(x\right), \quad v\left(-\infty, t\right) = 0, \quad \int_{-\infty}^{1}v\left(x, T\right) dx = T^{\alpha-1}\mathrm{e}^{\beta\sqrt{\log T}} \end{equation} where $\alpha \in \left(0, 1\right)$ and $\beta$ is some positive constant.
I am wondering if Greens Theorem can be applied to tackle this problem. Is there a way to solve this problem ? Thanks.