I have been struggling with following Heat equation IBVP, \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 \left(- \infty, 1\right]. \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 = \beta T^{\alpha-1} \end{equation} where $\alpha \in \left(0, 1\right)$ and $\beta$ is some positive constant.

Some interesting observation about this problem, if we consider the semi-infinite plane $x \in \left(- \infty, 1\right]$ $\times$ $t \in \left[0, T\right]$, we know line integrals along three of the boundaries of this rectangle, as in $\int_{-\infty}^{1}v\left(x, 0\right) dx = 1$, $\int_{0}^{T}v\left(-\infty, t\right) dt = 0$ and the boundary condition with the integral equation above.

Also if we can deduce $v\left(1, t\right)$ and $\frac{\partial v\left(x, t\right)}{\partial x}|_{x = 1}$ then we can attack the problem using laplace transform.

I am wondering if there is version of Greens or Divergence Theorem which can be applied to tackle this problem.

Any help or guidance will be greatly appreciated. Thank you.

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