I have used Laplace Transformation and got the solution in Laplace domain, but it seems very complicated to be transformed back to the real-time domain. Is it possible to solve the problem with another technique?
$$\frac{\partial T}{\partial t }=a^2 \frac{\partial^2 T}{\partial x^2}$$
The initial condition is:
$$T(x,0) = 0.5 \left(\mbox{Erfc}\left(\frac{x - a}{b}\right)+ e^{x} * \mbox{Erfc}\left(\frac{x + a}{b}\right)\right)$$
The boundary conditions:
$$T(\infty,t)=0$$ $$\frac{\partial T(0,t)}{\partial x }=0$$