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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:

$$T(\infty,t)=0$$ $$\frac{\partial T(0,t)}{\partial x }=0$$

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  • $\begingroup$ Have you tried separation of variables? $\endgroup$
    – Multigrid
    Aug 12, 2018 at 17:55
  • $\begingroup$ Not yet. I will try with this technique. But how to get the arbitrary constants when x approaches infinity by this technique? $\endgroup$
    – Refaat Galal
    Aug 12, 2018 at 18:00
  • $\begingroup$ Great, I guess it should work $\endgroup$
    – Multigrid
    Aug 12, 2018 at 18:01

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