Are you sure, did you try substituting that solution in terms of Fourier series into the equation and seeing if it works. I am sure it does.

I think you might be referring to the Dirac Generalised function, because of a jump discontinuity. But I do not see how that makes my solution false.I'm afraid I cannot explain why.

But perhaps that is only for an infinite bar. My case here is a finite bar.But I know that for an unbounded bar: -infinity <x<+inf the unique solution is the well-knownGauss kernel

$\displaystyle u(x,t)= \frac{1}{\alpha \sqrt{2\pi t}}\, exp \left( - \frac{(x-x^*)^2}{2\alpha^2t}\right)$