PDE:Laplace Transforms

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solution $u_t=u_{xx}\,$

$u(x,0) = \partial(x)\,$
$-\infty<x<\infty\,$
$t>0\,$

solution $u_t-u_{xx}=0\,$

$u(x,0^+) = 0\,$
$u(at,t) = g(t)\,$
$at\le x<\infty\,$
$t>0\,$

solution $u_t=u_{xx}\,$

$u_x(0,t) = \alpha(t)\,$
$u(x,0) = 0\,$
$x>0\,$
$t>0\,$