[math]v_t(x, t)-v_{xx} (x, t)=0 , x\in \mathbb{R}, t>0, v (x, 0) = sin^2(3x), x\in \mathbb{R}[/math] can this be solved using separation of variables?

[mario99]

Would you call the above bdy. condition "homogeneous" or "non-homogeneous" ?

I would call it a nonhomogeneous initial condition, but initial conditions have nothing to do with your assumption. Only with nonhomogeneous boundary conditions, this assumption can be applied (or if the PDE is nonhomogeneous). In PDEs, the boundary conditions are conditions in which the substitution is done in the spatial variable such as [imath]x, y, \text{or} \ z[/imath]. On the other hand, a substitution done in the time variable, [imath]t[/imath], is called an initial condition.

 

[mario99]

The solution is:

[imath]\displaystyle v(x,t) = e^{-36t}\sin^2(3x) - \frac{1}{2}e^{-36t} + \frac{1}{2}[/imath]

Test it. Does it satisfy the PDE and its initial condition?