start with assumption u(x,t) = g(x) * h(t) ← this is a standard assumption for PDE.Given,
u t - uxx = xt ; 0 < x < Pi
u(x,0) = 1 ; & ux (0,t) = 0, & ux(pi,t) = 0
How to find the solution u using separation of variables?
Instead of "using words" - please share your mathematical work.I have no problem with the homogeneous part and for the inhomogeneous part, I opt for the change of function such that u(x,t) = 1/2 [t2]x + v(x,t)
and then vt - vxx =0 and then there is similar changes in the initial conditions. But what would be my subsequent steps?