sol = NDSolve[{D[u[x, t], t] - D[u[x, t], {x, 2}] + 4 D[u[x, t], x] -4 u[x, t] == (3 - 2 t) Exp[2 t] + (1 - 9 t) Exp[-x],
u[x, 0] == -1,
Derivative[1, 0][u][0, t] == -t,
Derivative[1, 0][u][300, t] == 0},
u[x, t], {x, 0, 300}, {t, 1.5, 100}]