number 1:

Ut(x,t) = a*a Uxx(x,t)
U(x,0) = 0 
U(l,t) = U0
Ux(0,t) = 0
x in [0,L] 
t>0



number 3:

Ut = Uxx + exp (-t)*cos(2x)
U(x,0) = cos (4x) + cos(2x)
t > 0
x in R




number 4:

Ut = Uxx
U(x,0) = exp(-alpha * x)
Ux(0,t) = 0
x in [0, infinity]
alpha >0