I must solve this initial boundary value problem by transforming it into homogeneous BCs.

---> My work

I set where S(x,t) is the steady state and U(x,t) is the transient

So i substitute s(x,t) into the BCs

These two equations can give usb(t)=1 and a(t)=2/1-h?

and therefore u(x,t)=2x/(1-h) + (1-x) + U(x,t) =1 - (1+h/1-h)x + U(x,t)

MY PROBLEM is, how did they get b(t)=1 and a(t) = 2/1-h.

I know i have have to make s(0,t) = to something likea(t) + b(t) = 1and s_x(1,t) =?something?so that i can plug it into the but I am not sure what s and s_x should be.

Can someone please tell me what they are so I can finally get a(t) and b(t).

Then I can finally finish solving this problem.

Thanks in advance!