the problem is
solve the following heat problem using FFCT:
A metal bar of length L, is at constant temperature of , at t=0 the end x=L is suddenly given the constant temperature of and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any point x of the bar at any time t>0, assume k=1

Equations used:
heat eq.

with the additional equations shown on the attached image.

my attempt:
my attempt goes like this:

and i dont know how to continue...

Attached Image is from this link:>>>