I am solving a heat diffusion problem wich is described by the differential equation: θΤ/θt=a*(θ^2Τ/ΘΤ^2). I will apply two diffrent temperatures on the ends of a metal rod. Solving the equation i conclude, according to the boundary conditions of course, at the function:

Τ(x,t)=sum i=1-infinity [e^-i^2 * Pi^2 *a*t*(160*(-1)^i /(i*Pi) + 40/(i*Pi))*sin(i*Pi*x)] +100 x.( I gues its not very readable!sorry)

i also use a numerical method, crank-nicolson. It uses the previous(T') accordinng to time temperatures and previous(Ti-1) and next(Ti+1) temperatures according to length

Ti=A*T'(i) + B[T(i-1) +T(i+1)+T'(i-1)+T'(i+1)] A,B are known and also temperatures on the ends are known( 0,100 degrees respectively) so if i cut the rod on n+1 i get nXn system to solve! So far so good. Where is my question?

Comparing the temperatures that i get from both methods are satisfactorily close,but only after 60-70 seconds. On the first tens of seconds, on the points next to the ends of the rod(10cm and 90 cm if the hole rod is 1m), i have 3-5 degrees difference between these two methods.Wich results are the most relyable? why do i find these differences?

thank you very much!