One-dimensional systems

(a) Series solutions for the temperature field

Transient temperature distributions may always be considered to be due to a perturbation of an initially steady state. In general the perturbation consists of a change in the state of the surroundings at a certain time, t = t_initial. For convenience, t_initial = 0. The temperature distribution is then completely determined for any time t > 0 if the initial distribution is known,¹

\[\label{eq1} T (r,0) = T_{I}(r)\quad (\textrm{initial condition, I.C.})\tag{1}\]

and if the state of the surroundings is given for any time greater than zero (boundary condition, B.C.) Generally the B.C.’s may be written