I’m studying PDE and am currently on heat equation subject . $$\frac{\partial u}{\partial t}=c^2 \frac{\partial ^2 u}{\partial x^2}$$
This book I’m reading worked on heat equation of a rod for different cases But I’m interested in one case of scenario this book didn’t cover.
For example when this rod has temperature of $0$ at one end and $\theta$ at the other obviously we will have : $$\lim u(x,t) =\frac{\theta}{L} x , t \rightarrow\infty $$
But What if the rod has constant heat of $\theta$ at on one end and $\alpha$ at the other end . What would this $u(x,t)$ would be . And obviously after some time the heat of each x will not change afterwards , and I really want to know how to calculate that $t$.
If you know the link for the proof i would be really interested and appreciated.
And even when i want to imagine the physical side of this scenario I cannot tell what would happen if $t \rightarrow\infty$