This is an example of a heat equation:
$u_t=u_{xx}$, where $0<x<1$, $t>0$
$u_x(0,t)=u_x(1,t)=0$ and $u(0,t)=g(t)$ .
The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.
I've found that $u$ is temperature as a function of space and time $u_t$ is the rate of change of temperature at a point over time and $u_{xx}$is the second spatial derivative(thermal conductions) of temperature in the x direction.
However, I can't imagine what we are doing, what changes when the region we deal with changes etc. That is the question.
(More generally: what is the idea behind solving a heat/wave equation? What do we want to do and how are we doing it? )
Let $u(x,t)=\sum\limits_{n=0}^\infty C(n,t)\cos n\pi x$ so that it automatically satisfies $u_x(0,t)=u_x(1,t)=0$ ,
Then $\sum\limits_{n=0}^\infty C_t(n,t)\cos n\pi x=-\sum\limits_{n=0}^\infty n^2\pi^2C(n,t)\cos n\pi x$
$\therefore C_t(n,t)=-n^2\pi^2C(n,t)$
$\dfrac{C_t(n,t)}{C(n,t)}=-n^2\pi^2$
$\int\dfrac{C_t(n,t)}{C(n,t)}dt=-\int n^2\pi^2~dt$
$\ln C(n,t)=-n^2\pi^2t+a(n)$
$C(n,t)=A(n)e^{-n^2\pi^2t}$
$\therefore u(x,t)=\sum\limits_{n=0}^\infty A(n)e^{-n^2\pi^2t}\cos n\pi x$
$u(0,t)=g(t)$ :
$\sum\limits_{n=0}^\infty A(n)e^{-n^2\pi^2t}=g(t)$
$\therefore u(x,t)=\sum\limits_{n=0}^\infty A(n)e^{-n^2\pi^2t}\cos n\pi x$ , where $A(n)$ is the solution of $\sum\limits_{n=0}^\infty A(n)e^{-n^2\pi^2t}=g(t)$