Understanding the heat equation I've been reading about the heat equation recently and while I do grasp the mathematical principles involved, I'm struggling a bit with understanding the more in-depth details in a more general way.
For example:
Taking the the partial differential equation: $$u_{xx}=u_t\quad x\in(0,1)$$
I want to know concrete details, like what $u_{xx}$ represents. Given that $$u(x,t)$$ models temperature diffusion given a time-frame and spatial coordinates am I correct in thinking, that in $$u(x,0)= f(x)\quad x\in(0,1)$$ $f(x)$ here represents the area of whatever object we're studying heat conduction in, projected on a 1D plane? If anyone has time or curiosities like me, feel free to leave any detail about this part of the subject or your interpratation of it.
*Side question: Has anyone plotted the 1D - Heat Equation for $f(x)= x(1-x)$?
I have arrived at the conclusion that $u(x,t)=\frac{2}{\pi^3}\sum_{k=1}^{\infty}\frac{(-2(-1)^k+2)}{k^3}e^{(-k\pi)^2t}sin(k\pi x)$ and I'm not sure wether the graph is supposed to explode for $t >0.01$ the way it does.
 A: First of all, $u(x,t)$ represents the temperature at the position $x$, and at time $t$, thus the condition $u(x,0)=f(x)$ represents the initiale state of your system, that is what is the temperature at $t=0$. About the meaning of the heat equation, usually it is written as $$ \frac{\partial T}{\partial t}=D\frac{\partial ^2T}{\partial x^2} $$
where $T$ is the temperature and $D=\frac{\lambda}{\rho c}$ is the diffusivity constant, $\lambda$ tells you how well your object diffuses heat (it would be high for iron and small for wood for instance), $\rho$ is the mass density and $c$ is the massic heat capacity which tells you how much heat you have to give to the system to change its temperature. Substituting $u(x,t)=T(\sqrt{D}x,t)$ gives you that $u$ satisfies your equation $$ \frac{\partial u}{\partial t}=\frac{\partial ^2u}{\partial x^2} $$
so you can suppose without loss of generality that $D=1$. Anyway, if you look at the physical proof of this equation, it is roughly the mathematical translation of "the difference of energy of a section between $x$ and $x+dx$, between $t$ and $t+dt$ is the difference of heat flows between $x$ and $x+dx$, that is the one that enters minus the one that leaves", which we can write as
$$ d^2U=S\Phi(x)dt-S\Phi(x+dx)dt=-S\frac{d\Phi}{dx}dxdt $$
where $S$ represents the area of the section through which the flow passes. Now, Fourier's law tells you that $\Phi=-\lambda\frac{\partial T}{\partial x}$, therefore $\lambda\frac{\partial^2T}{\partial x^2}$ represents the difference of heat flows per unite of space and time. On the other hand, $d^2U=dCdT$ where $dC$ represents the heat capacity of the section between $x$ and $x+dx$. This section is of volume $Sdx$ and therefore $dC=c\rho Sdx$, which leads in the end to the desired equation. The interpretation is
$$ \rho cdT=\lambda\frac{\partial ^2T}{\partial x^2}dt $$
where the LHS represents the change of temperature of the infinitesimal section between $t$ and $t+dt$, and the RHS represents the difference of heat flows as I said.
This equation can be solved without the use of Fourier series, its general solution is
$$ u(x,t)=\frac{1}{\sqrt{2\pi t}}\int_{-\infty}^{+\infty}f(s)e^{-\frac{(x-s)^2}{2t}}ds $$
I don't know what your graph looks like with the initial condition $f(x)=x(1-x)$ but it should look like this for $t=0.1$ for instance :

