Somebody asked me to try to explain better than the book the solution to the heat equation.

The equation is:

$\displaystyle \alpha ^2 u_{xx} = u_t$ which is satisfied for all $\displaystyle 0<x<L$ and $\displaystyle t>0$.

This is called "heat equation" because that is what the equation is about. Let $\displaystyle u(x,t)$ represent a point at $\displaystyle x$ and time $\displaystyle t$ then the heat (temperature) at that point in time is given by $\displaystyle u(x,t)$. The equation above says that if we are given a bar with length $\displaystyle L$ and a certain physical property (density, specific heat, ... ) which is the constant $\displaystyle \alpha^2$ (the reason why it is squared is because it is less messy when we take square roots later on). Then it is modeled by that equation.

Now just like ODE's which need to have aninitial value problemotherwise they have infinitely many solutions we have the same situation with PDE's we need to have an initial value called theboundary value problem.

So let $\displaystyle u(x,0)=f(x)$ this means that at $\displaystyle t=0$, i.e. initial state, the temperature distribution at every single point $\displaystyle x$ is given by $\displaystyle f(x)$.

Now, the simpliest case of the heat equation is a bar with isulated bar, mathematically it means the temperature at the ends isalways zero. Thus, $\displaystyle u(0,t) = u(L,t)$ for all $\displaystyle t$.

Thus, the heat equation together with the boundary value problem is written as:

$\displaystyle \left\{ \begin{array}{c}\alpha^2 u_{xx} = u_t \mbox{ for }0<x<L \mbox{ and }t>0 \\ u(x,0)=f(x) \mbox{ for }0\leq x\leq L \\ u(0,t)=u(L,t)=0 \mbox{ for }t\geq 0 \end{array} \right\}$

Does it make sense so far? That is just the meaning of the physics behind this problem.