Solution to Heat Equation

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 an **initial value problem** otherwise they have infinitely many solutions we have the same situation with PDE's we need to have an initial value called the **boundary 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 is *always 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.

Diffusion coefficient and Integrating factor

**Sorry I think my previous input just added to confusion. So I have removed it.**

State and time-dependent coefficients?

Quote:

Originally Posted by

**ThePerfectHacker** I am not exatcly sure what you are asking?

Are you asking me what do you do if this is not a **homogenous** boundary value problem? As in $\displaystyle u(x,0)\not =0 \mbox{ or }u(x,L)\not = 0$. ?

Or are you asking me if there is another method to solve the heat equation?

Sorry I haven't read your book ... But after this I will of course! You write u(x,L) which must be a typo? If I understand correctly you have explained how to find the solution using the separation of variables method for boundary conditions u(0,t)=u(L,t)=0, all t, "isolated bar", and "initial heat" given by u(x,0)=f(x) when the coefficient alpha is constant.

I want to know what we can say about the solutions to more complicated versions of the basic pde such as time- and state-dependent coefficients including a first order term, i e

$\displaystyle a(x,t) \, u_{xx}(x,t) +b(x,t) \, u_x (x,t) = u_t(x,t)

$ for u(x,0)=f(x).

Also I want to know how the solutions are affected by different boundary conditions on x=0 and x=L. And what solutions do we find in higher dimensions when x belongs to different geometric objects in 2-3 dimensions?

I'm not sure if these questions are relevant for this forum so please advise!