# Heat Equation

1. Nov 25, 2005

### 'AQF

Is there a straightforward proof for the existence of the one-dimensional linear heat equation
f=u_t_-a^2*u_xx_=0.
Is so, how?
Note: _t_ represents the subscript, i.e., the derivative t, and _xx_ represents the subscript xx.

Is the heat equation well posed? Can this proven? How?

2. Nov 25, 2005

### Staff: Mentor

Have you tried writing
$$\frac{\partial{u}}{\partial{t}}\,-\,a^2\,\frac{\partial^2{u}}{\partial{x}^2}\,=\,0$$

as

$$\frac{\partial{u}}{\partial{t}}\,=\,a^2\,\frac{\partial^2{u}}{\partial{x}^2}$$ ?

Then let u(x,t) = X(x)T(t).

Separation of variables. And what about initial and boundary conditions?

See also - http://csep1.phy.ornl.gov/pde/node6.html - regarding a well-posed problem.

Last edited: Nov 25, 2005
3. Nov 25, 2005

### 'AQF

So it is well posed?

4. Nov 25, 2005

### Staff: Mentor

Last edited: Nov 25, 2005
5. Nov 25, 2005

### HallsofIvy

Staff Emeritus
An equation does not constitute a well-posed problem.

If you are given initial and boundary conditions, u(x,0)= f(x),
u(a,t)= g(t), u(b,t)= h(t) for some fixed a and b, then the problem is well-posed.

6. Dec 2, 2005

### Feynman

Where are you working ?
in an open set of ]R^n?
what is your bondery conditions(Dirichlet, Neumann, Robin,..........)?

7. Dec 12, 2005

### TEAM78

Hello everybody
Im busy doing some stuff on the Heat equation and would like to know what is the heat equation used for in detail. I have trolled the net looking to find the practical applications of the heat equation in mechanical engineering with little success, can you guys help

Last edited: Dec 12, 2005
8. Dec 26, 2005

### Staff: Mentor

The heat equation, or more precisely the heat conduction equation, is used to define the temperature (scalar) field. Here are some sites:

http://mathworld.wolfram.com/HeatConductionEquation.html

http://en.wikipedia.org/wiki/Heat_equation

http://www.mathphysics.com/pde/ch20wr.html

Derivation of the heat equation - http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node20.html
Solution of the heat equation: separation of variables - http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node21.html

Introduction to the One-Dimensional Heat Equation
http://www.math.duke.edu/education/ccp/materials/engin/pdeintro/pde1.html

9. Jan 16, 2006

### Feynman

You must presise your Boundary conditions