Is the heat equation well posed?

['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?

 

[Astronuc]

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.

 

['AQF]

So it is well posed?

 

[Astronuc]

Refer to the discussion of "well-posed problem", and I think one will be able to determine whether a given problem is well-posed or not.

http://en.wikipedia.org/wiki/Well-posed_problem

See also - http://www.soton.ac.uk/~jhr/MA361/node38.html

 

[HallsofIvy]

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.

 

[Feynman]

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

 

[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

 

[Astronuc]

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

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

 

[Feynman]

You must presise your Boundary conditions