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Heat Equation

  1. Nov 25, 2005 #1
    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. jcsd
  3. Nov 25, 2005 #2

    Astronuc

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    Staff: Mentor

    Have you tried writing
    [tex]\frac{\partial{u}}{\partial{t}}\,-\,a^2\,\frac{\partial^2{u}}{\partial{x}^2}\,=\,0[/tex]

    as

    [tex]\frac{\partial{u}}{\partial{t}}\,=\,a^2\,\frac{\partial^2{u}}{\partial{x}^2}[/tex] ?

    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
  4. Nov 25, 2005 #3
    So it is well posed?
     
  5. Nov 25, 2005 #4

    Astronuc

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    Staff: Mentor

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

    HallsofIvy

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    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.
     
  7. Dec 2, 2005 #6
    Where are you working ?
    in an open set of ]R^n?
    what is your bondery conditions(Dirichlet, Neumann, Robin,..........)?
     
  8. Dec 12, 2005 #7
    Hello everybody :smile:
    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 :smile:
     
    Last edited: Dec 12, 2005
  9. Dec 26, 2005 #8

    Astronuc

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    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
     
  10. Jan 16, 2006 #9
    You must presise your Boundary conditions
     
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