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Heat eqtn with Generation term

  1. Mar 22, 2010 #1

    I am wondering how to solve this heat equation with a 'generation' term included. In one instance I am adding the generation term, in the other I am subtracting it.

    [tex] a\frac{\partial^{2}f }{\partial x^{2}}-\frac{\partial f}{\partial t} - \lambda f = 0[/tex]

    [tex] a\frac{\partial^{2}f }{\partial x^{2}}-\frac{\partial f}{\partial t} + \lambda f = 0[/tex]

    Any information on how to solve these/links to a table would be great :D
  2. jcsd
  3. Mar 22, 2010 #2


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    Of course, you need some initial/boundary conditions to have a well posed problem. I would try separation of variables.
  4. Mar 23, 2010 #3
    Is the system bounded or is it in the whole space (x-axis in our case)?

    A problem with boundary condition would be best solved with variable separation.
    An infinite problem would be best solved with a Fourier Transform.

    Both will give you an implicit form solution (as a sum or an integral) unless the initial conditions are specifically chosen.
  5. Mar 23, 2010 #4
    Look at Partial Differential Equations: Sources and Solutions by AD Snider.

    He talks about how to solve these sorts of problems.
    I think the solution is to use a combination of Green's functions and Laplace transforms, but I do not recall exactly.
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