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A 2nd Order PDE Using Similarity Method

  1. May 14, 2017 #1
    Hi All,

    Does anybody know how to solve the following PDE? I tried a similarity solution method where eta = y/f(x) (which I can do successfully without the C * U term) but was unsuccessful.

    upload_2017-5-14_2-54-19.png

    Thank you very much in advance!
     
  2. jcsd
  3. May 14, 2017 #2

    Ssnow

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    If it is a PDE, it will be ##A\frac{\partial U}{\partial x}=B\frac{\partial^2 U}{\partial y^2}+C\cdot U## ...
     
  4. May 14, 2017 #3
    True - Sorry about that. Please take the d's to mean partial differential. Thank you for that catch.
     
  5. May 14, 2017 #4

    Ssnow

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    If you assume that ##A\not=0## you can write your equation as ##\frac{\partial U}{\partial x}=\frac{B}{A}\frac{\partial^2 U}{\partial y^2}+\frac{C}{A}\cdot U## that is an example of diffusion reaction equation with ## R(U)=\frac{C}{A}U##, see


    https://en.wikipedia.org/wiki/Reaction–diffusion_system


    where you call ##x=t## and ##y=x##, here the reaction term is simply ##\frac{C}{A}U##...

    Ssnow
     
  6. May 14, 2017 #5
    Thank you very much for your response and observation. Are you possibly aware of any closed form solutions to the diffusion reaction eq with R(U) = CU/A?
     
  7. May 14, 2017 #6

    pasmith

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    The substitution [itex]u = e^{Cx/A}v[/itex] results in a standard diffusion equation for [itex]v[/itex].
     
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