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First order integro differential equation

  1. Sep 11, 2015 #1
    Can anyone help me to solve a differential equation?
    I want to solve

    ∂v(p,t)/∂t=-p^2 v(p,t)-sqrt(2/pi)∫v(p,t)[1-δ(t)R(t)]dp+sqrt(2/pi)[δ(t)R^2(t) C]
    with initial data v(p,0)=0

    where C is constant and the integration from zero to infinty
    Any suggestion please?

    Solution by volterra integral equation??
     
  2. jcsd
  3. Sep 11, 2015 #2

    BvU

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    Do I read this right ? You want to solve $${\partial v(p, t) \over \partial t} = - p^2 v(p, t) - \sqrt{2\over \pi} \int_0^\infty \ v(p,t)\ \left [ 1 - \delta(t) R(t) \right ] \, dp \ \ + \sqrt{2\over \pi} C\, \delta(t) \,R^2(t) \ \ ? $$with ## \ v(p, t) = 0\ ## and ##R(t)## a given function of time ?

    (where does it come from ? what do the symbols stand for ?)

    --

    Any link with earlier posts (that seem to have petered out somewhat :rolleyes: ) ?
     
    Last edited: Sep 14, 2015
  4. Sep 13, 2015 #3
    Dear BvU,
    Thank you for your replay, yes the equation is right.
    The field equation is diffusion equation with 2 free boundary conditions
    I applied the fourier transform for the diffusion and the boundary conditions and finally i got this first ODE
    I stuck on it ?
    any idea please?
     
  5. Sep 13, 2015 #4

    BvU

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    (Sorry for mistyping ##
    \ v(p, t) = 0\ ## -- should of course have been ##
    \ v(p, 0) = 0\ ## as you wrote).

    Pretty hefty ! And does the ##\delta(t)## represent a time-dependent coefficient or is it the Kronecker delta function (in which case the term with the fector C is a bit problematic) ?

    I hope someone more knowledgeable reads this and helps out, for me it's not obvious how to start with such a thing....
     
  6. Sep 14, 2015 #5
    Thank you BvU and ##\delta(t)## is represent a time-dependent.
    I hope someone can help me in this...
     
  7. Sep 14, 2015 #6

    BvU

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    I hope so too. My recollection of diffusion is that it gives equations like $${\partial u(x, t) \over \partial t} = {\partial^2 u\over \partial x^2}$$ so I have a hard time putting your equation into a context. But, as you say in your post #3, it is an intermediate situation in a solution procedure that involves Fourier transforms. I'll have to read up on that (little time for that o0)) and even then you probably have to spell out what you are doing from the beginning before I can be of any use, so we'll have to wait for help...

    Oh, and
    doesn't tell me much.
     
    Last edited: Sep 14, 2015
  8. Sep 14, 2015 #7

    BvU

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    If you can't wait that long, here's what I'm reading. Particularly pages 110 and further
     
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