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Math Help - 2nd order, linear, 1-D, nonhomogeneous reaction-diffusion equation

  1. #1
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    2nd order, linear, 1-D, nonhomogeneous reaction-diffusion equation

    I'm working on a reaction-diffusion equation of the form

    Pt = DPxx KP + H
    where Pt = dP/dt and Pxx = d2P/dx2

    This is for a semi-infinite continuum, where P(x=0) is held at Po, the initial P along the continuum is 0, and the flux out the end is 0 (dP/dx = 0). I'm not certain whether it is helpful to define a length of the continuum, L where the zero flux BC is defined, thus making the continuum finite.

    According to Carslan and Jaeger (1959) section 4.14, the solution strategy is identify

    P(x,t) = Q(x) + R(x,t)

    where Q is not a function of time, and R is. Put the non-homogeneous terms in Q to form a 2nd order non-homogeneous ODE, and leave R as a homogeneous PDE.

    ODE in Q:
    DQxx KQ + H = 0
    This is a steady state equation, which uses the mixed boundary conditions:
    P(0,t) = Po
    dP/dx(infinity, t) = 0
    PDE in R:
    Rt = DQxx KR
    This uses the initial condition:
    P(x,0) = 0.
    My questions in this forum are (a) did I set it up right and (b) I can't solve R



    I believe the solution for Q is as follows:
    (1) solve the homogeneous part of Q:
    Q (K/D) = 0
    find the roots of the quadratic equation
    ay + by + c = 0
    r = sqrt(K/D)

    Q = C1exp(-rx) + C2exp(+rx) + Ys

    (2) solve Ys, the non-homogenous part of Q by method of unknown coefficients:
    try Ys = Ax2 + Bx + C
    Ys = 2Ax + B
    Ys = 2A

    substitute into Q:
    Y - (K/D)Y = -(H/D)
    -(K/D)Ax2 (K/D)Bx (K/D)C + 2A = -(H/D)
    gather the terms:
    -(K/D)A = 0 -> A=0
    -(K/D)B = 0 -> B=0
    2A - -(K/D)C = -(H/D) -> C=(H/K)

    plug coefficients back into general soln for Ys:
    Ys = (H/K)

    Q = C1exp(-rx) + C2exp(+rx) + (H/K)

    Matlab's dsolve agrees qualitatively with this solution.
    The solution fo R is proving to be more difficult for me. According to Carslon and Jaeger, the strategy for R is to substitute R = Se-kt, yielding St = DSxx but I havent figured out how to do this one yet. Any advice?


    Thanks! Adam
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  2. #2
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    Re: 2nd order, linear, 1-D, nonhomogeneous reaction-diffusion equation

    A couple of questions.

    1) Are D, K and H all constant. If not, do you know the forms of them?

    2) Are you on a finite domain or semi-infinite domain? The solution procedures for each will be different.
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  3. #3
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    Re: 2nd order, linear, 1-D, nonhomogeneous reaction-diffusion equation

    (1) Great point. D, H, and K are all constant.
    (2) I'm not sure. I had originally conceptualized this as being infinite, but researching in more detail, I think it would be an easier question if it has a defined length L. In any event the length of the domain L is >> the length scale in D (say 5,000x).
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