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Math Help - Fourier transform to solve inhomogeneous PDE

  1. #1
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    Fourier transform to solve inhomogeneous PDE

    Express the solution to the inhomogeneous equation
    \frac{\partial u}{\partial t} = \kappa \frac{\partial^2u}{\partial x^2} + S(x,t)
    satisfying the initial condition
    u(x,0) = 0, -\infty < x < \infty,
    as an integral involving the source term S(x,t).

    The given solution is \frac{1}{\sqrt{2\pi}} \int^t_0 \left( \int^\infty_{-\infty} \frac{e^{-(x-x')^2/(4\kappa(t - t'))}}{\sqrt{2\kappa(t-t')}} S(x',t') dx'  \right) dt'.

    Attempt:

    Take Fourier transform with respect to x.
    \frac{\partial U}{\partial t} = - \kappa k^2 U + \mathcal{F} \{S(x,t)\}

    Solve the ODE by integrating factor.
    U = \frac{1}{e^{\kappa k^2 t}} \int^t_0 e^{\kappa k^2 t'} \mathcal{F} \{ S(x,t) \} dt'
    U = \int^t_0 e^{-\kappa k^2 (t - t')} \mathcal{F} \{ S(x,t) \} dt'

    Find inverse transform of U.
    u = \frac{1}{\sqrt{2\pi}} \int^\infty_{-\infty} \int^t_0 e^{-\kappa k^2 (t - t')} \mathcal{F} \{ S(x,t) \} dt' e^{-ikx} dk
    u = \frac{1}{\sqrt{2\pi}} \int^t_0 \int^\infty_{-\infty} e^{-\kappa k^2 (t - t')} \mathcal{F} \{ S(x,t) \} e^{-ikx} dk dt'

    Have I got above correct? Why should I use definite integral \int^t_0 instead of indefinite integral \int for solving U? What should I do next? (I know I am using k instead of x' in the given solution but I don't see how this is even close to the solution).

    Should I expand \mathcal{F} \{ S(x,t) \} like below?
    u = \frac{1}{\sqrt{2\pi}} \int^t_0 \int^\infty_{-\infty} e^{-\kappa k^2 (t - t')} \int^\infty_{-\infty} S(x,t) e^{ikx} dx e^{-ikx} dk dt'
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  2. #2
    Super Member
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    Re: Fourier transform to solve inhomogeneous PDE

    I think it would be easier if you used Duhamel's principle. Assuming you know how to solve the homogenous case.
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  3. #3
    Junior Member
    Joined
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    Re: Fourier transform to solve inhomogeneous PDE

    Quote Originally Posted by Jose27 View Post
    I think it would be easier if you used Duhamel's principle. Assuming you know how to solve the homogenous case.
    Thank you. But I have not studied Duhamel's principle before and it is not in my lecture notes. There must be another way to solve the problem.
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