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Applying initial conditions to functions that are part of an Eigenfunction Expansion

  1. Jan 6, 2012 #1

    ryan.j

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    Gold Member

    I'm trying to solve a non-linear time-dependent diffusion equation to find R(x,t). To do so, I'm positing that :

    R(x,t)=[itex]\sum^{J}_{1}[/itex]X[itex]_{i}[/itex](x)T[itex]_{i}[/itex](t)

    which allows me to arrive at something that looks like :

    dT[itex]_{i}[/itex]/dt=A[itex]_{i}[/itex]T[itex]_{i}[/itex](t)-B*T[itex]_{i}[/itex](t)[itex]^{2}[/itex]

    The problem I'm having, through sheer lack of knowledge, is ascribing initial conditions to T[itex]_{i}[/itex](t).

    I know that R(x,0) = 1. Taking the case where J = 3, for example, can I simply say that T[itex]_{i}[/itex](0) = 1/3? If not, is there a way to determine the initial conditions for each T[itex]_{i}[/itex](t), given that I know that they need to sum to 1?

    Thank you kindly for any help.
    -ryan
     
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