- #1
Graham Power
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Homework Statement
The PDE: ∂n/∂t + G∂n/∂L=0
The initial condition: n(0,L)=ns
The boundary condition: n(t,0)=B/G
The parameter B and G above are dependent upon process conditions and change at each time. They can be calculated with adequate experimental data.
Homework Equations
I know that the characteristic equations for this PDE are:
∂n/∂s=0 (1)
∂L/∂s=G (2)
∂t/∂s=1 (3)
Solving the above:
n=n0 (4)
t=s (5)
L=Gt+L0 (6)
The Attempt at a Solution
From what I can see, the above equations suggest that along a characteristic curve given by equation (6), the population density, n, at size L0, travels along the size axis with rate of growth, G.
The initial condition, n(0,L)=ns describes the population of particles over a given size range. So the initial data I have is a number of values of population density, n, at a number of sizes.
Does the solution suggest that each initial value of population density, n corresponding to an initial size, L0 will stay constant along equation (6) and correspond to a different size in the next time step? I hope I am clear in what I am saying here.