# Method of characteristics to solve the hyperbolic PDE- population balance equation

## 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 im saying here.