# Method of Characteristics for Hyperbolic PDE

1. Apr 16, 2013

### Graham Power

I am trying to build a program in Matlab to solve the following hyperbolic PDE by the method of characteristics

∂n/∂t + G(t)∂n/∂L = 0

with the inital and boundary conditions

n(t,0)=B(t)/G(t) and
n(0,L)=ns

Here ns is an intial distribution (bell curve) but I dont have a function to fit it. I just have different values of n at various sizes, L (from 0 to 1000μm).

I know that the method of characterisitics gives the solution

∂n/∂s=0
∂t/∂s=1
∂L/∂s=G(t)

which gives the characteristic curve as L-G(t)*t=L0. So the values of n @ any initial size, L0, remain constant along the characteristic curve.

My problem is how do I incorporate the original n values at time zero and various sizes into the equation. I want to say:

n@(L0)=n@(L-G(t)*t)

If the initial condition was given as a function of L0, say n(0,L)= f(L0)=sin(L0), then I could just sub in the characteristic equation and calculate n(t,L) as sin(L-G(t)*t). But I dont have a function that describes the initial condition, just vaues of n at various sizes, L as said above.

The other problem is the boundary condition, this should make up one section of the solution and the initial condition the other.

But again how do I actually say n@(t,0)=B(t)/G(t)=n@(L-G(t)*t).

Say if n(t,0)=g(t), so the boundary condition is a function of time explicitly, not the case for me, B and G are actually a function of concentration which is changing with time, then the solution for the PDE above subject to the boundary would be;

n(t,L)=g(t-(L/G(t))

So I get the method overall, Im just confused how I propagate each value when I dont have functions to describe the initial and boundary conditions, like f(L0) and g(t) shown above.

$$L=L_0+\int_0^t{G(\xi)d\xi}$$