Simulating pipeline behaviour using ODEs simplified from PDEs

Hi, I am working on my Final Year Project,which is related to the leakage detection system, and I am required to run simulation of pipeline using the theoretical partial differential equations.

I have simplified the PDEs to ordinary differential equations, and I would like to run the simulation in MATLAB, the simplified ODEs are shown as below,

P(i,j+1) - 0.5*c1*v(i+1,j+1) + 0.5*c1*v(1,j+1) = P(i,j) + 0.5*c1*v(i+1,j) - 0.5*c1*v(i,j)

c1 = -rou*(a^2)*(deltaT/deltaX)

v(i,j+1) + 0.5*c2*P(i+1,j+1) + 0.5*c2*P(1,j+1) = v(i,j) + 0.5*c2*P(i+1,j) - 0.5*c2*P(i,j)

c2=(1/rou)*(deltaT/deltaX)

while rou, a, deltaT and deltaX are constant.

Could anyone please give me advice to put the equations in MATLAB and run the simulation? Thanks!

[Zhang lu]

What is boundary condition of your PDEs ?