Finite difference method for a system of pde

Hello, I'm trying to solve a system of three PDEs using the partial difference method, but I'm having a lot of trouble understanding how to start.

The equations are:

F'' + f*F'+beta*(rho(e)/rho - F^2) + alpha*s^2 = 2*xi*(F*(dF/dxi) - (df/dxi)*F')

s'' + f*s'=2*xi*(F*(ds/dxi) - (df/dxi)*s') + alpha_1*F*s

G''/Pr + f*G' + 2*(1-(Pr)^(-1))*[E*F*F' + Qs*s*s']' = 2*xi*(F*(dG/dxi) - (df/dxi)*F)

Where beta, alpha, alpha_1, E, Qs should depend from a parameter x, but can be considered constant.

If anyone has any idea on how I can procede to find the solution, it will be very appreciated.

I'll also attach the file with the complete equations and the relations between the terms, if you want to take a look. I'm studying the stationary problem.

Thank you

[Torsten]

Why do you think you have a system pf PDEs ? Aren't these three 2nd order ODEs ?

[Gloria Mazzeo]

Well the primes indicate derivatives with respect to eta, so I have both derivatives of eta and xi. I'm sorry I didn't specify, I recognize it was misunderstandable

Answer 1

[Torsten]

These kind of transformations are usually made to transform a system of PDEs to a system of ODEs that can be solved more easily.

E.g. in this article

https://oldwww.ippt.pan.pl/Repository/am-v49-no1-pp111-127.pdf

it's more obvious that equations (3.1) without the differentiation with respect to time are "simple" 2nd order differential equations in the independent variable "eta". My guess is that this is similar in your article because both start with the unsteady compressible flow equations.

To solve a system of 2nd order ordinary differential equations, use "bvp4c" or "bvp5c".