Dealing with boundary conditions in system of ODEs

[582153236]

Homework Statement

I'm trying to plot the steady state concentration of y_A vs. x, y_B vs x and y_u vs x using centered finite difference method.

Homework Equations

The Attempt at a Solution

τ represents the dimensionless time variable, so steady state would mean that the left hand side of each of the differential equations is 0.

I began this problem by replacing the derivatives that appear in the differential equations with the finite difference approximations for them. For example, y_A:

I also used the boundary conditions:

but I don't know at which point, I am taking y_A on the right side from. In the below equation should y_A on the right side be y_A,1 or something else (since the below equation estimates the derivative at the node n=1?

More importantly, how do I deal with this type of boundary condition in Matlab? I am used to dealing with boundary conditions such as y_A(0,t)=0 for which I can simply initialize Y(1)=0 in MATLAB but in this case I'm given a derivative boundary condition.

 

[SteamKing]

These aren't ODEs; they're PDEs.

 

[Chestermiller]

What you do is add a fictitious point at -1 such that:

$$\frac{y_{1}-y_{-1}}{2h}=-(1-y_0)$$

Combining this with $\frac{(y_{-1}+2y_0+y_1)}{h^2}$ gives:

$$\frac{(y_{-1}-2y_0+y_1)}{h^2}=\frac{2(y_1-y_0)}{h^2}-\frac{2(1-y_0)}{h}$$

You solve the time-dependent equation at x = 0, and use this in that equation. This is a standard way of handling that type of flux boundary condition.

Chet