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Coupled differential equations

Posted 9 years ago

Hello, I am working on a system of three coupled differential equations describing the diffusion and complexation of a metal and a ligand. I am looking for the solution of the functions f[x,t], g[x,t] and h[x,t] on the region S < x < R. The system is the following:

eqnsNC = { D[f[x, t],t]== DM * D[f[x, t], x,x]-kf * f[x, t] * h[x, t]+kdis * g[x, t], D[h[x, t], t]==DL * D[h[x, t], x,x]-kf * f[x, t] *h[x, t]+kdis * g[x, t],D[g[x, t], t]==DML * D[g[x, t], x,x]+kf * f[x, t] * h[x, t]-kdis *g[x, t]};

with the boundary conditions:

 bc={f^(1,0)[R,t]==g^(1,0)[R,t]==h^(1,0)[R,t]==0, f[x,0]==g[x,0]==h[x,0]==0, f[S,t]==MS * (1-Exp[-1000 t]), g[S,t]==(kf * LS* MS * (1-Exp[-1000 t]))/kdis, h[S,t]==LS * (1-Exp[-1000 t])};

and some constants:

 DM = 5*10^-6;   DL = 5*10^-6; DML = 5*10^-6; kf = 10^3; kdis = 63.8; MS = 0.25; LS = 0.2;  R = 0.12; NN = 0.1; S = 0.05; tf = 10000;

I can then solve this system using the command:

{fsol2, gsol2, hsol2} =  NDSolveValue[{eqnsNC, bc}, {f, g, h}, {x, S, R}, {t, 0, tf}]

The next step that is difficult for me is too add the extra condition that the function f[x,t] is identically 0 over the region NN <= x <R. Therefore I have 2 different systems of equations over the regions S < x < NN and over the region NN<= x < R. When S < x < NN, the equations are as above:

eqnsNC = { D[f[x, t],t]== DM * D[f[x, t], x,x]-kf * f[x, t] * h[x, t]+kdis * g[x, t], D[h[x, t], t]==DL * D[h[x, t], x,x]-kf * f[x, t] *h[x, t]+kdis * g[x, t],D[g[x, t], t]==DML * D[g[x, t], x,x]+kf * f[x, t] * h[x, t]-kdis *g[x, t]};

and when NN<= x < R the system reduces to

eqnsNCR = { D[h[x, t], t]==DL * D[h[x, t], x,x]+kdis * g[x, t], D[g[x, t], t]==DML * D[g[x, t], x,x]-kdis *g[x, t]};

However, the boundary conditions are only given in x=S and x=R, but in x=NN I only have the condition f[NN,t]==0.

I don't know how to implement this situation and how to define 2 different sets of equations on adjacent domains with no explicit boundary conditions at the border.

Does anyone have an idea? thank a lot for your answers!

POSTED BY: Maud J
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