Solve two species coupled linear reaction-functional diffusion

Hi everyone,

I am looking to solve some coupled pdes with neumann zero-flux bc and random ic. Ideally I would like to have a 2d or 3d solution but 1d will suffice for now. The two species involve a functional diffusion model which considers the difference in diffusion between the two species. This difference is then considered by a sigmoidal function to define positive or negative flow. There is also exponential decay of each of the species.

du/dt = - a1*u + 1/(1+exp(a2*d2u/dx2-a3*d2v/dx2)) - 1/2

dv/dt = - a4*v + 1/(1+exp(a5*d2u/dx2-a6*d2v/dx2)) - 1/2

Does anyone know of any off the shelf tools in matlab that can do this? I have the code running in mathematica but it is painfully slow.

Thanks for your help in advance :)

Answer 1

[Zhang lu]

you can convert the pde function as follow

d2u/dx2=[a5*ln(1/(du/dt+a1*u+1/2)-1)-a2*ln(1/(dv/dt+a1*u+1/2)-1)]/(a2*a6-a3*a5)

d2v/dx2=[a6*ln(1/(du/dt+a1*u+1/2)-1)-a3*ln(1/(dv/dt+a1*u+1/2)-1)]/(a2*a6-a3*a5)

Then, i think you can solve it .

[Isaac]

Thanks for the reply. However, I am not exactly sure how this helps as the functional is now on the d/dt instead d2/dx2. Can you please point me in the right direction

[Zhang lu]

what is boundary condition ?

[Isaac]

Hi Zhang,

Sorry for the late reply. I am looking to solve this with zero flux boundary conditions.

Thanks for your help.