Solve PDE system - interface problem between two domains

I am trying to solve a PDE system in one space dimension using pdepe function. I have two equations, each of them applied to different domains and I need to calculate the time evolution in the interface, I am attaching the equations, boundary conditions and initial conditions.

I do not see how could I define a boundary condition in the interface and also different domains for each equations. Is it possible to do it in this way? If not, could anybody indicate me an alternative to solve the problem?

Thanks

[Bill Greene]

No boundary or interface condition is needed where the material properties are discontinuous. The first argument to pdefun is x; you simply evaluate the coefficients based on the value of x. The pdepe documentation discusses this issue specifically: "Discontinuities in c and/or s due to material interfaces are permitted provided that a mesh point is placed at each interface."

Answer 1

[Torsten]

If the condition you state at x=delta is correct, you can solve the two PDEs one after the other since u1 and u2 are not coupled. I assume that u_i(t) is a function that is known a priori.

Best wishes

Torsten.

[MFV]

Thank you for your reply. Now I realize I did not outline the problem clearly, actually u_i(t) is the function I need to determine.

[Torsten]

What exactly are your transmission conditions between Domain 1 and Domain 2 for u1 and u2 ?

Best wishes

Torsten.

[MFV]

I am not sure I understood you properly. Between Domain 1 and Domain 2, u1 and u2 must be equal. Also, it must be equal:

-B_1·(∂u_1/∂x)+K·C·u_1 = -B_2·(∂u_2/∂x)+K·u_2

Thank you

[Torsten]

If those two conditions are your transmission conditions, you can proceed as Bill Greene suggests: You can solve both equations in the union of Domain 1 and Domain 2 and set the PDE coefficients depending on the actual x-position:

if x<=Delta
  c=...;
  s=...;
  f=...;
else
  ...
end if

One grid point should be placed at x=Delta.

The transmission conditions will be automatically satisfied.

Best wishes

Torsten.

[Torsten]

In your code, I guess f is your function, but what about c and s?

No. c, f and s is the usual notation for parts of the PDE equation that is to be solved by "pdepe".

Take a look at the documentation under

https://uk.mathworks.com/help/matlab/ref/pdepe.html

Example

"Solve Heat Equation in Cylindrical Coordinates"

Section

Code Equation