# Exchange Problem with PDEPE solver

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Thomas on 14 Jan 2015
Commented: Thomas on 15 Jan 2015
I want to use the PDEPE (or whatever else) solver to describe an exchange model between two different layers. Within the layers I want diffusion to occurr.
I want to start with a "hot" sphere surrounded by a cold spherical shell. Between those two reservoirs I want an exchange of the form d/dt U(leftborder) = -k*(U(leftborder)-U(rightborder). Within the layers I want a propagation (of heat for example) via diffusion.
My problems are:
(1) how to i solve a problem of two partial differential equations that are only coupled at their border in matlab. As far as i understand i can only solve coupled equations that have the same domain, i.e. the same xmesh.
(2) how can I implement a time dependent boundary condition like the one stated above.
Thanks for any help!
Best regards.

Torsten on 14 Jan 2015
There is no ready-to-use solver in MATLAB for this kind of problem.
You will have to discretize the PDEs in space and solve the resulting system of ODEs for the temperatures in the grid points by a solver for ordinary differential equations (e.g.ODE15s).
What second transmission condition should apply at the interface between the two layers ? Continuity of heat flux ?
Best wishes
Torsten.
##### 2 CommentsShowHide 1 older comment
Torsten on 14 Jan 2015
You have two unknowns at the interface: U(leftborder) and U(rightborder). So you need two equations to solve for them.
Usually, if e.g. material properties change at the interface, the transmission conditions are continuity of temperature (Tleft=Tright) and continuity of heat flux (Lambda_left*dT_left/dx=Lambda_right*dT_right/dx).
In your case, only one condition is stated.
Best wishes
Torsten.

Bill Greene on 14 Jan 2015
Torsten said:
"Usually, if e.g. material properties change at the interface, the transmission conditions are continuity of temperature (Tleft=Tright) and continuity of heat flux (Lambda_left*dT_left/dx=Lambda_right*dT_right/dx)."
If, in fact, these are the conditions at the material interface, solution using the pdepe function is straightforward. You have a single xmesh that spans both layers. In your function that calculates the pde coefficients, you use the value of the x-variable that is passed in to calculate the flux, i.e, if x < x_interface, f= k1*dudx, otherwise f = k2*dudx.
Bill
Thomas on 15 Jan 2015
Hm, okay it looks like i will have to do a numeric solution then! Thanks for the help!