Solving PDE involving boundary condition with partial derivatives with respect to space and time.

Question

Lei Huang 2015-12-16

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评论： Lei Huang 2019-4-1

采纳的回答： Torsten

Dear all,

I am trying to solve the following PDE (one-dimension diffusion):

with the following boundary condition :

I read the documentation here : http://www.mathworks.com/help/matlab/math/partial-differential-equations.html?refresh=true#f1-697925 and I saw that it works for boundary conditions that can be expressed as :

I don't know how to express my boundary condition under this form, since it contains two partial derivatives. How to address this problem properly?

Thank you.

PS: I have quite a few libraries available, but not the PDE toolbox.

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Cedric 2015-12-17

So your question is not how to fully solve your differential equation, but rather how to treat this kind of problems numerically with MATLAB?

Lei Huang 2015-12-17

编辑：Lei Huang 2015-12-17

Hi Torsten,

Thanks for your comments! I'm modeling the emission of chemicals from a material placed indoor and here's a diagram of the system:

The PDE describes the dynamics of the chemical concentration in the material. Initial condition is that the chemical is uniformly distributed throughout the material:

The first boundary condition assumes that there's no flux out of the base of the material:

The other boundary condition is imposed through a mass balance on the chemical in the room air:

And we assumed that equilibrium exists between the chemical concentration in the surface layer of the material and the room air:

Combining the above two equations we get the "strange" boundary condition that I mentioned.

The whole system is described in this article http://www.sciencedirect.com/science/article/pii/1352231094900973

I think this system is solvable because the author gave an analytical solution to it. Does anybody have experience solving this kind of problems numerically? Any hints will be appreciated!

Best, Lei

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Answer 1

Torsten 2015-12-18

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编辑：Torsten 2015-12-18

在 MATLAB Online 中打开

In principle, your boundary condition for C is

C|x=L = y*K_v

together with an ordinary differential equation for y that has to be solved simultaneously with your PDE

V*dy/dt = -D*A*dC/dx|x=L - Vdot*(y-y_in).

The solution of this ordinary differential equation is not possible within pdepe.

But it should be no problem to discretize the PDE in space and solve the resulting system of ordinary differential equations together with the ordinary differential equation for y using ODE15S, e.g. (method-of-lines).

Maybe you have access to COMSOL Multiphysics. With this program, you can easily couple the PDE with the above boundary ODE.

Best wishes

Torsten.

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CHORFi Salah 2019-1-27

Hi Lei,

I'm facing a similar but more complicated problem, can you provide your code to couple the two equations.

Thanks in advance.

Lei Huang 2019-4-1

Hi all,

I was able to discretize the system using Method-of-Lines and solve it numerically, but I've encountered instability problems with this solution. If interested please check out my new question here: https://www.mathworks.com/matlabcentral/answers/453710-instability-in-method-of-lines-discretization-of-a-1-d-diffusion-problem

Thanks, Lei

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Solving PDE involving boundary condition with partial derivatives with respect to space and time.

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