I am looking for an analytical solution of the 2nd order PDE as in the title. The problem would seem simple: I am referring to the case of the desorption of a volatile component from a solution layer in a closed vessel of finite geometry (p, T = normal).
I just try to find the time to establish a gas-liquid equilibrium. I mean vertical 1D model with no horizontal gradients. Unfortunately the literature research was ineffective. The only suggestion I found in a certain monograph does not make any sense. (Concentration disappears when time goes to infinity).
I'd be greateful for any suggestions.
Approximate solution for computation or semi-closed form (semi-analytic solution) is here : https://researchbank.swinburne.edu.au/file/e3f03925-ce94-419d-9f7c-444cf60a29bc/1/PDF%20%28Published%20version%29.pdf
Thank you Muhammad Ali very much for the answer and the actions taken. However, please note that in both cases you kindly suggested the models concern infinite or semi-finite objects described by Gaussians. These models are valid if the gaussian falls within the object's dimensions. The criterion is the standard deviation (sqrt (2Dt)). When the gaussian "tails" extend beyond the object dimension 0
Hello.
Your intuition is quite right :-).
Second Fick law is a parabolic second order partial differential equation (PDE). It can be solved by technique called separation of variables on finite domain. One have to suppose that solution C(x,t) consists of two independent functions X(x) and T(t) coupled this way: C(x,t)=X(x)T(t). Therefore ∂2C/∂x2=T∂2X/∂x2 and ∂C/∂t= X∂T/∂t. Using this assumption the second Fick equation splits in two ordinary differential equations (ODE) for T and X.
Solution then can be found in form of infinite Fourier series, therefore you have had the right feeling about it.
BUT, existence and uniqueness of PDE solution is strongly dependent on boundary conditions and initial condition. Although you already suggested the boundary conditions of Neumann type, the initial condition in form C(x,0)=f(x) is still missing
thank you very much Jan Simon (though a bit late) for valuable and accurate inspiration. I think I found what I was looking for with the variable separation method. It cost me a lot of sweat and my wife took offense at me a bit. Of course, the boundary conditions are decisive. I gave up the orthodox physical approach and, surprisingly, got a reasonable solution. However, I don't think I've discovered America.
Thank You again. Tomorrow I'm going to the flower shop. ;) br
A.
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