Re: question about NDSolve for Diffusion equation
- To: mathgroup at smc.vnet.net
- Subject: [mg102870] Re: question about NDSolve for Diffusion equation
- From: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>
- Date: Tue, 1 Sep 2009 03:50:37 -0400 (EDT)
Hi, Nguyen, If you put your first condition C(x,0)=0, then it must be zero also at x=0. This is consistent with your second condition, but is inconsistent with your third condition C(0,t)=Cs, since from this condition C(0,0) must be equal to Cs. For the same reason your second condition is inconsistent with the third one. You may argue that the third condition is formulated for t strictly larger than zero and is not applied at t=0, but in this case it is a "difficult" boundary condition with a discontinuity just in the point of the boundary (regarding space-time as a domain and t=0 as its boundary). It should be difficult to implement numerically. Let me propose you a more simple approach. First condition: C(0,t)=Cs at all t>=0. Second condition: C(x, 0)=Cs*f(x), where f(x) is some function obeying the following conditions: f(0)=1; f(x) monotonously decreases with x and f(infinity)=0. For example, you may try f(x)=Exp(-a*x) or f(x)=Exp[-a*x^2]. You may then look how the result depends upon the parameter a, and chose a values at which this dependence can be ignored. If in contrast you find that it cannot be ignored, then this means that such an initial distribution is crucial and must be taken into account. Physically such a statement is more realistic. It reflects the fact that as soon as you enter a sample into a measurement chamber with the diffusing agent, the diffusion starts even before your measurement starts, and when you begin measuring some distribution already exists. If my idea does not meet some peculiarities of your problem that are unknown to me at present, you have to think yourself of kinetics of establishing an equilibrium at the body boundary in your case, and to deduce the boundary conditions from this kinetics. As it was already mentioned you probably meant dC/dt, rather than the second derivative d''C/dt^2 ?? Have fun, Alexei Dear Moderator, again this is a question on Physics, rather than on Mathematica, sorry. Hello, I am trying to use the NDSolve of Mathematica to solve the diffusion equation which has diffusivity dependence to concentration. My work is dealing with phosphorous diffusion into silicon. The equation is d''C/dt^2 = d( C^2 *dC/dx)/dx where C= C(x,t). In my problem, the initial conditions should be : C(x,0)=0, C(0,0) = 0, C(0,t) = Cs if t > 0. If I used the NDSolve, there is a error message, the Initial conditions are not consistent. I dont known how to go further as I have checked again and again the initial conditions are fine. Your helps anf advice are gratefully acknowledged, Nam Nguyen. Stuttgart, Germany -- Alexei Boulbitch, Dr., habil. Senior Scientist IEE S.A. ZAE Weiergewan 11, rue Edmond Reuter L-5326 Contern Luxembourg Phone: +352 2454 2566 Fax: +352 2454 3566 Website: www.iee.lu This e-mail may contain trade secrets or privileged, undisclosed or otherwise confidential information. If you are not the intended recipient and have received this e-mail in error, you are hereby notified that any review, copying or distribution of it is strictly prohibited. Please inform us immediately and destroy the original transmittal from your system. Thank you for your co-operation.