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Complex PDE

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Hello,

I have a complex set of three PDEs which I am trying to simulate in MATLAB:

Here are the equations:

da/dt=k1*(1/(1+s^m))*(a^n/(1+a^n))-a-d1/(1+Ta)*a*(1-c)+D1/c*da/dx(dotproduct)dc/dx
ds/dt=k2*c*(a^n/(1+a^n))-d2*s*D2*d2s/dx2
dc/dt=d1/(1+Ta)*c*(1-c)+D3*d2c/dx2

I have simulated with the PDE form before when I do not have the last term of the first equation. However, this term D1/c*da/dx(dotproduct)dc/dx makes the problem very complex. Can I do the simulations with the PDE app or do I have to start from scratch on an m file? If I do my own m file, what would the syntax be for this set of PDEs?

Any help would be greatly appreciated as I am having much difficulty with this.

Thank you,

Stephen

5 Replies Last Post Feb 4, 2010, 11:27 a.m. EST
Ivar KJELBERG COMSOL Multiphysics(r) fan, retired, former "Senior Expert" at CSEM SA (CH)

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Posted: 1 decade ago Jan 27, 2010, 4:20 a.m. EST
Hi

Sorry but I have some problems to understand you, could you be somewhat more explicit about your variables:

I understand that you have :

space variables and time: x, t

dependent variables : a, c, s all functions of (x,t)

constants: k1, k2, d1, d2, D1, D2, D3, Ta, n, m

to use COMSOL notations: da/dt = at, da/dx = ax, d^2s/dx^2 = sxx, etc.

You rewrite your expression to the type "something = 0" to be plug-in compatible with COMSOL

Now remains the "ax(dotprduct)cx" but if a=a(x,t) and the same for c, then isn't the "dotproduct" simply the scalar product ? If not isnt it just to write it out related to the variables I may have missed ?

I'm missing something essential here I believe ;)

Good luck
Ivar
Hi Sorry but I have some problems to understand you, could you be somewhat more explicit about your variables: I understand that you have : space variables and time: x, t dependent variables : a, c, s all functions of (x,t) constants: k1, k2, d1, d2, D1, D2, D3, Ta, n, m to use COMSOL notations: da/dt = at, da/dx = ax, d^2s/dx^2 = sxx, etc. You rewrite your expression to the type "something = 0" to be plug-in compatible with COMSOL Now remains the "ax(dotprduct)cx" but if a=a(x,t) and the same for c, then isn't the "dotproduct" simply the scalar product ? If not isnt it just to write it out related to the variables I may have missed ? I'm missing something essential here I believe ;) Good luck Ivar

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Posted: 1 decade ago Jan 27, 2010, 3:54 p.m. EST
Thank you so much for your response.

Your assumptions are all correct except T is a constant and a is the depedent variable. The term here in question is the dot product between the gradient of a and the gradient of c with D2/c being the coefficient. So, yes, this does give you a scalar, but this scalar is dependent on both space (since it depends on the gradients) and time (since both variables vary with time). My problem is that I do not know if there is a built-in module of this PDE type or if I need to do it from scratch. If I do it from scratch, I am not sure the proper notation as I have not seen anything like this in the manual.

Any help would be greatly appreciated,

Stephen
Thank you so much for your response. Your assumptions are all correct except T is a constant and a is the depedent variable. The term here in question is the dot product between the gradient of a and the gradient of c with D2/c being the coefficient. So, yes, this does give you a scalar, but this scalar is dependent on both space (since it depends on the gradients) and time (since both variables vary with time). My problem is that I do not know if there is a built-in module of this PDE type or if I need to do it from scratch. If I do it from scratch, I am not sure the proper notation as I have not seen anything like this in the manual. Any help would be greatly appreciated, Stephen

Ivar KJELBERG COMSOL Multiphysics(r) fan, retired, former "Senior Expert" at CSEM SA (CH)

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Posted: 1 decade ago Jan 27, 2010, 4:44 p.m. EST
Hi

great then the "Ta" is a typo and should be read "T*a" with "T" a constant.

so the three equations become:

at=k1*(1/(1+s^m))*(a^n/(1+a^n))-a-d1/(1+T*a)*a*(1-c)+D1/c*ax*cx
st=k2*c*(a^n/(1+a^n))-d2*s*D2*sxx
ct=d1/(1+T*a)*c*(1-c)+D3*cxx

space variables: x,t
dependent variables: a,s,c
k1,d1,T,D1,k2,d2,D2,D3,m,n all constants and with implicit:
a=a(x,t) at=da(x,t)/dt ax=da(x,t)/dx ...
s=s(x,t) st=ds(x,t)/dt sx=ds(x,t)/dx sxx=d^2s(x,t)/dx^2 ...
c=c(x,t) ct=dc(x,t)/dt cx=dc(x,t)/dx cxx=d^2c(x,t)/dx^2 ...


So far I understand it, one must now reorder the equations with some algebra to get them to fit the canonical coefficient form of COMSOL, unfortunately I have not been playing that much with arbitrary PDEs and COMSOL so I would need some time to sort it out (if possible), and it's getting late form me just now. Furthermore, I still havnt catched the reason why the dotproduct, in this case makes it so much worse (I find it awfull enough as is, OK ;)

In anycase I would typically start to read through my copy of :

www.comsol.com/academic/books/mmwfem/

to be inspired. There are some good exercices therein starting from the PDE's and reordering them to be COMSOL compliant, so I can only propose that you borrow this book, or perhaps you have it already ?

Good luck
Ivar
Hi great then the "Ta" is a typo and should be read "T*a" with "T" a constant. so the three equations become: at=k1*(1/(1+s^m))*(a^n/(1+a^n))-a-d1/(1+T*a)*a*(1-c)+D1/c*ax*cx st=k2*c*(a^n/(1+a^n))-d2*s*D2*sxx ct=d1/(1+T*a)*c*(1-c)+D3*cxx space variables: x,t dependent variables: a,s,c k1,d1,T,D1,k2,d2,D2,D3,m,n all constants and with implicit: a=a(x,t) at=da(x,t)/dt ax=da(x,t)/dx ... s=s(x,t) st=ds(x,t)/dt sx=ds(x,t)/dx sxx=d^2s(x,t)/dx^2 ... c=c(x,t) ct=dc(x,t)/dt cx=dc(x,t)/dx cxx=d^2c(x,t)/dx^2 ... So far I understand it, one must now reorder the equations with some algebra to get them to fit the canonical coefficient form of COMSOL, unfortunately I have not been playing that much with arbitrary PDEs and COMSOL so I would need some time to sort it out (if possible), and it's getting late form me just now. Furthermore, I still havnt catched the reason why the dotproduct, in this case makes it so much worse (I find it awfull enough as is, OK ;) In anycase I would typically start to read through my copy of : http://www.comsol.com/academic/books/mmwfem/ to be inspired. There are some good exercices therein starting from the PDE's and reordering them to be COMSOL compliant, so I can only propose that you borrow this book, or perhaps you have it already ? Good luck Ivar

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Posted: 1 decade ago Jan 27, 2010, 4:55 p.m. EST
Thank you so much for your help. One small thing: st=k2*c*(a^n/(1+a^n))-d2*s*D2*sxx should be st=k2*c*(a^n/(1+a^n))-d2*s+D2*sxx (maybe it was a typo). Anyways, the book you mentioned: is this available for free online? Thank you again for your help, and I hope to hear from you regarding getting the equations in the canonical coefficient form of COMSOL.

Stephen


Hi

great then the "Ta" is a typo and should be read "T*a" with "T" a constant.

so the three equations become:

at=k1*(1/(1+s^m))*(a^n/(1+a^n))-a-d1/(1+T*a)*a*(1-c)+D1/c*ax*cx
st=k2*c*(a^n/(1+a^n))-d2*s*D2*sxx
ct=d1/(1+T*a)*c*(1-c)+D3*cxx

space variables: x,t
dependent variables: a,s,c
k1,d1,T,D1,k2,d2,D2,D3,m,n all constants and with implicit:
a=a(x,t) at=da(x,t)/dt ax=da(x,t)/dx ...
s=s(x,t) st=ds(x,t)/dt sx=ds(x,t)/dx sxx=d^2s(x,t)/dx^2 ...
c=c(x,t) ct=dc(x,t)/dt cx=dc(x,t)/dx cxx=d^2c(x,t)/dx^2 ...


So far I understand it, one must now reorder the equations with some algebra to get them to fit the canonical coefficient form of COMSOL, unfortunately I have not been playing that much with arbitrary PDEs and COMSOL so I would need some time to sort it out (if possible), and it's getting late form me just now. Furthermore, I still havnt catched the reason why the dotproduct, in this case makes it so much worse (I find it awfull enough as is, OK ;)

In anycase I would typically start to read through my copy of :

www.comsol.com/academic/books/mmwfem/

to be inspired. There are some good exercices therein starting from the PDE's and reordering them to be COMSOL compliant, so I can only propose that you borrow this book, or perhaps you have it already ?

Good luck
Ivar


Thank you so much for your help. One small thing: st=k2*c*(a^n/(1+a^n))-d2*s*D2*sxx should be st=k2*c*(a^n/(1+a^n))-d2*s+D2*sxx (maybe it was a typo). Anyways, the book you mentioned: is this available for free online? Thank you again for your help, and I hope to hear from you regarding getting the equations in the canonical coefficient form of COMSOL. Stephen [QUOTE] Hi great then the "Ta" is a typo and should be read "T*a" with "T" a constant. so the three equations become: at=k1*(1/(1+s^m))*(a^n/(1+a^n))-a-d1/(1+T*a)*a*(1-c)+D1/c*ax*cx st=k2*c*(a^n/(1+a^n))-d2*s*D2*sxx ct=d1/(1+T*a)*c*(1-c)+D3*cxx space variables: x,t dependent variables: a,s,c k1,d1,T,D1,k2,d2,D2,D3,m,n all constants and with implicit: a=a(x,t) at=da(x,t)/dt ax=da(x,t)/dx ... s=s(x,t) st=ds(x,t)/dt sx=ds(x,t)/dx sxx=d^2s(x,t)/dx^2 ... c=c(x,t) ct=dc(x,t)/dt cx=dc(x,t)/dx cxx=d^2c(x,t)/dx^2 ... So far I understand it, one must now reorder the equations with some algebra to get them to fit the canonical coefficient form of COMSOL, unfortunately I have not been playing that much with arbitrary PDEs and COMSOL so I would need some time to sort it out (if possible), and it's getting late form me just now. Furthermore, I still havnt catched the reason why the dotproduct, in this case makes it so much worse (I find it awfull enough as is, OK ;) In anycase I would typically start to read through my copy of : http://www.comsol.com/academic/books/mmwfem/ to be inspired. There are some good exercices therein starting from the PDE's and reordering them to be COMSOL compliant, so I can only propose that you borrow this book, or perhaps you have it already ? Good luck Ivar [/QUOTE]

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Posted: 1 decade ago Feb 4, 2010, 11:27 a.m. EST
I was wondering if you had any additional thoughts on this. We're really struggling trying to find a form which COMSOL can handle.

Stephen
I was wondering if you had any additional thoughts on this. We're really struggling trying to find a form which COMSOL can handle. Stephen

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