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From: Wallace Yang on 5 Aug 2010 18:13
Hi:

I'm new to the pdetool and was wondering if it can solve the following (electrostatic) problem:

There's an infinite rod of uniform charge density inside a solution. Normally, the poisson equation governs the behaviour of electrostatics. However, the charge distribution of mobile ions within the solution gets attracted to the charged rod and requires a slight alteration of the poisson equation, turning it into the poisson-boltzmann equation.

div(e(r)grad(V(r)))= rho(r) + summation(q[i]c[i]exp(-q[i]V(r)/kT)

where e(r) is the relative permitivity, V(r) is voltage, rho(r) is charge density, q[i], c[i] are the charge and density of ion i, and kT are constants.

I wish to find V(r) for all r. Is it possible to use pdetool? If not, what other pde tools are out there?

Thanks.
From: Torsten Hennig on 5 Aug 2010 22:35
> Hi:
>
> I'm new to the pdetool and was wondering if it can
> solve the following (electrostatic) problem:
>
> There's an infinite rod of uniform charge density
> inside a solution. Normally, the poisson equation
> governs the behaviour of electrostatics. However, the
> charge distribution of mobile ions within the
> solution gets attracted to the charged rod and
> requires a slight alteration of the poisson equation,
> turning it into the poisson-boltzmann equation.
>
> div(e(r)grad(V(r)))= rho(r) +
> summation(q[i]c[i]exp(-q[i]V(r)/kT)
>
> where e(r) is the relative permitivity, V(r) is
> voltage, rho(r) is charge density, q[i], c[i] are the
> charge and density of ion i, and kT are constants.
>
> I wish to find V(r) for all r. Is it possible to use
> pdetool? If not, what other pde tools are out there?
>
> Thanks.

The pde-toolbox can handle nonlinear elliptic pdes
of the form
-div(c(u)grad(u)) + a(u)*u = f(u) ;
so formally, the solution of your pde is possible.

Best wishes
Torsten.
From: Torsten Hennig on 5 Aug 2010 22:55
> > Hi:
> >
> > I'm new to the pdetool and was wondering if it can
> > solve the following (electrostatic) problem:
> >
> > There's an infinite rod of uniform charge density
> > inside a solution. Normally, the poisson equation
> > governs the behaviour of electrostatics. However,
> the
> > charge distribution of mobile ions within the
> > solution gets attracted to the charged rod and
> > requires a slight alteration of the poisson
> equation,
> > turning it into the poisson-boltzmann equation.
> >
> > div(e(r)grad(V(r)))= rho(r) +
> > summation(q[i]c[i]exp(-q[i]V(r)/kT)
> >
> > where e(r) is the relative permitivity, V(r) is
> > voltage, rho(r) is charge density, q[i], c[i] are
> the
> > charge and density of ion i, and kT are constants.
> >
> > I wish to find V(r) for all r. Is it possible to
> use
> > pdetool? If not, what other pde tools are out
> there?
> >
> > Thanks.
>
> The pde-toolbox can handle nonlinear elliptic pdes
> of the form
> -div(c(u)grad(u)) + a(u)*u = f(u) ;
> so formally, the solution of your pde is possible.
>
> Best wishes
> Torsten.

Or do you have to determine the other quantities
with the pde-toolbox, too ?
If yes: How do the corresponding equations look like ?

Best wishes
Torsten.
From: Wallace Yang on 6 Aug 2010 13:30
Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote in message <1428430446.65619.1281077778709.JavaMail.root(a)gallium.mathforum.org>...
> > > Hi:
> > >
> > > I'm new to the pdetool and was wondering if it can
> > > solve the following (electrostatic) problem:
> > >
> > > There's an infinite rod of uniform charge density
> > > inside a solution. Normally, the poisson equation
> > > governs the behaviour of electrostatics. However,
> > the
> > > charge distribution of mobile ions within the
> > > solution gets attracted to the charged rod and
> > > requires a slight alteration of the poisson
> > equation,
> > > turning it into the poisson-boltzmann equation.
> > >
> > > div(e(r)grad(V(r)))= rho(r) +
> > > summation(q[i]c[i]exp(-q[i]V(r)/kT)
> > >
> > > where e(r) is the relative permitivity, V(r) is
> > > voltage, rho(r) is charge density, q[i], c[i] are
> > the
> > > charge and density of ion i, and kT are constants.
> > >
> > > I wish to find V(r) for all r. Is it possible to
> > use
> > > pdetool? If not, what other pde tools are out
> > there?
> > >
> > > Thanks.
> >
> > The pde-toolbox can handle nonlinear elliptic pdes
> > of the form
> > -div(c(u)grad(u)) + a(u)*u = f(u) ;
> > so formally, the solution of your pde is possible.
> >
> > Best wishes
> > Torsten.
>
> Or do you have to determine the other quantities
> with the pde-toolbox, too ?
> If yes: How do the corresponding equations look like ?
>
> Best wishes
> Torsten.

Thank you for your help, Torsten. For now, I will assume that I have the other quantities. How would I be able to specify that V=0 at r = infinity?

Thanks.
From: Torsten Hennig on 7 Aug 2010 01:55
> Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote
> in message
> <1428430446.65619.1281077778709.JavaMail.root(a)gallium.
> mathforum.org>...
> > > > Hi:
> > > >
> > > > I'm new to the pdetool and was wondering if it
> can
> > > > solve the following (electrostatic) problem:
> > > >
> > > > There's an infinite rod of uniform charge
> density
> > > > inside a solution. Normally, the poisson
> equation
> > > > governs the behaviour of electrostatics.
> However,
> > > the
> > > > charge distribution of mobile ions within the
> > > > solution gets attracted to the charged rod and
> > > > requires a slight alteration of the poisson
> > > equation,
> > > > turning it into the poisson-boltzmann equation.
>
> > > >
> > > > div(e(r)grad(V(r)))= rho(r) +
> > > > summation(q[i]c[i]exp(-q[i]V(r)/kT)
> > > >
> > > > where e(r) is the relative permitivity, V(r) is
> > > > voltage, rho(r) is charge density, q[i], c[i]
> are
> > > the
> > > > charge and density of ion i, and kT are
> constants.
> > > >
> > > > I wish to find V(r) for all r. Is it possible
> to
> > > use
> > > > pdetool? If not, what other pde tools are out
> > > there?
> > > >
> > > > Thanks.
> > >
> > > The pde-toolbox can handle nonlinear elliptic
> pdes
> > > of the form
> > > -div(c(u)grad(u)) + a(u)*u = f(u) ;
> > > so formally, the solution of your pde is
> possible.
> > >
> > > Best wishes
> > > Torsten.
> >
> > Or do you have to determine the other quantities
> > with the pde-toolbox, too ?
> > If yes: How do the corresponding equations look
> like ?
> >
> > Best wishes
> > Torsten.
>
> Thank you for your help, Torsten. For now, I will
> assume that I have the other quantities. How would I
> be able to specify that V=0 at r = infinity?
>
> Thanks.

You can either choose a large R at which you specify
V=0 or you can transform your equation such that
r=0 maps to r~=0 and r = infinity maps to an
r~ < infinity (e.g. r~ = r/(1+r))
I didn't check whether the second possibility will
result in an equation that the pde-toolbox can
handle.

Best wishes
Torsten.
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