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Re: Diffusion Model using NDSolve - Advice needed

• To: mathgroup at smc.vnet.net
• Subject: [mg99426] Re: Diffusion Model using NDSolve - Advice needed
• From: schochet123 <schochet123 at gmail.com>
• Date: Tue, 5 May 2009 06:01:30 -0400 (EDT)
• References: <gtnk8c$3pa$1@smc.vnet.net>

The standard solution to this problem is to map each of the slabs into
some fixed domain and formulate the problem in that domain. For
example, if you have slabs a[j]<x<a[j+1] for j=1, ,,,, n then the
transformations T[j_][x_]=(x-a[j])/(a[j+1]-a[j]) map slab j  into
[0,1], and their inverses
Tinv[j][y_]=a[j]+(a[j+1]-a[j])y map the domain [0,1] back to the
slabs.

Let u[j][x] denote the value of the solution u in slab j. Use the
chain rule to calculate the equation satisfied by U[j][y_]=u[j][Tinv[j]
[y]] and the boundary conditions, which now are all set at the
boundaries y=0 and y=1 of the computational domain.

After running sol=NDSolve[...] and obtaining solutions Unum[j][y_]=U[j]
[y]/.sol[[1]] for j=1,...,n you can map the solutions back into the
slabs and use Piecewise to combine them into a single solution u.

Steve

On May 4, 11:47 pm, gwhollywood <thedramat... at hotmail.com> wrote:
> Hey all!! I will try and be really brief. If you think you may know how t=
o help, but don't understand what I'm saying, please ask! I'm desperate for=
 advice.
>
> I want to use NDSolve to solve the 1-D Diffusion Equation for a "composit=
e slab" with THREE LAYERS, each having an arbitrary thickness and diffusivi=
ty.
>
> I can solve the problem easily using NDSolve for the case of a single lay=
er.
>
> However I am having a lot of trouble figuring out how to specify the prob=
lem for three layers. There should be a separate solution for each layer on=
 its respective part of the domain (the total thickness).
>
> There are six total boundary conditions. The most important are the four =
that appear "within" the slab at the two interfaces. They require matching =
of the flux (which is proportional to the gradient), and proportionality of=
 the concentrations (therefore the solution is not necessarily continuous a=
t the interfaces).
>
> So it kinda ends up being a piecewise solution - one part valid for a cer=
tain section of the composite slab - know what I mean???
>
> I have tried entering all the equations (3 second order pde's, three init=
ial conditions, six boundary conditions) in NDSolve but I immediately get t=
he error that some of the boundary conditions specified are NOT at the edge=
 of domain (which is obviously true since they are specified within the lay=
er). Hence one of the three solutions is only valid within its own section =
of the slab.
>
> I am having an awful time trying to figure out how to pose this problem w=
ithin Mathematica. I am fairly proficient with the program in general.
>
> Any thoughts??