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Re: Diffusion Model using NDSolve - Advice needed
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[] 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??