PDE with RecurrenceTable

*To*: mathgroup at smc.vnet.net*Subject*: [mg124740] PDE with RecurrenceTable*From*: Alexei Boulbitch <Alexei.Boulbitch at iee.lu>*Date*: Fri, 3 Feb 2012 02:11:36 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

Dear Community, I am trying to make a simple numeric FEM solution of parabolic PDE. It seems that RecurrenceTable function is designed exactly for such a job. Indeed, in the Help/RecurrenceTable/Scope/Partial Difference Equations one finds an example. My problem is that this example is only one, and I would say, it is not basic enough. My question: do you know some other examples of the use of the RecurrenceTable for this sort of equations?? I would like to explain: I already went through the MathGroup archive and have seen numerous posts recommending various sophisticated FEM packages. My question is not about them. I want to learn to make simple programs of this sort myself to fast test an equation at hand. For example, here is a classical equation of temperature conductivity: (\[PartialD]u(x,t))/\[PartialD]t=a^2*(\[PartialD]^2u(x,t))/\[PartialD]x^2 with a=0.1, the boundary conditions: u[0,t]==1 and u[1,t]==0 and the initial condition u[x,0]== Cos[3 \[Pi]*x/2]; using the explicit method on the rectangular lattice, taken from a textbook: u[j,k+1]=\[Sigma]*u[j+1,k]+(1-2*\[Sigma])*u[j,k]+\[Sigma]*u[j-1,k]; \[Sigma]=(a^2*\[Tau])/h^2; Tau and h are temporal and spatial step sizes. This is the code: a = 0.1; h = 0.1; \[Tau] = 0.0001; \[Sigma] = a^2*\[Tau]/h^2; lst2 = RecurrenceTable[{u[j, k + 1] == \[Sigma]*u[j + 1, k] + (1 - 2*\[Sigma])* u[j, k] + \[Sigma]*u[j - 1, k], u[j, 0] == Cos[3 Pi*j/20.], u[0, k] == 1, u[10, k] == 0}, u, {j, 0, 10}, {k, 0, 100}]; Show[{ ListPlot3D[lst2, AxesLabel -> {Style["t", 16, Italic], Style["x", 16, Italic], Style["u", 16, Italic]}, PlotStyle -> Blue], Plot3D[0, {t, 0, 100}, {x, 0, 10}, PlotStyle -> {Yellow, Opacity[0.4]}] }] The solution obtained this way, however, does not show evolution. It is clear that with increasing t the temperature, u, should forget its initial form and approach to a straight line. What is wrong? Thank you, Alexei Alexei BOULBITCH, Dr., habil. IEE S.A. ZAE Weiergewan, 11, rue Edmond Reuter, L-5326 Contern, LUXEMBOURG Office phone : +352-2454-2566 Office fax: +352-2454-3566 mobile phone: +49 151 52 40 66 44 e-mail: alexei.boulbitch at iee.lu<mailto:alexei.boulbitch at iee.lu>