Re: Specifiying finiteness condition using NDSolve

*To*: mathgroup at smc.vnet.net*Subject*: [mg85245] Re: Specifiying finiteness condition using NDSolve*From*: bghiggins at ucdavis.edu*Date*: Mon, 4 Feb 2008 03:10:16 -0500 (EST)*References*: <fnrnir$ohl$1@smc.vnet.net>

On Jan 30, 9:50=A0pm, pradeep <prad... at purdue.edu> wrote: > Hello all, > I am trying to solve a PDE for unsteady laminar flow through a circular > tube using NDSolve, but am having difficulty in implementing one of the > boundary conditions which states that the velocity at all time, at r=0 > (centre of the tube), is FINITE. How do I specify the fact that the > function i am solving for has a value that is finite at a specific point > in the domain in mathematica? > > Thanks in advance! > > Pradeep > Purdue University Pradeep, I presume you are trying to solve the following linear PDE, where x is the radial coordinate D[u[t, x], t] == 1 + D[u[t, x], x, x] + D[u[t, x], x]/x and the difficulty is the term with 1/x which becomes indeterminate at x=0. If you apply l'Hopital's rule you find that the appropriate PDE at x=0 is D[u[t, x], t] == 1 + 2 D[u[t, x], x, x] at x=0 Thus to solve this system of PDEs we construct the following helper functions g1[x_ /; x > 0] := 1 g1[x_ /; x == 0] := 2 g2[x_ /; x > 0] := 1/x g2[x_ /; x == 0] := 0 sol = NDSolve[{D[u[t, x], t] == 1 + g1[x] D[u[t, x], x, x] + g2[x] D[u[t, x], x], u[0, x] == 0, Derivative[0, 1][u][t, 0] == 0, u[t, 1] == 0}, u, {t, 0, 10}, {x,= 0, 1}] And here is a plot of the solution at various time intervals Plot[Evaluate[{u[0.02, x], u[0.08, x], u[0.1, x], u[0.4, x], u[1, x], u[5, x]} /. sol], {x, 0, 1}] Cheers, Brian