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Numerical solution of coupled nonlinear PDEs

  • To: mathgroup at smc.vnet.net
  • Subject: [mg112092] Numerical solution of coupled nonlinear PDEs
  • From: "Dominic" <miliotodc at rtconline.com>
  • Date: Mon, 30 Aug 2010 06:19:16 -0400 (EDT)

Hi guys,

For the functions a(t,y), b(t,y), and x(t,y),I have the PDE system:

a'=a''+a x''+x'a'

b'=b''-bx''-x'b'

x''=b-a

Sorry I can't show the system more clearly here but on the left side the 
partial is respect to t for a and b and it's with respect to y for the 
x(t,y), and on the right, all partials are with respect to y,

and I'd like to obtain a numerical solution for any simple IBVP using 
NDSolve, for example in the domain 0<t<1 and 0<y<1 with all initial 
conditions set to y  This is the code I'm using:

mysol = NDSolve[{D[a[t, y], t] ==
D[a[t, y], {y, 2}] + a[t, y] D[x[t, y], {y, 2}] +
D[x[t, y], y] D[a[t, y], y]
,
D[b[t, y], t] ==
D[b[t, y], {y, 2}] - b[t, y] D[x[t, y], {y, 2}] -
D[x[t, y], y] D[b[t, y], y]
,
D[x[t, y], {y, 2}] == b[t, y] - a[t, y]
,
a[0, y] == y, a[t, 0] == 0, a[t, 1] == 1,

b[0, y] == y, b[t, 0] == 0, b[t, 1] == 1,

x[0, y] == y, x[t, 0] == 0, x[t, 1] == 1},

{a, b, x}, {t, 0, 1}, {y, 0, 1}, MaxSteps -> 10000]

However, I receive boundary-value errors and singular errors.

Can someone help me set up this system in any way for any reasonable 
domain with any reasonable initial and bondary values to obtain a 
non-trivial solution?

Thanks guys,

Dominic


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