Difficulty in obtaining a numerical solution for a partial

*To*: mathgroup at smc.vnet.net*Subject*: [mg114282] Difficulty in obtaining a numerical solution for a partial*From*: Dilip Ahalpara <dpa1951 at gmail.com>*Date*: Tue, 30 Nov 2010 04:01:54 -0500 (EST)

Dear all, I have a partial differential equation (PDE), which when I try to solve using NDSolve (to get U(x,t) in a given range, say x in [0,20] and t in [0,10]), it blows up. The PDE for U(x,t) is, Ut + (2/U) Ux Uxx - Uxxx = 0 where Ux=delU/delx, Ut=delU/delt, Uxx=del^2U/delx^2 etc. As initial condition, a profile U(x,0) is fed, which is obtained from a known function satisfying the PDE, obtained from (c0/2) * (Sech[Sqrt[c0]/2 * (x-x0-c0*t)])^2 and the initial profile is placed around x0, with a height specified by c0. The boundary condition is set to match at x0Left and x0Right. To show that my Mathematica program has a correct logic, I first solve a PDE Ut + 6 U Ux + Uxxx = 0 using the same initial and boundary conditions -- and it works nicely as expected. I include below the Mathematica program. ---------- (* Define the differntial equations*) diffEqnLHS1 = D[u1[x, t], t] + 6*u1[x, t]*D[u1[x, t], x] + D[u1[x, t], {x, 3}] diffEqnLHS2 = D[u2[x, t], t] + D[u2[x, t], x] D[u2[x, t], {x, 2}]*2/u2[x, t] - D[u2[x, t], {x, 3}] template[x0_, c0_, x_, t_] := 0.5*c0*(Sech[0.5*Sqrt[c0]*(x - x0 - c0*t)])^2; cValue = 1.2; {x0Left, x0Right} = {0.0, 20.0}; (*Solution space*) x0Template = 5.0; (*Placing initial profile at x0*) c0Template = 3;(*with height c0*) initCondition = template[x0Template, c0Template, x, 0] + template[x0Template + (x0Right - x0Left), c0Template, x, 0] + template[x0Template - (x0Right - x0Left), c0Template, x, 0]; Print[N[initCondition /. x -> x0Left, 10], " ", N[initCondition /.x -> x0Right, 10], " ", (initCondition /. x -> x0Left) == (initCondition /.x -> x0Right)]; soln1 = NDSolve[diffEqnLHS1 == 0 && u1[x, 0] == initCondition && u1[x0Left, t] == u1[x0Right, t], u1, {t, 0, 10}, {x, x0Left, x0Right}]; // Timing Animate[Plot[Evaluate[u1[x, t] /. soln1[[1]]], {x, x0Left, x0Right}, PlotRange -> {0, 2}, GridLines -> Automatic], {t, 0, 10}] soln2 = NDSolve[diffEqnLHS2 == 0 && u2[x, 0] == initCondition && u2[x0Left, t] == u2[x0Right, t], u2, {t, 0, 1}, {x, x0Left, x0Right}]; // Timing Animate[Plot[Evaluate[u2[x, t] /. soln2[[1]]], {x, x0Left, x0Right}, PlotRange -> {0, 2}, GridLines -> Automatic], {t, 0, 1}] ---------- As can be seen by running this code, the Animate for soln1 works quite ok -- it shows the initial profile propagating to the right (increasing x) without changing the shape of the profile. However when I try to solve another PDE of my interest, i.e. corresponding to soln2, NDSolve does not generate a well behaved solution. My questions: [1] What is going wrong while solving the second PDE? [2] How to obtain a numerical solution for the second PDE correctly? Thanks for your time. Dilip