Re: PDE & Complex solving problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg36491] Re: PDE & Complex solving problem*From*: Tom Burton <tburton at brahea.com>*Date*: Mon, 9 Sep 2002 00:29:30 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

On 9/8/02 12:22 AM, in article aleu3o$557$1 at smc.vnet.net, "Shz Shz Oon" <OONSSHZ at hitachi.com.my> wrote: > 1.) How to find the General Solution for below's partial differential > equation? > (y + u) du/dx + y (du/dy) = x - y > ** I use "d" to represent the partial differential symbol. > Can it be solved by function NDSolve in mathematica 4.1? How? It's been a while, but I suspect that the presence of the nonlinear convective term u du/dx alone makes a general solution unlikely. It's easy to get a numerical solution for a particular set of boundary conditions, but you are not guaranteed a solution. For instance, the following choice of a and b are skirting failure, as the diagnostic contour plot shows. Tom Burton Needs["Graphics`Colors`"] \!\(\(soln = With[{a = \(-1\), b = 1.27, e = 0.001}, solutions = NDSolve[{\((y + u[x, y])\)\ \[PartialD]\_x u[x, y] + y\ \[PartialD]\_y u[x, y] == x - y, u[x, b] == 0, u[a, y] == 0}, u, {x, a, 10}, {y, b, 10}]; \[IndentingNewLine]Plot3D[ Evaluate[u[x, y] /. \[InvisibleSpace]First[solutions]], {x, a, 10}, {y, b, 10}, PlotPoints \[Rule] 50]; \[IndentingNewLine]ContourPlot[ Evaluate[y + u[x, y] /. \[InvisibleSpace]First[solutions]], {x, a, 10}, {y, b, 10}, PlotPoints \[Rule] 50, ColorFunction \[Rule] \((If[#1 > e, Green, Red] &)\)]\[IndentingNewLine]];\)\)