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Re: PDE & Complex solving problem

On 9/8/02 12:22 AM, in article aleu3o$557$1 at, "Shz Shz Oon"
<OONSSHZ at> 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


\!\(\(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]];\)\)

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