       Partial differential equation

• To: mathgroup at smc.vnet.net
• Subject: [mg14458] Partial differential equation
• From: Wei Zhang <zhangw at eng.umd.edu>
• Date: Wed, 21 Oct 1998 03:32:55 -0400
• Organization: University of Maryland, College Park
• Sender: owner-wri-mathgroup at wolfram.com

```Hello!

I got problems solving  the following partial differential equation
using NDSolve, it has taken more than 6hours without coming out
anything. Anyone can help me please? I am new with Mathematica...is
there any 'standard' packet that i can use to solve nonlinear partial
differentialequations numerically?

thanks a lot,
wei

\!\(\(\[Rho]0 = 2.65*10\^\(-6\); \n\[Beta] = 3.9*10\^\(-3\); \nH = 10;
\n
D\_t = 2.38; \nL = 40*10\^\(-4\); \nT\_0 = 200; \nf = 1000; \n
e = 1.602*10\^\(-19\); \nZ = 4; \nD\_b = 3*10\^\(-14\); \n
D\_l = 1.5*10\^\(-16\); \nB = 50*10\^9/10\^4; \n
\[CapitalOmega] = 10\^\(-20\); \nk = 1.38*10\^\(-23\); \n\)\)

\!\(j = 2*\(10\^6\) Sin[6.28*f*t]\n
a = \((\((H\/L - \(j\^2\) \[Rho]0\ \[Beta])\)/D\_t)\)\^\(1/2\); \n
T = T\_0 +
\(\(\(j\^2\) \[Rho]0\ \)\/\(\(a\^2\) D\_t\)\)
\((1 - Cosh[a\ x]\/Cosh[\(a\ L\)\/2])\); \n
\[Rho] = \[Rho]0 \((1 + \[Beta] \((T - T\_0)\))\); \)

2000000 Sin[6280. t]

\!\(\*
RowBox[{
RowBox[{
RowBox[{"solution", "=",
RowBox[{"NDSolve", "[",
RowBox[{
RowBox[{"{",
RowBox[{
\(\[PartialD]\_t u[x, t] ==
\((\(Z\ e\ \[Rho]\ \ j\ D\_b\)\/\(k\ T\)\ )\)
\[PartialD]\_x\ u[x, t] +
\((\(B\ \[CapitalOmega]\ D\_b\)\/\(k\ T\))\)
\[PartialD]\_\(x, x\)u[x, t]\), ",", \(u[x, 0] ==
0\),
",", " ",
RowBox[{
RowBox[{\(\((Z\ e\ \[Rho]0\ j\ )\) u[0, t]\), "-",
RowBox[{\((B\ \[CapitalOmega]\ )\),
RowBox[{
SuperscriptBox["u",
TagBox[\((1, 0)\),
Derivative],
MultilineFunction->None], "[", \(0, t\),
"]"}]}]}],
"==", "0"}], ",",
RowBox[{
RowBox[{\(\((Z\ e\ \[Rho]0\ j\ )\)\ u[L, t]\), "-",
RowBox[{\((B\ \[CapitalOmega]\ )\),
RowBox[{
SuperscriptBox["u",
TagBox[\((1, 0)\),
Derivative],
MultilineFunction->None], "[", \(L, t\),
"]"}]}]}],
"==", "0"}]}], "}"}], ",", "u", ",", \({x, \ 0, \
L}\),
",", " ", \({t, 0, 10000}\)}], "]"}]}], ";"}], "\n"}]\)

Plot3D[Evaluate[u[x,t]/.First[solution]],{x,0,L},{t,0,10000}];

```

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