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a better solution for a fourth order differential equation
*To*: mathgroup at smc.vnet.net
*Subject*: [mg70480] a better solution for a fourth order differential equation
*From*: Roger Bagula <rlbagula at sbcglobal.net>
*Date*: Tue, 17 Oct 2006 02:59:22 -0400 (EDT)
Harmonic Poisson type equation:
(d'am-(4*Pi*L)*T/Phi)*(d'am+(4*Pi*Lstar)*T/Phi)*Phi=0
L=2/(3+2*w)
Lstar=2/(3-2*w)
L*LStar=Abs[L]^2
In this model :
Abs[L]^2=4/(9-4*(w2))=4/(9-4*Phi)
Which should allow a solution for the scalar field Phi in terms of the
Energy density.
4d symmetrical linear partition as x->[x[i],{i,1,4}]: (scale of energy
density as one)
y''''[x]*(4*y[x]-9)+48*Pi*y''[x]-64*Pi^2=0
Solves numerically in Mathematica with suitable boundary conditions
I'd like to sove it better with more realistic energy densities and
boundary conditions.
Mathematica Workbook:
\!\(\*
RowBox[{\(Clear[Phi, x, deq, L, Lstar, w, y]\),
"\n", \(p4 = D[y[x], {x, 4}]\), "\n", \(p2 = D[y[x], {x, 2}]\),
"\n", \(p1[x_] = D[y[x], {x, 1}]\), "\n", \(p1[0]\),
"\n", \(p3[x_] = D[y[x], {x, 3}]\), "\n", \(p3[0]\),
"\[IndentingNewLine]", "\n", \(Expand[\((A - L*K)\)*\((A - Lstar*K)\)]\),
"\n", \(w = Sqrt[y[x]]\), "\n", \(L = 2/\((3 + 2*w)\)\),
"\n", \(Lstar = 2/\((3 - 2*w)\)\),
"\n", \(deq =
Expand[FullSimplify[
ExpandAll[\((p4 - p2\ K\ L - p2\ K\ Lstar +
K\^2\ L\ Lstar)\)*\((\(-9\) + 4\ y[x])\)]]]\),
"\n", \(Solve[\(-4\)\ K\^2 - 9\ x + 4\ A0\ x \[Equal] 0, x]\),
"\n", \(DSolve[{deq == 0, y[0] \[Equal] 0}, y[x], x]\),
"\n", \(K = 4*Pi\), "\n",
RowBox[{\(z[x_]\), "=",
RowBox[{\(y[x]\), "/.",
RowBox[{"NDSolve", "[",
RowBox[{
RowBox[{"{",
RowBox[{\(deq == 0\),
",", \(y[0] \[Equal] 1/\((6.6732*10^\((\(-8\))\))\)\), ",",
RowBox[{
RowBox[{
SuperscriptBox["y", "\[Prime]",
MultilineFunction->None], "[", "0", "]"}], "\[Equal]",
"0"}], ",",
RowBox[{
RowBox[{
SuperscriptBox["y", "\[Prime]\[Prime]",
MultilineFunction->None], "[", "0", "]"}],
"==", \(-\(K\/3\)\)}], ",",
RowBox[{
RowBox[{
SuperscriptBox["y",
TagBox[\((3)\),
Derivative],
MultilineFunction->None], "[", "0", "]"}],
"\[Equal]", \(\(-4\)*K2/9\)}]}], "}"}], ",", \(y[x]\),
",", \({x, \(-10\), 10}\)}], "]"}]}]}],
"\n", \(Plot[\(-ArcTanh[2*x]\) +
1/\((6.6732*10^\((\(-8\))\))\), {x, \(-2\), 2}]\),
"\n", \(Plot[\(-\((\(-ArcTanh[2*x]\) + 1/\((6.6732*10^\((\(-8\))\))\) -
z[x])\)\), {x, \(-2\), 2}]\),
"\n", \(e = 4.80325*10^\((\(-10\))\)\), "\n", \(c = 2.997925*1010\),
"\n", \(mp = 1.6726140*10^\((\(-24\))\)\),
"\n", \(me = 9.10958*10^\((\(-28\))\)\),
"\n", \(r[m_] = e2/\((m*c2)\)\), "\n", \(r[me]\), "\n", \(v[r[me]]\),
"\n", \(r[mp]\), "\n", \(v[r[mp]]\), "\n", \(v[r_] = 4*Pi*r3/3\),
"\n", \(Scalemp = mp*c2/v[r[mp]]\), "\n", \(Scaleme = me*c2/v[r[me]]\),
"\n", \(hbar = 1.0545919*10^\((\(-27\))\)\),
"\n", \(alphaE = \((hbar*c)\)/e\), "\n", \(Scalemp/alphaE\),
"\n", \(Scaleme/alphaE\),
"\n", \(ScaleA = 28*mp*c2/v[137.03608*r[me]]\)}]\)
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