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MathGroup Archive 2001

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Solving a system of nonlinear equations containing integrals

  • To: mathgroup at smc.vnet.net
  • Subject: [mg27166] Solving a system of nonlinear equations containing integrals
  • From: anthozique at my-deja.com
  • Date: Fri, 9 Feb 2001 03:10:15 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

I would like to find roots for the following system of equations (my
unknowns are a and b, Lambda and Sigma are just parameters):

\!\(eq1\  = \
    b\^2\ \[Lambda]\ Log[2] +
      1\/6\ \((1 + Log[4])\) - \(1\/\[Sigma]\^2\) \((2\ a\^2\ *\
            Integrate[\[ExponentialE]\^\(\(\(-r\^2\) - \
\[Rho]\^2\)\/\[Sigma]\^2\)\ r\ \[Rho]\ BesselI[
                  0, \(2\ r\ \[Rho]\)\/\[Sigma]\^2]\ Sech[r\/b]\^2\
Sech[\
\[Rho]\/b]\^2, {r, 0, \[Infinity]}, {\[Rho], 0, \[Infinity]}])\)\)

and

\!\(eq2\  = \ \(-b\^2\)\ \[Lambda]\ Log[2] +
      1\/\(b\ \[Sigma]\^2\)*\((2\ a\^2\ *\
            Integrate[
              r\^2\ *\[ExponentialE]\^\(\(\(-r\^2\) - \
\[Rho]\^2\)\/\[Sigma]\^2\)\ *\[Rho]\ *
                BesselI[0, \(2\ r\ \[Rho]\)\/\[Sigma]\^2]*\
                Sech[\[Rho]\/b]\^2\ *Sech[r\/b]\^2\ *Tanh[r\/b], {r,
                0, \[Infinity]}, {\[Rho], 0, \[Infinity]}])\)\)

with e.g. Lambda=1 and Sigma=0.1

As you can see, this system contains some nasty equations involving
integrals and bessel functions. I think I have tried nearly all kind of
combinations of options that mathematica can offer, particularly
combinations of FindRoot and NIntegrate, e.g. (with Lambda=1 and
Sigma=0.1)
FindRoot[{eq1,eq2},{a,2.16},{b,0.75},...] where eq1=....NIntegrate[]...
and eq2=....NIntegrate[]......
I couldn't get any satisfactory result (without any warnings). Could
anyone suggest me the best way to find roots for this system ?

















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