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MathGroup Archive 2001

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How can I acceralate calcution by improving coding?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg30314] How can I acceralate calcution by improving coding?
  • From: "Toshiyuki \(Toshi\) Meshii" <meshii at mech.fukui-u.ac.jp>
  • Date: Wed, 8 Aug 2001 01:34:09 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Hello,

I want to accelerate calculation speed of a function u1[r, t], which is a
transient temperature distribution of a hollow cylinder with inner/outer
radii of a and b, respectively.

(* u is in an asymptotic form using 0 th order Bessel function of 1st and
2nd kind.
The eigenvalue rho is a root of the f[\Rho]==0. *)

f[\[Rho]_] = (\[Rho]*BesselJ[1, \[Rho]*a] + \[Alpha]0*BesselJ[0, \[Rho]*a])*
     BesselY[1, \[Rho]*b] - (\[Rho]*BesselY[1, \[Rho]*a] +
\[Alpha]0*BesselY[0, \[Rho]*a])*
     BesselJ[1, \[Rho]*b];

(* The concrete values used for numerical evaluation are as follows. *)

a = 0.005; b = 0.015;\[Kappa]= 0.0130/3600;\[Alpha]0=10000000/10.9;\[Omega]=
  2\[Pi];

(* n th root \[Rho]n and constant \[Beta]n are obtained as follows *)

nmax=96;

ddn = Table[FindRoot[ f[d] == 0, {d, 100+(j-1) 314.2},
                        MaxIterations->100],{j,1,nmax}];

Subscript[\[Rho], n_] := ddn[[n,1,2]];

Subscript[\[Beta], n_] :=
   BesselJ[1, Subscript[\[Rho], n]*b]/BesselY[1, Subscript[\[Rho], n]*b];

(* Now, let's defined u[r,t] *)


Subscript[R, n_][r_] := BesselJ[0, Subscript[\[Rho], n]*r] -
    Subscript[\[Beta], n]*BesselY[0, Subscript[\[Rho], n]*r];

  u[r_, t_] = Sin[\[Omega]*t] +
    Sum[((\[Omega]*(\[Kappa]*Subscript[\[Rho], n]^2*
          (E^((-\[Kappa])*Subscript[\[Rho], n]^2*t) -
           Cos[\[Omega]*t]) - \[Omega]*Sin[\[Omega]*t]))/
       ((\[Kappa]*Subscript[\[Rho], n]^2)^2 + \[Omega]^2))*
      ((2*\[Alpha]0*a)/Subscript[\[Rho], n]^2)*
      ((Subscript[R, n][a]*Subscript[R, n][r])/
       ((b*Subscript[R, n][b])^2 -
        (1 + (\[Alpha]0/Subscript[\[Rho], n])^2)*
         (a*Subscript[R, n][a])^2)), {n, 1, nmax}];

(* Can anyone help to accelarate calculation ?
The following Plot3D takes 180 sec on a Pentium III 800 MHz machine.
I tried to ues Module, but it did not contribute in speed up*)

Timing[Plot3D[u[r, t], {t, 0, 2}, {r, a, b}, PlotRange -> {-1, 1},
   AxesLabel -> {" t sec", "  r m", "u K  "},
   FaceGrids -> {{0, -1, 0}, {1, 0, 0}, {0, 1, 0}, {-1, 0, 0}},
   PlotPoints -> 50]]

-Toshi



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