Derivative and Integration of NDSolve solution

• To: mathgroup at smc.vnet.net
• Subject: [mg121098] Derivative and Integration of NDSolve solution
• From: Jiwan Kim <hwoarang.kim at gmail.com>
• Date: Sun, 28 Aug 2011 04:04:45 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Hello, Mathgroup.

By solving the coupled differential equation, I got Te[z,t] and Tl[z,t]
solution in the following code.
Then, I wanted to get the Eta[z,t] using NIntegrate function. But it is not
working.
For the detail explanation, Eta[z,t] is the integration function of
dTl[z,t]/dt at z=|z-v(t-tt)|,t=tt.
Could you help me..? plz...

Jiwan.

Remove["Global`*"];
\[Rho] = 8910;(* mass density : kg/m^3 *)
v = 4.08;(* sound velocity : nm/ps *)
\[Beta] = 1.3 10^-5;(* linear expansion : /K *)
B = 1.8 10^11; (* bulk modulus : Pa *)
c = 3 10^5; (* light speed : nm/ps *)
\[Lambda] = 800; \[Omega] =
2 \[Pi] c/\[Lambda]; (* light wavelength : nm *)
Ce = 1.065 10^3; (* electron heat cap. at 300 K : 3.19 10^5 J/m^3K *)
\
Cl = 3.95 10^6; (* lattice heat cap. : J/m^3K = 26.1 J/mol.K *)
g = 4.4 10^5; (* coupling constant : 4.4 10^17 W/m^3.K *)
K = 91 10^6; (* thermal conductivity : 91 W/m.K -> 91 10^18 *)
\[Xi]1 = 13.5; (* pump absorption depth: nm *)
\[Xi]2 = 14.5; (* probe absorption depth: nm *)
R = 0.4; (* reflection at interface *)
\[Eta]0 = 1;
I0 = 1.05 10^10; (* 2.77 10^13 J/m^2.pulse(ps) -> 2.77 10^22 *)
PulseWidth = 0.2 ; (* 200 fs *)

S[t_] = I0 Exp[-t^2/(2 PulseWidth)^2];
pow[z_, t_] = 1/\[Xi]1 S[t] Exp[-z/\[Xi]1]; (* W/m^3 *)
L = 1000; (* sample thickness : nm *)
solution =
NDSolve[{Ce Te[z, t] D[Te[z, t], t] ==
K D[Te[z, t], z, z] - g (Te[z, t] - Tl[z, t]) + pow[z, t],
Cl D[Tl[z, t], t] == g (Te[z, t] - Tl[z, t]),
Te[z, -2] == Tl[z, -2] == 300, (D[Te[z, t], z] /. z -> L) ==
0, (D[Te[z, t], z] /. z -> 0) == 0}, {Te, Tl}, {z, 0, L}, {t, -2,
20}, MaxSteps -> Infinity, MaxStepSize -> {0.5, 0.02}][[1]]
Plot[{Te[z, t], Tl[z, t]} /. solution /. z -> 0, {t, -2, 20},
PlotRange -> All]
\[Eta][z_, t_] = -(3 B \[Beta])/(2 \[Rho] v^2)
NIntegrate[
Sign[z -
v (t - tt)] (D[Tl[z, t], t] /.
solution /. {z -> Abs[z - v (t - tt)], t -> tt}), {tt, -1000,
1000}]
Plot[\[Eta][z, t] /. z -> 1, {t, 0, 100}]
--
--------------------------------------------------------------------------
Institute of Physics and Chemistry of Materials Strasbourg (IPCMS)
Department of Ultrafast Optics and Nanophotonics (DON)
23 rue du Loess, B.P. 43,
67034 STRASBOURG Cedex 2, France

```

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