Re: Derivative and Integration of NDSolve solution

• To: mathgroup at smc.vnet.net
• Subject: [mg121193] Re: Derivative and Integration of NDSolve solution
• From: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>
• Date: Sat, 3 Sep 2011 08:06:08 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <j3ct1i\$5m0\$1@smc.vnet.net>

```Try

fun[z_, t_, tt_] =  Sign[z - v (t - tt)] (D[Tl[z, t], t] /.
solution /. {z -> Abs[z - v (t - tt)], t -> tt});
\[Eta][z_,   t_] := -(3 B \[Beta])/(2 \[Rho] v^2) NIntegrate[  fun[z,
t, tt], {tt, -1000, 1000}]
Plot[\[Eta][1, t], {t, 0, 100}]

Cheers -- Sjoerd

On Aug 28, 10:08 am, Jiwan Kim <hwoarang.... at gmail.com> wrote:
> 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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