Nonhomogeneous Wave Equation with NDSolve

*To*: mathgroup at smc.vnet.net*Subject*: [mg121677] Nonhomogeneous Wave Equation with NDSolve*From*: Jiwan Kim <hwoarang.kim at gmail.com>*Date*: Sun, 25 Sep 2011 05:42:24 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

Hello, mathgroup. I want to ask you how to solve the nonhomogeneous wave equation. I know that usual wave equation is very simple to handle. However, problem in this code is a source function, G[z,t] and the boundary condition. Experiment conditions are as following. There is 1-dim slab {z,0,L}. If the heat pulse exert on left edge z=0, we got the evolution of electron (Te[z,t]) and lattice temperature (Tl[z,t]) in the slab. I have got these temperature profile using NDSolve. And the G[z,t] of the wave equation is proportional to the Tl[z,t]. I think that here is the problem. The source function of the wave equation is given by dG[z,t]/dz. However, dG[0,t]/dz is not possible because z=0 is left boundary of the slab. I think that this kind of problem is well known one. How can I solve this. Is there anything that I am wrong..? Here is the code. Plz, help me. Thank you in advance.. (* Properties of Nickel, Converted to nm and ps *) Remove["Global`*"]; \[Rho] = 8910;(* mass density : kg/m^3 *) v = 4.08;(* sound velocity : nm/ps *) \[Beta] = 1.34 10^-5;(* linear expansion : /K *) B = 1.8 10^11; (* bulk modulus : Pa *) \[Gamma] = 1.065 10^3; (* 6000 electron heat cap. at 300 K : 3.19 10^5 J/m^3K *) \ Cl = 3.95 10^6; (* lattice heat cap. : 3.95 10^6 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 *) I0 = 3 10^10; (* 6.1 10^13 J/m^2.pulse(ps) -> 6.6 10^22 *) pulwth = 0.15 ; (* 150 fs *) \[Sigma] = pulwth/(2 (2 Log[2])^0.5); S[t_] := I0 Exp[-t^2/(2 \[Sigma]^2)]; pow[z_, t_] := 1/\[Xi]1 S[t] Exp[-z/\[Xi]1]; (* W/m^3 *) L = 400; (* sample thickness : nm *) solution = NDSolve[{\[Gamma] 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, -5] == Tl[z, -5] == 300, Te[L, t] == 300, (D[Te[z, t], z] /. z -> 0) == 0}, {Te, Tl}, {z, 0, L}, {t, -5, 50}, MaxSteps -> Infinity, MaxStepSize -> {0.3, 0.03}, Method -> "ImplicitRungeKutta"][[1]] Plot3D[Tl[z, t] /. solution, {z, 0, 100}, {t, -1, 50}, PlotRange -> All] G[z_, t_] = 3 \[Beta] B/\[Rho] ( Tl[z, t] - 300) /. solution; Plot3D[G[z, t], {z, 0, L}, {t, -5, 50}, PlotRange -> All] solution2 = NDSolve[ D[u[z, t], t, t] == v^2 D[u[z, t], z, z] - D[G[z, t], z], v^2 (D[u[z, t], z] /. z -> 0) - G[0, t] == 0, u[L, t] == 0, u[z, -5] == 0, {u}, {z, 0, L}, {t, -5, 50}, MaxSteps -> Infinity, Method -> "ImplicitRungeKutta"][[1]] Plot3D[u[z, t] /. solution2, {z, 0, L}, {t, -5, 50}] -- ----------------------------------------------------------------------------- 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