Stochastic excitable elements

The finite element method (FEM) has become the dominant computational method in structural engineering. In general, the input parameters in the standard FEM assume deterministic values. In earthquake engineering, at least the excitation is often random. However, considerable uncertainties might be involved not only in the excitation of a structure but also in its material and geometric properties. A rational treatment of these uncertainties needs a mathematical concept similar to that underlying the standard FEM. Thus, FEM as a numerical method for solving boundary value problems has to be extended to stochastic boundary value problems. The extension of the FEM to stochastic boundary value problems is called stochastic finite element method (SEEM). 

The stochastic Liouville equations are readily solved for the time-dependent density matrix elements pu (e.g., through Laplace transforms) the latter may then be used in turn to develop expressions for the polarized fluorescence or absorption difference signals. The initial values of the density matrix elements under 5-function pulsed excitation are given by... 

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