Partial differential equations frequency domain

The lowest level of abstraction, here called the geometry level, is the closest to physical reality, in which the physics is described by partial differential equations. This level is the domain of finite-element, boundary-element or related methods (e.g., [7-9]). Due to their high accuracy, these methods are well suited for calculating, for example, the distribution of stresses, distortions and natural resonant frequencies of MEMS structures. But they also entail considerable computational effort. Thus, these methods are used to solve detailed problems only when needed, whereas simulations of complete sensor systems and, in particular, transient analyses are carried out using methods at higher levels of abstraction. 

This transfer function can now be studied in the frequency domain. It should be noted that these are linear partial differential equations and that the process of frequency domain analysis is appropriate. The range of values of e = 0.01 to 0.2, M = 5 to 20, and R = 0.75 have been established [Grant and Cotton, 1991] in a numerical finite difference solution of the governing equations. Having established these values the frequency response can be completed. 

Step 1. Setting up the model equations on the particle scale. These equations are generally nonlinear partial differential equations (PDEs). For analysis in the frequency domain, it is most convenient to use nondimensional concentrations and temperatures, dehned as relative deviations from their steady-state values. 

This transformation of a partial differential equation into an ordinary differential equation illustrates a general advantage of working in the frequency domain. Solutions are of the form... 

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