Laws partial differential-algebraic equation

The general method for solving Eqs. (11) consists of transforming the partial differential equations with the help of Fourier-Laplace transformations into a set of linear algebraic equations that can be solved by the standard techniques of matrix algebra. The roots of the secular equation are the normal modes. They yield the laws for the decays in time of all perturbations and fluctuations which conserve the stability of the system. The power-series expansion in the reciprocal space variables of the normal modes permits identification of relaxation, migration, and diffusion contributions. The basic information provided by the normal modes is that the system escapes the perturbation by any means at its disposal, regardless of the particular physical or chemical reason for the decay. 

This second-order partial differential equation, abusively called Pick s second law (but it is convenient to do so) represents the basic algebraic model of transient diffusion. 

Now all the information required for solving the Fick s law is known. We will use an operator approach. When a Laplace transform is applied to a linear differential equation, it gives an operator algebraic equation. Similarly, a linear differential equation is obtained from the partial differential equation (5.10) ... 

Dynamic models of chemical processes consist of ordinary differential equations (ODE) and/or partial differential equations (PDE), plus related algebraic equations. In this book we will restrict our discussion to ODE models, with the exception of one PDE model considered in Section 2.4. For process control problems, dynamic models are derived using unsteady-state conservation laws. 

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