Parabolic systems

P. NeittaanmSki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994)... 

For the same type of catalyst we have observed in a recirculation laboratory reactor multiplicity, periodic and chaotic behavior. Unfortunately, so far we are not able to suggest such a reaction rate expression which would be capable of predicting all three regimes [8]. However, there is a number of complex kinetic expressions which can describe periodic activity. One can expect that such kinetic expressions combined with heat and mass balances of a tubular nonadiabatic reactor may give rise to oscillatory behavior. Detailed calculations of oscillatory behavior of singularly perturbed parabolic systems describing heat and mass transfer and exothermic reaction are apparently beyond, the capability of both standard current computers and mathematical software. 

Only some of the important works for distributed systems control shall be reviewed here. Since Butkovskii results require the explicit solution of the system equations, this restricts the results to linear systems. This drawback was removed by Katz (1964) who formulated a general maximum principle which could be applied to first order hyperbolic systems and parabolic systems without representing the system by integral equations. Lurie (1967) obtained the necessary optimality conditions using the methods of classical calculus of variations. The optimization problem was formulated as a Mayer-Bolza problem for multiple integrals. 

A. I. VOLPERT, V. A. VOLPERT, AND V. A. YohFEKI, Traveling Wave Solutions of Parabolic Systems, American Mathematical Society, Providence, RI, 1994. 

A large flat aperture is most often associated with a narrow pencil beam. However, we shall in Chapter 7 consider antennas with omnidirectional pattern and low visibility in the backward direction over a broad band. Further discussed in Chapter 8 is how to design a feed for a parabolic cylinder that will produce an RCS about 6dB lower than with no feed at all. The design of such a feed is closely related to the omnidirectional design. We emphasize, however, that parabolic systems never can attain the inherently low RCS level encountered for the flat aperture over a broad frequency band. 

U. Nowak. PDEXIM - A Softwctre Package for the Numericcd Solution of Parabolic Systems in One Space Dimension (cJso in this publication). 

Example 5.2. Consider the parabolic system with diffusion and nonlinearity of the Van der Pol type [14]... 

Singularly perturbed parabolic systems of equations of the type... 

The barrier thickness yQ evidently has the character of a screening length. How, then, does it differ from the conventional Debye lei th The two calculations are based on somewhat different models. That for the Debye ler th sets no limit to the local carrier concentration the barrier calculation which leads to eqn. (1) does set such a limit, namely the charge density accommodated in donors, and this assumed to be constant throughout the barrier. Accordingly, the Debye length defines an exponentially decaying potential contour, whereas the present barrier system yields a parabolic system. 

Volpert, A. L, Volpert Vit.A., Volpert VIA. Traveling Wave Solution of Parabolic Systems. AMS Books Online, p. 455,2003. 

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