By Shantanu Das
In this ebook, not just are mathematical abstractions mentioned in a lucid demeanour, but in addition numerous sensible functions are given rather for process identity, description after which effective controls. The reader will get a sense of the huge applicability of fractional calculus within the box of technology and engineering. With this e-book, a starter can comprehend the strategies of this rising box with a minimum attempt and easy mathematics.
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Extra info for Functional fractional calculus for system identification and controls
This introduction chapter has given the thought that there is a wonderful universe of mathematics staying within the boundary of one complete differentiation and one complete integration. The science maturity will absorb the richness in this fractional calculus may be in coming years of twenty-first century. The chapter gives the idea that this fractional calculus is as rigorous as its counterpart classical integer order differentiation and integration, with subject’s richness for scientific research for future.
The base function is the Gamma function, which generalizes the factorial expression, used in multiple differentiation and repeated integrations, in integer order calculus. The Mittag- Leffler function is the basis function of fractional calculus, as the exponential function is to the integer order calculus. Several modifications of the Mittag-Leffler functions are introduced which are developed since 1903, for study of the fractional calculus. 1 Gamma Function One of the basic functions of the fractional calculus is Euler’s Gamma function.
Because the wires are long and insulated, they will be treated as “semi-infinite” heat conductors. 1 represents one such wire of thermocouple S. Das, Functional Fractional Calculus for System Identification and Controls. C Springer 2008 35 36 3 Observation of Fractional Calculus in Physical System Description Tsurf (t) x=0 x = −∞ x = +∞ Thermocouple Direction of heat flow Fig. 1 Heat flow in semi-infinite wire thermocouple pair. The thick line in Fig. 1 represents the semi-infinite heat conductor, the thermocouple wire measuring the temperature at x = 0 the furnace wall, called as Tsur f (t), which dynamically varies with the time.