By Krzysztof J. Latawiec, Marian Łukaniszyn, Rafal Stanislawski

This quantity offers chosen facets of non-integer, or fractional order platforms, whose research, synthesis and purposes have more and more develop into a true problem for varied examine groups, starting from technological know-how to engineering. The spectrum of purposes of the fractional order calculus has particularly increased, in reality it might be challenging to discover a science/engineering-related topic zone the place the fractional calculus had no longer been included. The content material of the fractional calculus is ranged from natural arithmetic to engineering implementations and so is the content material of this quantity. the amount is subdivided into six elements, reflecting specific elements of the fractional order calculus. the 1st half incorporates a unmarried invited paper on a brand new formula of fractional-order descriptor observers for fractional-order descriptor continous LTI structures. the second one half presents new components to the mathematical concept of fractional-order structures. within the 3rd a part of this quantity, a number of latest ends up in approximation, modeling and simulations of fractional-order platforms is given. The fourth half provides new strategies to a few difficulties in controllability and regulate of non-integer order structures, particularly fractional PID-like keep watch over. The 5th half analyzes the soundness of non-integer order structures and a few new effects are provided during this very important admire, specifically for discrete-time structures. the ultimate, 6th a part of this quantity provides a spectrum of functions of the noninteger order calculus, starting from bi-fractional filtering, particularly of electromyographic signs, throughout the thermal diffusion and advection diffusion strategies to the SIEMENS platform implementation. This volume's papers have been all subjected to stimulating reviews and discussions from the energetic viewers of the RRNR'2014, the sixth convention on Non-integer Order Calculus and Its purposes that was once equipped through the dep. of electric, keep an eye on and machine Engineering, Opole college of expertise, Opole, Poland.

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**Additional resources for Advances in Modelling and Control of Non-integer-Order Systems: 6th Conference on Non-integer Order Calculus and Its Applications, 2014 Opole, Poland**

**Example text**

R. Remark 1. 6 that was proved in [10]. 4 Example In electrodynamics, if the Lagrangian represents a charged particle interacting with a electromagnetic ﬁeld, the Lagrangian density for the electromagnetic ﬁeld (see [5, 7]) has the following form: L= 1 (E2 − H2 ), 8π (8) where E and H are the electric ﬁeld vector and the magnetic ﬁeld vector, respectively. Let t = (t0 , t1 , t2 , t3 , t4 ) and A(t) = (A1 (t), A2 (t), A3 (t)), A0 (t) be a vector potential and a scalar potential, respectively. They are deﬁned by setting E = ∇A0 − ∂A , ∂t0 H = curlA.

Handbook of Diﬀerential Equations Ordinary Diﬀerential Equations. Elsevier, North Holland (2005) 8. : Fractional h-diﬀerence equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5(1), 110–121 (2011) 9. : A suﬃcient condition of viability for fractional diﬀerential equations with the Caputo derivative. Journal of Mathematical Analysis and Applications 381 (2011) 10. : The theory of discrete fractional calculus: Development and application. PhD thesis, University of Nebraska - Lincoln (2011) 11.

1007/978-3-319-09900-2_3 26 M. Klimek and M. Blasik version of SLPs yields solutions which correspond to real eigenvalues and are orthogonal eigenfunctions. The aim of this paper is to propose a simple method of proof of the existence result for a discrete spectrum of a regular FSLP with vanishing Dirichlet and fractional Neumann boundary conditions. We extend here an approach developed in [12] in the case of a regular FSLP with derivatives in the range (0,1). By using the methods of functional analysis we also prove that the respective inﬁnite countable set of eigenfunctions yields a basis in the space of square-integrable functions.