Advances in Fractional Calculus: Theoretical Developments by Satwinder Jit Singh, Anindya Chatterjee (auth.), Jocelyn

By Satwinder Jit Singh, Anindya Chatterjee (auth.), Jocelyn Sabatier, Om Prakash Agrawal, J. A. Tenreiro Machado (eds.)

In the final twenty years, fractional (or non integer) differentiation has performed an important position in quite a few fields reminiscent of mechanics, electrical energy, chemistry, biology, economics, regulate idea and sign and photo processing. for instance, within the final 3 fields, a few vital concerns reminiscent of modelling, curve becoming, filtering, trend reputation, aspect detection, identity, balance, controllability, observability and robustness are actually associated with long-range dependence phenomena. related development has been made in different fields in this article. The scope of the booklet is hence to give the cutting-edge within the examine of fractional structures and the appliance of fractional differentiation.

As this quantity covers contemporary purposes of fractional calculus, it will likely be of curiosity to engineers, scientists, and utilized mathematicians.

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1 Introduction Lorenzo and Hartley (LH) [1,2] have clearly established the importance of timedependent initialization function in taking into account the history of a system which evolves according to a differential equation of fractional order. They have considered both the Riemann–Liouville (RL) and the Grunwald formulations of fractional calculus [3–6] in developing the initialization function [7]. This paper examines the Caputo fractional derivative [8,9] with the objective of determining the inferred initialization, that is, the history function associated with the Caputo fractional derivative from the perspective of the Lorenzo–Hartley scheme.

The stability, accuracy, and computational speeds for these algorithms are examined. Numerical simulations exhibit that the choice of a numerical scheme will depend on the problem considered and the performance criteria selected. Keywords Fractional differential equations, fractional derivatives, numerical schemes for fractional differential equations, Volterra integral equation, Gr¨ unwald Letnikov approximation. 1 Introduction Fractional derivatives (FDs) and fractional integrals (FIs) have received considerable interest in recent years.

The approximation that f D,D(–x) # 0 in Eq. (11) improves as D approaches 2 and consequently the results of using Eqs. (14) and (15) become more accurate. When the value of D deviated further from 2, the results from Eqs. (14) and (15) become less accurate. However, the total number of real zeros in this case can be easily enumerated by a brute-force technique described later. 42 but in this range ED,1(–x) has only one real zero. 42 when at most 3 significant digits in D are specified. As D gets closer to 2, D can be specified to an increasing number of significant digits.

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