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Quasilinearization method for finite systems of nonlinear RL fractional differential equations

Denton Zachary, Ramirez Juan Diego

Opuscula Mathematica

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2020

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Vol. 40, no. 6

667–-683

EN

In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order 0 < q < 1. Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given.

2

On the existence of optimal consensus control for the fractional Cucker-Smale model

Almeida R., Kamocki R., Malinowska A. B., Odzijewicz T.

Archives of Control Sciences

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2020

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Vol. 30, No. 4

625--651

EN

This paper addresses the nonlinear Cucker-Smale optimal control problem under the interplay of memory effect. The aforementioned effect is included by employing the Caputo fractional derivative in the equation representing the velocity of agents. Sufficient conditions for the existence of solutions to the considered problem are proved and the analysis of some particular problems is illustrated by two numerical examples.

3

Monotone method for Riemann-Liouville multi-order fractional differential systems

Denton Z.

Opuscula Mathematica

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2016

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Vol. 36, no. 2

189--206

EN

In this paper we develop the monotone method for nonlinear multi-order N-systems of Riemann-Liouville fractional differential equations. That is, a hybrid system of nonlinear equations of orders qi where 0 < qi < 1. In the development of this method we recall any needed existence results along with any necessary changes. Through the method's development we construct a generalized multi-order Mittag-Leffler function that fulfills exponential-like properties for multi-order systems. Further we prove a comparison result paramount for the discussion of fractional multi-order inequalities that utilizes lower and upper solutions of the system. The monotone method is then developed via the construction of sequences of linear systems based on the upper and lower solutions, and are used to approximate the solution of the original nonlinear multi-order system.

4

Monotone iterative technique for finite systems of nonlinear Riemann-,Lliouville fractional differential equations

Denton Z., Vatsala A.S.

Opuscula Mathematica

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2011

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Vol. 31, no. 3

327-339

EN

Comparison results of the nonlinear scalar Riemann-Liouville fractional differential equation of order q, 0 < q ≤ 1, are presented without requiring Hölder continuity assumption. Monotone method is developed for finite systems of fractional differential equations of order q, using coupled upper and lower solutions. Existence of minimal and maximal solutions of the nonlinear fractional differential system is proved.

5

Linear dynamic system identification in the frequency domain using fractional derivatives

Janiczek T., Janiczek J.

Metrology and Measurement Systems

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2010

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Vol. 17, nr 2

279-287

EN

This paper presents a study of the Fourier transform method for parameter identification of a linear dynamic system in the frequency domain using fractional differential equations. Fundamental definitions of fractional differential equations are briefly outlined. The Fourier transform method of identification and their algorithms are generalized so that they include fractional derivatives and integrals.