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Lie symmetry, convergence analysis, explicit solutions, and conservation laws for the time-fractional modified Benjamin-Bona-Mahony equation

Al-deiakeh Rawaya, Alquran Marwan, Ali Mohammed, Qureshi Sania, Momani Shaher, Malkawi Abed Al-Rahman

Journal of Applied Mathematics and Computational Mechanics

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2024

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Vol. 23, nr 1

19-31

EN

Lie symmetry analysis is considercd as one of the most powerful techniques that has been used for analyzing and extracting various types of solulions to partial differential equations. Conservation laws reflect important aspects of the behavior and pcoperties of physical systems. This paper focuses on the investigation of the (1+1)-dimensional time-fractional modified Benjamin-Bona-Mahony equation (mBBM) incorporating Riemann-Louville derivatives (RLD). Through the application of Lie symmetry analysis, ihe study cxplores similarity reductions and transforms the problem into a nonlinear ordinary differential equation with fractional order. A power series solution is obtained using the Erdelyi-Kober fractional operator, and the convergence of the solutions is analyzed. Furthemore, novel conservation laws for the time-fractional mBBM equation are established. The findings of the current work contribute to a deeper understanding of the dynamics of this fractional evolution equation and provide valuable insights into its behavior.

2

The existence of mild solutions and approximate controllability for nonlinear fractional neutral evolution systems

Echchaffani Zoubida, Aberqi Ahmed, Karite Touria, Leiva Hugo

International Journal of Applied Mathematics and Computer Science

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2024

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Vol. 34, no. 1

15--27

EN

The existence of mild solutions and approximate controllability for Riemann-Liouville fractional neutral evolution systems with nonlocal conditions of a fractional order is investigated. The Laplace transform and semigroup theory are the tools used to prove the existence. In turn, approximate controllability is proved on the basis of a Nemytskii operator, a Mittag-Leffler function and certain hypotheses using fixed point theorems, as well as the construction of a Cauchy sequence. An example is provided to highlight the main results.

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Time-optimal control of linear fractional systems with variable coefficients

Matychyn Ivan, Onyshchenko Viktoriia

International Journal of Applied Mathematics and Computer Science

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2021

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Vol. 31, no. 3

375--386

EN

Linear systems described by fractional differential equations (FDEs) with variable coefficients involving Riemann–Liouville and Caputo derivatives are examined in the paper. For these systems, a solution of the initial-value problem is derived in terms of the generalized Peano–Baker series and a time-optimal control problem is formulated. The optimal control problem is treated from the convex-analytical viewpoint. Necessary and sufficient conditions for time-optimal control similar to that of Pontryagin’s maximum principle are obtained. Theoretical results are supported by examples.

4

Interpretation of Fractional Derivatives as Reconstruction from Sequence of Integer Derivatives

Tarasov V. E.

Fundamenta Informaticae

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2017

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Vol. 151, nr 1/4

431--442

EN

In this paper, we propose an “informatic” interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders as reconstruction from infinite sequence of standard derivatives of integer orders. The reconstruction is considered with respect to orders of derivatives.