Linear conformable differential system and its controllability

Younas, Awais; Abdeljawad, Thabet; Batool, Rida; Zehra, Anum; Alqudah, Manar A.

doi:10.1186/s13662-020-02899-0

Cited by 18 publications

(12 citation statements)

References 24 publications

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“…Recently, the conformable fractional Gauss hypergeometric function and a class of conformable fractional differential equations through that function were treated in [15]. Further interesting ideas on the conformable derivative can be found in the work by [16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

“…Note that some authors have demonstrated that the conformable derivative is not the same as a fractional order derivative, but it is a first-order derivative multiplied by an additional factor (see for example [23]). Hence, Definition 1 seems to be a natural extension of the conventional order derivative to noninteger order loosing memory effect.…”

Section: Introductionmentioning

confidence: 99%

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Further Developments of Bessel Functions via Conformable Calculus with Applications

Abul-Ez

Zayed

Youssef

2021

Journal of Function Spaces

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The theory of Bessel functions is a rich subject due to its essential role in providing solutions for differential equations associated with many applications. As fractional calculus has become an efficient and successful tool for analyzing various mathematical and physical problems, the so-called fractional Bessel functions were introduced and studied from different viewpoints. This paper is primarily devoted to the study of developing two aspects. The starting point is to present a fractional Laplace transform via conformable fractional-order Bessel functions (CFBFs). We establish several important formulas of the fractional Laplace Integral operator acting on the CFBFs of the first kind. With this in hand, we discuss the solutions of a generalized class of fractional kinetic equations associated with the CFBFs in view of our proposed fractional Laplace transform. Next, we derive an orthogonality relation of the CFBFs, which enables us to study an expansion of any analytic functions by means of CFBFs and to propose truncated CFBFs. A new approximate formula of conformable fractional derivative based on CFBFs is provided. Furthermore, we describe a useful scheme involving the collocation method to solve some conformable fractional linear (nonlinear) multiorder differential equations. Accordingly, several practical test problems are treated to illustrate the validity and utility of the proposed techniques and examine their approximate and exact solutions. The obtained solutions of some fractional differential equations improve the analog ones provided by various authors using different techniques. The provided algorithm may be beneficial to enrich the Bessel function theory via fractional calculus.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Further Developments of Bessel Functions via Conformable Calculus with Applications

Abul-Ez

Zayed

Youssef

2021

Journal of Function Spaces

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“…In 2018 Zhao et al [10] introduced a set of Maxwell's equations using the Generalized Conformable Fractional Derivative (GCFD), which obeys the classical properties of the integer-order derivative. Several works have recently appeared on conformal derivatives, e.g., [14][15][16][17][18][19]. Clearly showing that it is a field of great growth, despite its recent creation.…”

Section: Introductionmentioning

confidence: 99%

A two-index generalization of conformable operators with potential applications in engineering and physics

et al. 2021

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We developed a somewhat novel fractional-order calculus workbench as a certain generalization of the Khalil’s conformable derivative. Although every integer-order derivate can naturally be consistent with fully physical-sense problem’s quotation, this is not the standard scenario of the non-integer-order derivatives, even aiming physics systems’s modelling, solely.We revisited a particular case of the generalized conformable fractional derivative and derived a differential operator, whose properties overcome those of the integer-order derivatives, though preserving its clue advantages.Worthwhile noting, that two-fractional indexes differential operator we are dealing, departs from the single-fractional index framework, which typifies the generalized conformable fractional derivative. This distinction leads to proper mathematical tools, useful in generalizing widely accepted results, with potential applications to fundamental Physics within fractional order calculus. The later seems to be especially appropriate for exercising the Sturm-Liouville eigenvalue problem, as well as the Euler-Lagrange equation and to clarify several operator algebra matters.

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“…Furthermore, authors in [18] worked on the controllability of a fractional linear timeinvariant neutral dynamical system. Especially, numerous works from the above citations [15,[19][20][21] have described the complete controllability of conformable systems. In particular, Wang et al [19] investigated the controllability of conformable semilinear systems.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, Wang et al [19] investigated the controllability of conformable semilinear systems. In addition, authors in [20] described the linear conformable differential system and its controllability.…”

Section: Introductionmentioning

confidence: 99%

Controllability of Differential Systems with the General Conformable Derivative

et al. 2021

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In this paper, the controllability of differential systems with the general conformable derivative is studied. By elaborating the rank criterion and the conformable Gram criterion, sufficient and necessary conditions to investigate that a linear general conformable system is null completely controllable are given. We obtain a full generalization to the general conformable fractional-order system case. In addition, Krasnoselskii’s fixed point theorem to obtain a complete controllability result for a semilinear general conformable system is applied.

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Linear conformable differential system and its controllability

Cited by 18 publications

References 24 publications

Further Developments of Bessel Functions via Conformable Calculus with Applications

Further Developments of Bessel Functions via Conformable Calculus with Applications

A two-index generalization of conformable operators with potential applications in engineering and physics

Controllability of Differential Systems with the General Conformable Derivative

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