On existence and stability results to a class of boundary value problems under Mittag-Leffler power law

Ali, Gauhar; Shah, Kamal; Abdeljawad, Thabet; Khan, Hasib; Rahman, Ghaus ur; Khan, Aziz

doi:10.1186/s13662-020-02866-9

Cited by 9 publications

(5 citation statements)

References 15 publications

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“…The scriptUscriptH stability concept is initiated by the authors Ulam and Hyers [44, 45], and it has a significant effect in the fractional differential equations field [17, 46, 47]. Throughout this section, we will discuss scriptUscriptH stability, scriptUscriptHscriptR stability and their generalizations for the solution of the system (1.2)–(1.3).…”

Section: Stability Of Solutionmentioning

confidence: 99%

“…Indeed, the most optimal emulative operator among a nonsingular kernel operator is that which depends on Mittag-Leffler function, which is called Atangana-Baleanu-Caputo (ABC) operator [12]. In view of this, many authors employed ABC derivative to study fractional differential equations and modeling of the infectious diseases, we refer to these works [15][16][17][18][19][20]. Particularly, Alnahdi et al [21], studied the existence, uniqueness and continuous dependence of solutions of the nonlinear implicit fractional differential equation with nonlocal conditions involving the ABC fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, the most optimal emulative operator among a nonsingular kernel operator is that which depends on Mittag–Leffler function, which is called Atangana–Baleanu–Caputo (ABC) operator [12]. In view of this, many authors employed ABC derivative to study fractional differential equations and modeling of the infectious diseases, we refer to these works [15–20]. Particularly, Alnahdi et al.…”

Section: Introductionmentioning

confidence: 99%

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Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator

Ayari

Thabet²

2023

AJMS

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PurposeThis paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results.Design/methodology/approachThis paper considered theoretical and numerical methodologies.FindingsThis paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures.Originality/valueThe novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

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Section: Stability Of Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator

Ayari

Thabet²

2023

AJMS

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“…In this connection, the existence and uniqueness of a solution studied by various techniques, such as monotone iterative technique, were discussed in [22]; topological degree theory was the concern in [23]; a method of successive approximation was developed in [24]; fixed point index theory was used in [25], and tools of fixed point theory (FPT) were used in [26]. For more details, see the recent work cited as [27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Formulation, Solution’s Existence, and Stability Analysis for Multi-Term System of Fractional-Order Differential Equations

2022

Self Cite

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In the recent past, multi-term fractional equations have been studied using symmetry methods. In some cases, many practical test problems with some symmetries are provided to demonstrate the authenticity and utility of the used techniques. Fractional-order differential equations can be formulated by using two types of differential operators: single-term and multi-term differential operators. Boundary value problems with single- as well as multi-term differential operators have been extensively studied, but several multi-term fractional differential equations still need to be formulated, and examination should be done with symmetry or any other feasible techniques. Therefore, the purpose of the present research work is the formulation and study of a new couple system of multi-term fractional differential equations with delay, as well as supplementation with nonlocal boundary conditions. After model formulation, the existence of a solution and the uniqueness conditions will be developed, utilizing fixed point theory and functional analysis. Moreover, results related to Ulam’s and other types of functional stability will be explored, and an example is carried out to illustrate the findings of the work.

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“…Since approximately the year 2000, fractional calculus theory has gained substantial popularity and significance. It has tremendously attractive applications in diverse and widespread fields of physics and engineering, such as rheology, viscoelasticity, electromagnetic theory, diffusive transport, fluid flow, and electrical networks (Ali et al, 2020; Aydogan et al, 2020; Kilbas et al, 2006; Mainardi, 2010; Podlubny, 1999; Samko et al, 1993; Zaslavsky, 2005). Fractional order models are often more accurate than classical integer order descriptions because fractional order derivatives and integrals embed the description of the memory and hereditary properties of different substances (Baleanu and Mendes Lopes, 2019).…”

Section: Introductionmentioning

confidence: 99%

A numerical approach for solving fractional optimal control problems with mittag-leffler kernel

Jafari

Ganji

Sayevand

et al. 2021

Journal of Vibration and Control

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In this work, we present a numerical approach based on the shifted Legendre polynomials for solving a class of fractional optimal control problems. The derivative is described in the Atangana–Baleanu derivative sense. To solve the problem, operational matrices of AB-fractional integration and multiplication, together with the Lagrange multiplier method for the constrained extremum, are considered. The method reduces the main problem to a system of nonlinear algebraic equations. In this framework by solving the obtained system, the approximate solution is calculated. An error estimate of the numerical solution is also proved for the approximate solution obtained by the proposed method. Finally, some illustrative examples are presented to demonstrate the accuracy and validity of the proposed scheme.

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On existence and stability results to a class of boundary value problems under Mittag-Leffler power law

Cited by 9 publications

References 15 publications

Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator

Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator

Formulation, Solution’s Existence, and Stability Analysis for Multi-Term System of Fractional-Order Differential Equations

A numerical approach for solving fractional optimal control problems with mittag-leffler kernel

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