Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings

Kaabar, Mohammed K. A.; Refice, Ahmed; Souid, Mohammed Said; Martínez, Francisco; Etemad, Sina; Siri, Zailan; Rezapour, Shahram

doi:10.3390/math9141693

Cited by 16 publications

(2 citation statements)

References 31 publications

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“…Nevertheless, in addition to numerous published papers on fractional constant order problems, few studies on the existence theory have been done in relation to variable order problems [13][14][15][16][17][18][19]. Hence, investigation of this interesting and general topic makes all our findings worthy.…”

Section: Introductionmentioning

confidence: 93%

Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

et al. 2021

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In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin’s continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin’s continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results.

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Section: Introductionmentioning

confidence: 93%

Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

et al. 2021

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“…10 Kaabar, Refice, Souid, Martínez, Etemad, Siri, and Rezapour [45] Established the stability criteria for the solution of the fractional boundary problem under variable order.…”

Section: S Nomentioning

confidence: 99%

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

et al. 2023

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This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.

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Impulsive Fractional Differential Equations with Retardation and Anticipation

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Karapınar

Lazreg

et al. 2023

Synthesis Lectures on Mathematics &Amp; Statistics

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Existence and U-H-R Stability of Solutions to the Implicit Nonlinear FBVP in the Variable Order Settings

Cited by 16 publications

References 31 publications

Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

Impulsive Fractional Differential Equations with Retardation and Anticipation

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