Stability analysis of nonlinear fractional differential equations with Caputo and Riemann-Liouville derivatives

Khan, Aziz; Syam, Muhammed I.; Zada, Akbar; Khan, Hasib

doi:10.1140/epjp/i2018-12119-6

Cited by 52 publications

(32 citation statements)

References 27 publications

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“…Suppose that T 1 > a is large enough so that x(t) > 0 for t ≥ T 1 . Hence, (10) implies that f 1 (t, x) > 0 for t ≥ T 1 .…”

Section: Oscillation Of Hadamard Fractional Differential Equations Inmentioning

confidence: 97%

“…The oscillation theory for fractional differential and difference equations was studied by some authors (see [1][2][3][4][5][6][7][8]). The study of oscillation and other qualitative properties of fractional dynamical systems such as stability, existence, and uniqueness of solutions is necessary to analyze the systems under consideration [9][10][11][12][13]. The analysis of the memory of certain fractional dynamical systems has gained a high impact in the last few years [14,15].…”

Section: Introductionmentioning

confidence: 99%

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On the oscillation of Hadamard fractional differential equations

Abdalla

Abdeljawad

2018

Adv Differ Equ

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Hadamard fractional derivatives are nonlocal fractional derivatives with singular logarithmic kernel with memory, and hence they are suitable to describe complex systems. In this paper, sufficient conditions are established for the oscillation of solutions fractional differential equations in the frame of left Hadamard fractional derivatives of order α ∈ C, Re(α) ≥ 0. The results are also obtained for fractional Hadamard derivatives in the Caputo setting. Examples are provided to illustrate the applicability of the main results.

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“…Suppose that T 1 > a is large enough so that x(t) > 0 for t ≥ T 1 . Hence, (10) implies that f 1 (t, x) > 0 for t ≥ T 1 .…”

Section: Oscillation Of Hadamard Fractional Differential Equations Inmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

On the oscillation of Hadamard fractional differential equations

Abdalla

Abdeljawad

2018

Adv Differ Equ

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“…With the help of recent development on the HUS, [37][38][39][40] we analyze the suggested ABC fractional-order L-V reaction-diffusion model (6) and initiate the following definition of HUS:…”

Section: Hyers-ulam Stabilitymentioning

confidence: 99%

Existence of solution for a fractional‐order Lotka‐Volterra reaction‐diffusion model with Mittag‐Leffler kernel

Khan

et al. 2019

Math Methods in App Sciences

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In the literature, many researchers have studied Lotka‐Volterra (L‐V) models for different types of studies. In order to continue the study, we consider a fractional‐order L‐V model involving three different species in the Atangana‐Baleanu‐Caputo (ABC) sense of fractional derivative. This new model has potentials for a large number of research‐oriented studies. The first point that arises is whether the new model has a solution or not. Therefore, to answer this question, we consider the existence and uniqueness (EU) of the solutions and then Hyers‐Ulam (HU) stability for the proposed L‐V model.

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“…Fractional differential equations (FDEs) present new models for many applications in physics, biomathematics, environmental issues, control theory, image processing, chemistry, mechanics, and so on [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Recently, researchers focus on studying various aspects of fractional differential equations, such as stability analysis, existence, multiplicity, and uniqueness of solutions .…”

Section: Introductionmentioning

confidence: 99%

“…Some authors studied the existence and uniqueness of solutions for fractional differential equations with Caputo or Riemann-Liouville derivatives based on the Banach contraction principle and investigate the stability results for various fractional problems [4,5,16,17]. Others studied the existence and multiplicity results of positive solutions or the iterative scheme.…”

Section: Introductionmentioning

confidence: 99%