Monotone Iterative Method for ψ-Caputo Fractional Differential Equation with Nonlinear Boundary Conditions

Baitiche, Zidane; Derbazi, Choukri; Alzabut, Jehad; Samei, Mohammad Esmael; Kaabar, Mohammed K. A.; Siri, Zailan

doi:10.3390/fractalfract5030081

Cited by 37 publications

(9 citation statements)

References 35 publications

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“…In addition, the aforesaid techniques have been widely used to deal with FDEs subject to initial and boundary conditions. Some significant results can be found in [25,[30][31][32][33][34][35][36][37]. We demonstrate that the said method is well known because it not only produces constructive proof for existence theorems but it also yields various comparison results, which are powerful tools to investigate the qualitative properties of solutions.…”

Section: Introductionmentioning

confidence: 89%

Extremal Solutions of Generalized Caputo-Type Fractional-Order Boundary Value Problems Using Monotone Iterative Method

et al. 2022

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The aim of this research work is to derive some appropriate results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) under nonlinear boundary conditions (NBCs). The aforesaid results are derived by using the monotone iterative method, which exercises the procedure of upper and lower solutions. Two sequences of extremal solutions are generated in which one converges to the upper and the other to the corresponding lower solution. The method does not need any prior discretization or collocation for generating the aforesaid two sequences for upper and lower solutions. Further, the aforesaid techniques produce a fruitful combination of upper and lower solutions. To demonstrate our results, we provide some pertinent examples.

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Section: Introductionmentioning

confidence: 89%

Extremal Solutions of Generalized Caputo-Type Fractional-Order Boundary Value Problems Using Monotone Iterative Method

et al. 2022

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“…Anjum et al [44] applied Li-He's modified homotopy perturbation approach to solve the microelectromechanical system. Baitiche et al [45] used the monotone iterative method for fractional DEs with non-linearity at the boundary. Do et al [46] extended Chebyshev wavelets to two-dimensional fractional DEs.…”

Section: Introductionmentioning

confidence: 99%

Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

et al. 2023

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Physical phenomena and natural disasters, such as tsunamis and floods, are caused due to dispersive water waves and shallow waves caused by earthquakes. In order to analyze and minimize damaging effects of such situations, mathematical models are presented by different researchers. The Wu–Zhang (WZ) system is one such model that describes long dispersive waves. In this regard, the current study focuses on a non-linear (2 + 1)-dimensional time-fractional Wu–Zhang (WZ) system due to its importance in capturing long dispersive gravity water waves in the ocean. A Caputo fractional derivative in the WZ system is considered in this study. For solution purposes, modification of the homotopy perturbation method (HPM) along with the Laplace transform is used to provide improved results in terms of accuracy. For validity and convergence, obtained results are compared with the fractional differential transform method (FDTM), modified variational iteration method (mVIM), and modified Adomian decomposition method (mADM). Analysis of results indicates the effectiveness of the proposed methodology. Furthermore, the effect of fractional parameters on the given model is analyzed numerically and graphically at both integral and fractional orders. Moreover, Caputo, Caputo–Fabrizio, and Atangana–Baleanu approaches of fractional derivatives are applied and compared graphically in the current study. Analysis affirms that the proposed algorithm is a reliable tool and can be used in higher dimensional fractional systems in science and engineering.

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“…where G σ c is the Caputo fractional derivative of order 1 < σ < 2 and ℘ is a given function [3]. Some authors have been utilized comparative strategies through K MNC technique to diverse sorts of FBVPs [4,5]. In 2016, Kiataramkul et al studied the generalized Langevin and Sturm-Liouville fractional di erential equations (GL-SLFDEs) of Hadamard type, with antiperiodic boundary conditions ( APBCs) of the form…”

Section: Introductionmentioning

confidence: 99%

Investigation of the Generalized Proportional Langevin and Sturm–Liouville Fractional Differential Equations via Variable Coefficients and Antiperiodic Boundary Conditions with a Control Theory Application Arising from Complex Networks

Boutiara

Kaabar

Siri

et al. 2022

Mathematical Problems in Engineering

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This article studies the existence theory of an innovation type of generalized proportional fractional differential equations with the assistance of the technique of Kuratowski measure on noncompactness combined with the fixed-point theorem of Mönch. Also, we use Lebesgue’s dominated convergence theorem and Arzelá–Ascoli fixed point theorem on existence and uniqueness results. An application is also presented by employing two illustrative iks, which enrich our outcomes.

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Monotone Iterative Method for ψ-Caputo Fractional Differential Equation with Nonlinear Boundary Conditions

Cited by 37 publications

References 35 publications

Extremal Solutions of Generalized Caputo-Type Fractional-Order Boundary Value Problems Using Monotone Iterative Method

Extremal Solutions of Generalized Caputo-Type Fractional-Order Boundary Value Problems Using Monotone Iterative Method

Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

Investigation of the Generalized Proportional Langevin and Sturm–Liouville Fractional Differential Equations via Variable Coefficients and Antiperiodic Boundary Conditions with a Control Theory Application Arising from Complex Networks

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