Numerical solvability of generalized Bagley–Torvik fractional models under Caputo–Fabrizio derivative

Hasan, Shatha; Djeddi, Nadir; Al‐Smadi, Mohammed; Al‐Omari, Shrideh; Momani, Shaher; Fulga, Andreea

doi:10.1186/s13662-021-03628-x

Cited by 10 publications

(4 citation statements)

References 35 publications

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“…Later, the RPS approach was developed for handling various kinds of FDEs [26][27][28][29]. This approach produces solutions to the given problem in the convergent generalized Taylor's series formula without involving discretization, linearization, or perturbation [30][31][32]. It may be applied directly to given problems by selecting an appropriate value for the initial guess approximation.…”

Section: Introductionmentioning

confidence: 99%

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Approximate Analytic–Numeric Fuzzy Solutions of Fuzzy Fractional Equations Using a Residual Power Series Approach

et al. 2022

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In this article, we consider a reliable analytical and numerical approach to create fuzzy approximated solutions for differential equations of fractional order with appropriate uncertain initial data by the means of a residual error function. The concept of strongly generalized differentiability is utilized to introduce the fuzzy fractional derivatives. The proposed method provides a systematic scheme based on generalized Taylor expansion and minimization of the residual error function, so as to obtain the coefficients values of a fractional series based on the given initial data of triangular fuzzy numbers in the parametric form. The obtained approximated solutions are provided within an appropriate radius to the requisite domain in the form of rapidly convergent fractional series according to their parametric form. The method’s performance and applicability are verified by applying it on some numerical examples. The impact of -levels and fractional order is presented quantitatively and graphically, showing the coincidence between the exact and the fuzzy approximated solutions. Moreover, for reliability and accuracy, our obtained results are numerically compared with the exact solutions and with results obtained using other methods described in the literature. This indicates that the proposed approach overcomes the difficulties that appear in other approaches to create fractional series solutions for varied uncertain natural problems arising within the fields of applied physics and engineering.

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Section: Introductionmentioning

confidence: 99%

“…In view of the last described FS technique, we took into account ϕ 1r (0) = ω 0 = r + 1 and ϕ 2r (0) = ρ 0 = 3 − r. Suppose that the k-th approximated solutions for FIVPs (30) have the following FS expansions form…”

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confidence: 99%

Approximate Analytic–Numeric Fuzzy Solutions of Fuzzy Fractional Equations Using a Residual Power Series Approach

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“…The complex Ginzburg-Landau equation is a type of nonlinear Schrödinger equation, which governs the evolution of certain amplitude of instability pulses in a wide variety of dissipative system. In the literature, many chemical and physical phenomena are described by using the complex Ginzburg-Landau equation, in dynamic phase transitions, surface waves in viscous liquids, processes in optics, laser physics, superconductivity, modeling of Bose-Einstein condensation, spatially extended nonequilibrium systems and other phenomena, to mention but a few (see, e.g., [1][2][3][4][5][6]). The aforementioned model is also studied by many researchers from several aspects, including phase dynamics and modulation instability.…”

Section: Introductionmentioning

confidence: 99%

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

et al. 2022

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In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well.

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“…The proposed algorithm is straightforward, accurate and powerful for creating a series of solutions for different models that occur in applied mathematics without terms of perturbation, discretization, and linearization. For more information about advanced different and approximate methods, refer to [ 30 , 31 , 32 , 33 , 34 , 35 ] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense

Bataineh

Alaroud

Al‐Omari

et al. 2021

Entropy

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Fuzzy differential equations provide a crucial tool for modeling numerous phenomena and uncertainties that potentially arise in various applications across physics, applied sciences and engineering. Reliable and effective analytical methods are necessary to obtain the required solutions, as it is very difficult to obtain accurate solutions for certain fuzzy differential equations. In this paper, certain fuzzy approximate solutions are constructed and analyzed by means of a residual power series (RPS) technique involving some class of fuzzy fractional differential equations. The considered methodology for finding the fuzzy solutions relies on converting the target equations into two fractional crisp systems in terms of ρ-cut representations. The residual power series therefore gives solutions for the converted systems by combining fractional residual functions and fractional Taylor expansions to obtain values of the coefficients of the fractional power series. To validate the efficiency and the applicability of our proposed approach we derive solutions of the fuzzy fractional initial value problem by testing two attractive applications. The compatibility of the behavior of the solutions is determined via some graphical and numerical analysis of the proposed results. Moreover, the comparative results point out that the proposed method is more accurate compared to the other existing methods. Finally, the results attained in this article emphasize that the residual power series technique is easy, efficient, and fast for predicting solutions of the uncertain models arising in real physical phenomena.

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Numerical solvability of generalized Bagley–Torvik fractional models under Caputo–Fabrizio derivative

Cited by 10 publications

References 35 publications

Approximate Analytic–Numeric Fuzzy Solutions of Fuzzy Fractional Equations Using a Residual Power Series Approach

Approximate Analytic–Numeric Fuzzy Solutions of Fuzzy Fractional Equations Using a Residual Power Series Approach

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense

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