Numerical solution of fractal-fractional Mittag–Leffler differential equations with variable-order using artificial neural networks

Zúñiga-Aguilar, C. J.; Gómez-Aguilar, J. F.; Ugalde, Hector M. Romero; Escobar-Jiménez, R.F.; Fernández‐Anaya, Guillermo; Alsaadi, Fawaz E.

doi:10.1007/s00366-020-01229-y

Cited by 15 publications

(2 citation statements)

References 46 publications

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“…Given these facts, it is reasonable to conclude that non-linear domestic wave are frequently observed in the sea, and it would be beneficial if simplified evolution equations could be effectively applied to illustrate strongly nonlinear dispersive waves in various areas of ecology, solid-state physics and engineering. The characteristics of wave, interactions wave, shallow-water wave, optical fibres, internal gravity waves (found in both the sea and the atmosphere), and stratum waves are all characterized by the nonlinear nature of dispersive waves [57][58][59][60][61][62][63][64][65][66] respectively.…”

Section: Introductionmentioning

confidence: 99%

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

Kumar,

Fartyal

2023

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In this Paper, fractional homotopy perturbation transform method (FHPTM) is used to obtain a variety of approximate solutions for the nonlinear fractional generalized Pochhammer-Chree equation which describes certain interesting higher-dimensional waves in harmonic studies, electrical, fluid mechanics, and other fields. By apply the caputo fabrizio derivative via laplace transform technique, we investigate all concerned wave models that have been used in the examination for the propagation of harmonic waves in a cylindrical rod and several problems in fluid mechanics and wave theory in physics. Banach's fixed point hypothesis is tested for governing fractional-order model in order to establish the existence and uniqueness of the achieved solution. We considered the model in terms of arbitrary order with three cases and introduced corresponding numerical simulations to demonstrate and validate the effectiveness of the proposed algorithm. By assigning appropriate values to free parameters, dynamical wave structures of some approximate solutions are graphically demonstrated using 2D and 3D fig. This method can also be used to approximate the solutions of other well-known equations in engineering physics, quantum field , and other applied sciences. Furthermore, various simulations are used to demonstrate the physical behaviors of the acquired solution with respect to fractional integer order.

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Section: Introductionmentioning

confidence: 99%

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

Kumar,

Fartyal

2023

Opt Quant Electron

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“…Therefore, numerical solutions to those VO-FDEs are indispensable especially for practical applications. A great variety of numerical methods have been investigated for VO-FDEs in recent times (Kumar and Chaudhary, 2017; Awotunde et al , 2015; Povstenko and Klekot, 2016; Zeng et al , 2015; Yang et al , 2018; Cai et al , 2018; Sweilam and Assiri, 2013; Kamenia et al , 2017; Sun et al , 2009; Sheng et al , 2011; Straka, 2018; Sierociuk et al , 2015; Xu et al , 2018; Solís-Pérez and Gómez-Aguilar, 2020; Gómez-Aguilar and Atangana, 2019; Ghanbari and Gómez-Aguilar, 2018; Zúñiga et al , 2021; Guo and Wang, 2021; Li et al , 2021). Likewise, various types of wavelets are being studied for problems having greater computational load lately.…”

Section: Introductionmentioning

confidence: 99%

Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations

Ocak

Polat

2021

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PurposeMulti-term variable-order fractional differential equations (VO-FDEs) are powerful tools in accurate modeling of transient-regime real-life problems such as diffusion phenomena and nonlinear viscoelasticity. In this paper the Chebyshev polynomials of the fourth kind is employed to obtain a numerical solution for those multi-term VO-FDEs.Design/methodology/approachTo this end, operational matrices for the approximation of the VO-FDEs are obtained using the Fourth kind Chebyshev Wavelets (FKCW). Thus, the VO-FDE is condensed into an algebraic equation system. The solution of the system of those equations yields a coefficient vector, the coefficient vector in turn yields the approximate solution.FindingsSeveral examples that we present at the end of the paper emphasize the efficacy and preciseness of the proposed method.Originality/valueThe value of the paper stems from the exploitation of FKCWs for the numerical solution of multi-term VO-FDEs. The method produces accurate results even for relatively small collocation points. What is more, FKCW method provides a compact mapping between multi-term VO-FDEs and a system of algebraic equations given in vector-matrix form.

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Theoretical Investigations on the Multistability, Quasiperiodicity and Synchronization of the Driven Chua’s Circuit

Sivaganesh

Srinivasan

2023

Circuits Syst Signal Process

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Numerical solution of fractal-fractional Mittag–Leffler differential equations with variable-order using artificial neural networks

Cited by 15 publications

References 46 publications

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics

Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations

Theoretical Investigations on the Multistability, Quasiperiodicity and Synchronization of the Driven Chua’s Circuit

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