Soliton Solutions of Generalized Third Order Time‐Fractional KdV Models Using Extended He‐Laplace Algorithm

Qayyum, Mubashir; Ahmad, Efaza; Afzal, Sidra; Acharya, Saraswati

doi:10.1155/2022/2174806

Cited by 8 publications

(4 citation statements)

References 43 publications

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“…υ n (ξ, η, ς, μ) and 􏽥 υ(ξ, η, ς, μ), defned in a Banach space (B[0, T], ‖.‖). Ten, the series solution 􏽥 υ in (19) converges into the solution of ( 11), whenever cϵ(0, 1).…”

Section: Methodology Of Fuzzy He-laplace Algorithmmentioning

confidence: 99%

“…Te Sturm-Liouville diferential equation is studied by Charandabi et al in [18] using α − ψ− contractible mappings to prove the existence of solutions. Qayyum et al performed numerical analysis on generalized time-fractional KDV models in [19][20][21]. Etemad et al used Krasnoselskii's fxed point theorem to investigate a new class of nonlinear fractional diferential equations [22].…”

Section: Introductionmentioning

confidence: 99%

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New Solutions of Fuzzy-Fractional Fisher Models via Optimal He–Laplace Algorithm

Qayyum

Tahir

Acharya

2023

International Journal of Intelligent Systems

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Fuzzy differential equations have gained significant attention in recent years due to their ability to model complex systems in the presence of uncertainty or imprecise information. These equations find applications in various fields, such as biomathematics, horological processes, production inventory models, epidemic models, fluid models, and economic investments. The Fisher model is one such example, which studies the dynamics of a population with uncertain growth rates. This study proposes a dual parametric extension of the He–Laplace algorithm for solving time-fractional fuzzy Fisher models. The proposed methodology uses triangular fuzzy numbers to introduce uncertainty in highly nonlinear fractional differential equations under the generalized Hukuhara differentiability concept. The obtained solutions are validated against existing results in crisp form and are found to be more accurate. The results are analyzed by finding solutions with different values of spatial coordinate ξ , time η , and ς -cut for both upper and lower bounds, and they are presented graphically. The analysis reveals that the uncertain probability density function gradually decreases at left and right boundaries when the fractional order is increased. The study provides valuable insights into the behavior of population growth under uncertain conditions and demonstrates the effectiveness of the proposed methodology in solving fuzzy-fractional models.

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Section: Methodology Of Fuzzy He-laplace Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New Solutions of Fuzzy-Fractional Fisher Models via Optimal He–Laplace Algorithm

Qayyum

Tahir

Acharya

2023

International Journal of Intelligent Systems

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“…Baleanu and Jassim [44] extended the modifed fractional homotopy perturbation technique on Helmholtz and coupled Helmholtz equations. Qayyum et al [45,46] utilized the He-Laplace method to solve generalized third-and ffth-order timefractional KdV models. Fractional Navier-Stokes equations are investigated by Jena and Chakraverty [47] through homotopy perturbation Elzaki transform.…”

Section: Introductionmentioning

confidence: 99%

New Solutions of Time- and Space-Fractional Black–Scholes European Option Pricing Model via Fractional Extension of He-Aboodh Algorithm

Qayyum,

Ahmad

2024

Journal of Mathematics

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The current study explores the space and time-fractional Black–Scholes European option pricing model that primarily occurs in the financial market. To tackle the complexities associated with solving models in a fractional environment, the Aboodh transform is hybridized with He’s algorithm. This facilitates in improving the efficiency and applicability of the classical homotopy perturbation method (HPM) by ensuring the rapid convergence of the series form solution. Three cases that are time-fractional scenario, space-fractional scenario, and time-space-fractional scenario are observed through graphs and tables. 2D graphical analysis is performed to depict the behaviour of a given option pricing model for varying time, stock price, and fractional parameters. Solutions of the European option pricing model at various fractional orders are also presented as 3D plots. The results obtained through these graphs unfold the interchange between time- and space-fractional derivatives, presenting a comprehensive study of option pricing under fractional dynamics. The competency of the proposed scheme is illustrated via solutions and errors throughout the fractional domain in tabular form. The validity of the He-Aboodh results is exhibited by comparison with existing errors. Analysis shows that the proposed methodology (He-Aboodh algorithm) is a valuable scheme for solving time-space-fractional models arising in business and economics.

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“…The approximate solution is calculated by taking inverse Laplace-Carson transform and then adding the results.Case 4: Fuzzy-Fractional ˜˜  Circuit Consider the fuzzy-fractional ˜˜  circuit model[50] given in equation (22) =   and σ = ˜1. The triangular fuzzy numbers in model have intervals  = (0, 5, 10), =(10,15,20),  =(10,15,20), and Ã0 = (0, 0.1, 0.2). By utilizing definition 7 we can express them in parametric form as…”

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Fuzzy-fractional modeling and simulation of electric circuits using extended He-Laplace-Carson algorithm

Qayyum,

Ahmad

2024

Phys. Scr.

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In literature, most of the electric circuits are found to be at integer order, which are unable to capture the involved uncertainties and memory effects. To understand these effects, the current study is focused on modeling and analysis of fuzzy-fractional LC circuit, RC circuit, RL circuit, and RLC circuit. Triangular fuzzy numbers are utilized to introduce uncertainties in circuits. The memory effects are considered through fractional derivative in Caputo sense. A hybrid homotopy perturbation method (HPM) is established for solution purpose, in which standard HPM is combined with Laplace-Carson transformation in fuzzy-Caputo sense. This leads to an excellent and well-organized approach to calculate numerical solutions in the aspect of fuzzy logic and fractional calculus. The results obtainedthrough proposed approach are validated by comparing them with already existing results in literature. Residual errors are also calculated across the domain of r at fractional order to determine the convergence and stability of electric circuit models. To study the dynamics of inductor, resistor, and capacitor on current along with fractional and fuzzy parameters involved in the electric circuits, contour and three-dimensional plots are constructed. Several plots specifying the fuzzy membership function of models across various parameter values are also demonstrated. The results achieved through this study indicate that He-Laplace-Carson method can assist engineers and researchers when dealing with electric circuits characterized by fuzzy and fractional dynamics.

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Soliton Solutions of Generalized Third Order Time‐Fractional KdV Models Using Extended He‐Laplace Algorithm

Cited by 8 publications

References 43 publications

New Solutions of Fuzzy-Fractional Fisher Models via Optimal He–Laplace Algorithm

New Solutions of Fuzzy-Fractional Fisher Models via Optimal He–Laplace Algorithm

New Solutions of Time- and Space-Fractional Black–Scholes European Option Pricing Model via Fractional Extension of He-Aboodh Algorithm

Fuzzy-fractional modeling and simulation of electric circuits using extended He-Laplace-Carson algorithm

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