A Robust Analytical Method for Regularized Long Wave Equations

Jani, Haresh P.; Singh, Twinkle

doi:10.1007/s40995-022-01380-9

Cited by 8 publications

(2 citation statements)

References 23 publications

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“…Using AT-HPM, Manimegalai et al [39] solved strongly nonlinear oscillators with great success. Jani and Singh [40] found it had obvious advantages over the decomposition method, Yasmin [41] revealed the dynamic behavior of the fractional convection-reaction-diffusion process, and Jani and Singh [42] extended it to the soliton theory.…”

Section: Introductionmentioning

confidence: 99%

The Aboodh transformation-based homotopy perturbation method: new hope for fractional calculus

2023

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Fractional differential equations can model various complex problems in physics and engineering, but there is no universal method to solve fractional models precisely. This paper offers a new hope for this purpose by coupling the homotopy perturbation method with Aboodh transform. The new hybrid technique leads to a simple approach to finding an approximate solution, which converges fast to the exact one with less computing effort. An example of the fractional casting-mold system is given to elucidate the hope for fractional calculus, and this paper serves as a model for other fractional differential equations.

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Section: Introductionmentioning

confidence: 99%

The Aboodh transformation-based homotopy perturbation method: new hope for fractional calculus

2023

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“…It may face problems in dealing with highly nonlinear fractional differential equations in obtaining a curve of approximation solution. The approximate analytical methods such as Homotopy perturbation method (HPM), Adomian decomposition method (ADM), Homotopy analysis method (HAM), Variational iteration method (VIM), and Differential transform method (DTM) [25][26][27][28][29][30][31][32][33] can play an active role to overcome such situations. Some analytical approaches provide direct convergence solutions, whereas some techniques, such as HAM, have a control parameter h that gives faster and better solutions for nonlinear FFDEs.…”

Section: Introductionmentioning

confidence: 99%

Solution of generalized fractional Jaulent–Miodek model with uncertain initial conditions

Sartanpara,

Meher,

Nikan

et al. 2023

AIP Advances

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This paper analyses a coupled system of generalized coupled system of fractional Jaulent–Miodek equations, including uncertain initial conditions with fuzzy extension. In this regard, an extension of the homotopy with a generalized integral algorithm is adopted for a class of time-fractional fuzzy Jaulent–Miodek models by mixing the fuzzy q-homotopy analysis algorithm with a generalized integral transform and Caputo fractional derivative. The triangular fuzzy numbers (TFNs)are expressed in double parametric form using κ-cut and r-cut and utilized to explain the uncertainties arising in the initial conditions of highly nonlinear differential equations with generalized Hukuhara differentiability (gH-differentiability). The TFNs are controlled by the κ-cut and r-cut, and the variability of uncertainty is examined using a “triangular membership function” (TMF). The results are analyzed by finding the solutions for different spatial coordinate values of time with κ-cut and r-cut for both lower and upper bounds and validated through numerical and graphical representations in crisp cases. Finally, it can be seen that the uncertain probability density function rapidly decreases at the left and right edges when the fractional order is increased, and it is observed that the obtained solutions are more accurate than the existing results through the Hermite wavelet method in the literature.

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Approximate‐analytical iterative approach to time‐fractional Bloch equation with Mittag–Leffler type kernel

Akshey,

Singh

2024

Math Methods in App Sciences

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The paper aims to analyze the fractional Bloch equation with Caputo and Atangana–Baleanu–Caputo (ABC) derivative having a nonsingular kernel. The fixed point theorem is used to prove the existence and uniqueness of the proposed equation. Furthermore, the approximate‐analytical solution of the proposed equation is obtained by Aboodh transform iterative method (ATIM) in the form of a convergent series. The technique (ATIM) combines the new iterative method with the Aboodh transform. The convergence analysis of the approximate solution is discussed using the Cauchy convergence theorem. The validity of the ATIM is shown by numerical simulation and graphs. Also, the preference for nonsingular kernel over singular kernel is shown with the help of tables and graphs. For integer order, the obtained solutions from Caputo and ABC derivative are compared with the exact solutions and published work.

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A Robust Analytical Method for Regularized Long Wave Equations

Cited by 8 publications

References 23 publications

The Aboodh transformation-based homotopy perturbation method: new hope for fractional calculus

The Aboodh transformation-based homotopy perturbation method: new hope for fractional calculus

Solution of generalized fractional Jaulent–Miodek model with uncertain initial conditions

Approximate‐analytical iterative approach to time‐fractional Bloch equation with Mittag–Leffler type kernel

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