Numerical study of fractional phi-4 equation

Y. Massoun; C. Cesarano; A. K Alomari; A. Said; Y. Massoun; C. Cesarano; A. K Alomari; A. Said

doi:10.3934/math.2024418

AIMS Mathematics

2024, Volume 9, Issue 4: 8630-8640. doi: 10.3934/math.2024418

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Numerical study of fractional phi-4 equation

Y. Massoun ^{1
,
,},
C. Cesarano ^{2
,
,},
A. K Alomari ³,
A. Said ⁴

1.
Department of Mathematics, Faculty of Sciences, University of Algiers, Algeria
2.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele Ⅱ, 39, 00186 Roma, Italy
3.
Department of Mathematics, Faculty of Science, Yarmouk University, Irbid 211-63, Jordan
4.
Department of Mathematics, Faculty of Sciences, University of Djilali Bounaama, Khemise Miliana, Algeria

Received: 08 November 2023 Revised: 08 February 2024 Accepted: 23 February 2024 Published: 29 February 2024
MSC : 26A33, 65-XX, 55P10

In this paper, we established an analytical solution for the fractional phi-4 model within the Caputo derivative using the homotopy analysis method. This equation known for its nonlinear characteristics often describes various physical phenomena like solitons, wave propagation, and field theories. The fractional version introduces fractional derivatives, making it even more challenging. The homotopy analysis method can effectively handle these nonlinearities. Our objective was to illustrate the reliability and accuracy of our proposed algorithm, which we achieved through a comparative analysis against results obtained using the Yang transform decomposition method. Using the residual error to determine the optimal value of the convergence control parameter $ \hbar $, the results presented underscored the remarkable efficiency and accuracy of this approach.
- phi-4 equation,
- Caputo fractional derivative,
- homotopy analysis method
Citation: Y. Massoun, C. Cesarano, A. K Alomari, A. Said. Numerical study of fractional phi-4 equation[J]. AIMS Mathematics, 2024, 9(4): 8630-8640. doi: 10.3934/math.2024418

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Abstract

In this paper, we established an analytical solution for the fractional phi-4 model within the Caputo derivative using the homotopy analysis method. This equation known for its nonlinear characteristics often describes various physical phenomena like solitons, wave propagation, and field theories. The fractional version introduces fractional derivatives, making it even more challenging. The homotopy analysis method can effectively handle these nonlinearities. Our objective was to illustrate the reliability and accuracy of our proposed algorithm, which we achieved through a comparative analysis against results obtained using the Yang transform decomposition method. Using the residual error to determine the optimal value of the convergence control parameter $ \hbar $, the results presented underscored the remarkable efficiency and accuracy of this approach.

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