A mathematical approach to ABC fractional modeling of the Klein–Gordon equation

Fouzia Chita; Souad Ayadi; Mohammed Boukedroun; Meltem Erden Ege; Ozgur Ege; Mohammed Rabih; Fouzia Chita; Souad Ayadi; Mohammed Boukedroun; Meltem Erden Ege; Ozgur Ege; Mohammed Rabih

doi:10.3934/math.2026970

AIMS Mathematics

2026, Volume 11, Issue 8: 24060-24089. doi: 10.3934/math.2026970

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A mathematical approach to ABC fractional modeling of the Klein–Gordon equation

1.
Mathematics Department, Faculty of Matter Sciences and Computer Science, University of Djilali Bounaama Khemis Miliana, Khemis Miliana 44225, Algeria
2.
Acoustics and Civil Engineering Laboratory, Physics Department, Faculty of Material Sciences and Computer Science, Khemis Miliana University, Ain Defla 44225, Algeria
3.
Independent Researcher, Izmir 35000, Turkey
4.
Department of Mathematics, Ege University, Bornova, Izmir 35100, Turkey
5.
Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia

Received: 17 June 2026 Revised: 25 July 2026 Accepted: 29 July 2026 Published: 06 August 2026
MSC : 26D10, 34A12, 34K20, 45G10, 47H10

This work investigated a fractional Klein–Gordon model formulated with the Atangana–Baleanu (AB) fractional derivative. The physical motivation for employing this fractional operator was discussed, emphasizing its ability to describe time-dependent processes with memory effects. The mathematical model was established, and the existence and uniqueness of its solution were examined using fixed-point theory. In addition, the Laplace multistage telescoping decomposition method (LMTDM) was introduced and applied to obtain approximate analytical solutions. The effectiveness of the proposed approach was evaluated through a comparison of absolute errors with those obtained using the Laplace Adomian decomposition method (LADM). The numerical results indicated that the LMTDM provides accurate approximations and exhibits good convergence properties, making it a reliable technique for solving fractional Klein–Gordon equations.
Citation: Fouzia Chita, Souad Ayadi, Mohammed Boukedroun, Meltem Erden Ege, Ozgur Ege, Mohammed Rabih. A mathematical approach to ABC fractional modeling of the Klein–Gordon equation[J]. AIMS Mathematics, 2026, 11(8): 24060-24089. doi: 10.3934/math.2026970

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Abstract

This work investigated a fractional Klein–Gordon model formulated with the Atangana–Baleanu (AB) fractional derivative. The physical motivation for employing this fractional operator was discussed, emphasizing its ability to describe time-dependent processes with memory effects. The mathematical model was established, and the existence and uniqueness of its solution were examined using fixed-point theory. In addition, the Laplace multistage telescoping decomposition method (LMTDM) was introduced and applied to obtain approximate analytical solutions. The effectiveness of the proposed approach was evaluated through a comparison of absolute errors with those obtained using the Laplace Adomian decomposition method (LADM). The numerical results indicated that the LMTDM provides accurate approximations and exhibits good convergence properties, making it a reliable technique for solving fractional Klein–Gordon equations.

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