Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

Analytical, Numerical, and PINN Analysis of the ABC Fractional GK Equation: A Compatibility-Aware Benchmark Study

Document Type : Original Article

Authors
1 Department of Mathematics, Begum Rokeya University, Rangpur-5404, Bangladesh
2 Bangladesh Army University of Engineering & Technology, Natore, Bangladesh
3 Department of Mathematics, Bangladesh Army University of Engineering & Technology, Natore, Bangladesh
4 Department of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh
Abstract
In this study, we develop a unified analytical-numerical-machine-learning framework for a time-fractional Gardner-Kawahara (GK) equation governed by the Atangana-Baleanu (AB) derivative in the Caputo sense. Also, we introduce compatibility-aware forced benchmark so that the nonlocal Mittag-Leffler (ML) memory law, the prescribed initial state, and the source term remain mutually consistent at the initial time. We apply three independent solution routes applied to the same problem: a Laplace-Adomian decomposition method (LADM), a Fourier pseudo spectral discretization combined with an AB predictor-corrector history treatment, and a physics-informed neural network (PINN) formulated through the equivalent Atangana-Baleanu integral equation. For the analytical solution, we investigate a six-component LADM expansion to be convergent over the selected short-time interval; at the lowest order considered, α=0.85, the last retained component is below 1% of the total solution and the normalized integral residual is approximately 3.10×〖10〗^(-3), in which the independent numerical solution is grid converged, with coarse-to-refined relative differences of order 〖10〗^(-8)–〖10〗^(-7), and agrees with LADM at t=0.10 with relative L_2 errors ranging from 4.42×〖10〗^(-8) for α=1 to 2.92×〖10〗^(-3) for α=0.85. For the representative fractional case α=0.90, the PINN reproduces the numerical wave field with a full space-time relative L_2 error of 1.07% and a final-time relative error of 1.26%. Across the analytical and numerical solutions, this paper shows that decreasing the fractional order strengthens the forced transient response, demonstrating the effect of ML memory on the nonlinear dispersive dynamics. In this research, the principal contribution is not a new isolated solver, but a compatibility-aware, three-way cross-validation framework for an Atangana-Baleanu-Caputo (ABC) fractional GK model, connecting semi-analytical decomposition, high-accuracy spectral computation, and physics-informed learning within one reproducible benchmark.
Keywords
Subjects

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Articles in Press, Accepted Manuscript
Available Online from 14 November 2026

  • Receive Date 11 October 2026
  • Accept Date 10 November 2026
  • First Publish Date 14 November 2026
  • Publish Date 14 November 2026