Authors:
Ameena K. Essa,Fatema S. Al-Juboori,Asmaa Saddam Jaafar,Hasanain Jalil Neamah Alsaedi,Adel S.Hussain,Rana Aziz Yousif Almuttalibi,Haider Mshali,DOI NO:
https://doi.org/10.26782/jmcms.2026.08.00008Keywords:
Bagley–Torvik equation; fractional differential-algebraic equations; Caputo derivative; physics-informed neural networks; operator consistency; convergence; Volterra operator.,Abstract
This revised study develops a consistency-aware modified physics-informed neural network (PINN) formulation for fractional Bagley–Torvik differential–algebraic equations (fractional DAEs). The revision addresses the central mathematical issue raised by the review: a hereditary Caputo operator and an instantaneous algebraic constraint cannot be treated as unrelated penalty terms without a compatibility argument. The proposed formulation therefore introduces a coupled residual map, an implicit-function condition for the algebraic variable, and a reduced fractional operator obtained by eliminating the algebraic state locally. A consistency theorem is established showing that, under explicit Lipschitz, nonsingularity, approximation-density, and residual-stability assumptions, a sequence of network pairs whose composite residual tends to zero converges to the coupled solution manifold. The corresponding Volterra operator is shown to be continuous and compact under standard boundedness conditions, while a local contraction condition provides existence and uniqueness for the reduced initial-value formulation. The numerical material supplied with the original manuscript is retained and reanalyzed without introducing unreported computational values. For Example 1, the reported absolute-error data give MAE = 6.2454 × 10⁻³ and RMSE = 9.5014 × 10⁻³; for Example 2, the corresponding values are MAE = 1.1672 and RMSE = 2.0072. The revised discussion consequently distinguishes pointwise approximation quality from operator consistency and avoids unsupported claims of unconditional convergence. The resulting framework provides a mathematically controlled basis for applying modified PINNs to fractional Bagley–Torvik DAEs.Refference:
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