Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators

Anil Kumar Menaria

doi:doi:10.11648/j.ajam.20261404.17

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Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators

Anil Kumar Menaria^*

Published in American Journal of Applied Mathematics (Volume 14, Issue 4)

Received: 1 July 2026 Accepted: 11 July 2026 Published: 14 August 2026

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Abstract

Anisotropic cosmological models play an important role in describing the evolution of the early universe, where directional expansion and spatial curvature cannot be neglected. Classical Bianchi models, however, are formulated through local differential equations and therefore do not account for hereditary effects that may influence the interaction between geometry and matter over cosmic time scales. Motivated by the growing use of nonlocal fractional operators in mathematical physics, this work develops a memory-dependent Bianchi type-IV cosmological model by incorporating the Atangana-Baleanu-Caputo fractional derivative into the reduced Einstein evolution equations. The resulting formulation preserves the characteristic curvature contribution through the explicit A^-2 geometric potential and leads to a closed nonlinear dynamical system governing the directional scale factors and directional Hubble parameters. The analytical study is carried out by transforming the fractional model into an equivalent Volterra integral equation, from which existence and uniqueness of solutions are established using Banach's fixed-point theorem. Additional theoretical results provide admissibility conditions for the energy density together with a sufficient criterion for the asymptotic decay of anisotropy under suitable expansion assumptions. To investigate the influence of nonlocal memory, a predictor-corrector numerical scheme is implemented for several fractional orders. The computational results demonstrate that decreasing the fractional order strengthens hereditary effects, slows the decay of the shear scalar and anisotropy parameter, delays isotropization, and produces a measurable memory lag in the expansion dynamics while recovering the classical Einstein evolution as the fractional order approaches unity. These findings show that the proposed framework offers a mathematically consistent and computationally reproducible extension of Bianchi type-IV cosmology, providing a useful approach for investigating memory-driven anisotropic evolution within the setting of fractional gravitational dynamics.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 4)
DOI	10.11648/j.ajam.20261404.17
Page(s)	244-258
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Bianchi Type-IV, Atangana-Baleanu Derivative, Nonlocal Memory, Anisotropic Cosmology, \\Existence and Uniqueness, Isotropization

References

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[7]	A. Tateishi, H. Ribeiro, and E. Lenzi, "The role of fractional time-derivative operators on anomalous diffusion," Frontiers in Physics, vol. 5, Art. no. 52, 2017. https://doi.org/10.3389/fphy.2017.00052
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[12]	M. B. Jeelani, A. S. Alnahdi, M. A. Almalahi, M. S. Abdo, H. A. Wahash, and M. A. Abdelkawy, "Study of the Atangana-Baleanu-Caputo type fractional system with a generalized Mittag-Leffler kernel," AIMS Mathematics, vol. 7, no. 2, pp. 2001-2018, 2022. https://doi.org/10.3934/math.2022115
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Menaria, A. K. (2026). Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators. American Journal of Applied Mathematics, 14(4), 244-258. https://doi.org/10.11648/j.ajam.20261404.17

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ACS Style

Menaria, A. K. Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators. Am. J. Appl. Math. 2026, 14(4), 244-258. doi: 10.11648/j.ajam.20261404.17

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AMA Style

Menaria AK. Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators. Am J Appl Math. 2026;14(4):244-258. doi: 10.11648/j.ajam.20261404.17

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@article{10.11648/j.ajam.20261404.17,
  author = {Anil Kumar Menaria},
  title = {Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {4},
  pages = {244-258},
  doi = {10.11648/j.ajam.20261404.17},
  url = {https://doi.org/10.11648/j.ajam.20261404.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261404.17},
  abstract = {Anisotropic cosmological models play an important role in describing the evolution of the early universe, where directional expansion and spatial curvature cannot be neglected. Classical Bianchi models, however, are formulated through local differential equations and therefore do not account for hereditary effects that may influence the interaction between geometry and matter over cosmic time scales. Motivated by the growing use of nonlocal fractional operators in mathematical physics, this work develops a memory-dependent Bianchi type-IV cosmological model by incorporating the Atangana-Baleanu-Caputo fractional derivative into the reduced Einstein evolution equations. The resulting formulation preserves the characteristic curvature contribution through the explicit A-2 geometric potential and leads to a closed nonlinear dynamical system governing the directional scale factors and directional Hubble parameters. The analytical study is carried out by transforming the fractional model into an equivalent Volterra integral equation, from which existence and uniqueness of solutions are established using Banach's fixed-point theorem. Additional theoretical results provide admissibility conditions for the energy density together with a sufficient criterion for the asymptotic decay of anisotropy under suitable expansion assumptions. To investigate the influence of nonlocal memory, a predictor-corrector numerical scheme is implemented for several fractional orders. The computational results demonstrate that decreasing the fractional order strengthens hereditary effects, slows the decay of the shear scalar and anisotropy parameter, delays isotropization, and produces a measurable memory lag in the expansion dynamics while recovering the classical Einstein evolution as the fractional order approaches unity. These findings show that the proposed framework offers a mathematically consistent and computationally reproducible extension of Bianchi type-IV cosmology, providing a useful approach for investigating memory-driven anisotropic evolution within the setting of fractional gravitational dynamics.},
 year = {2026}
}

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TY  - JOUR
T1  - Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators
AU  - Anil Kumar Menaria
Y1  - 2026/08/14
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261404.17
DO  - 10.11648/j.ajam.20261404.17
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 244
EP  - 258
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261404.17
AB  - Anisotropic cosmological models play an important role in describing the evolution of the early universe, where directional expansion and spatial curvature cannot be neglected. Classical Bianchi models, however, are formulated through local differential equations and therefore do not account for hereditary effects that may influence the interaction between geometry and matter over cosmic time scales. Motivated by the growing use of nonlocal fractional operators in mathematical physics, this work develops a memory-dependent Bianchi type-IV cosmological model by incorporating the Atangana-Baleanu-Caputo fractional derivative into the reduced Einstein evolution equations. The resulting formulation preserves the characteristic curvature contribution through the explicit A-2 geometric potential and leads to a closed nonlinear dynamical system governing the directional scale factors and directional Hubble parameters. The analytical study is carried out by transforming the fractional model into an equivalent Volterra integral equation, from which existence and uniqueness of solutions are established using Banach's fixed-point theorem. Additional theoretical results provide admissibility conditions for the energy density together with a sufficient criterion for the asymptotic decay of anisotropy under suitable expansion assumptions. To investigate the influence of nonlocal memory, a predictor-corrector numerical scheme is implemented for several fractional orders. The computational results demonstrate that decreasing the fractional order strengthens hereditary effects, slows the decay of the shear scalar and anisotropy parameter, delays isotropization, and produces a measurable memory lag in the expansion dynamics while recovering the classical Einstein evolution as the fractional order approaches unity. These findings show that the proposed framework offers a mathematically consistent and computationally reproducible extension of Bianchi type-IV cosmology, providing a useful approach for investigating memory-driven anisotropic evolution within the setting of fractional gravitational dynamics.
VL  - 14
IS  - 4
ER  -

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Author Information

Anil Kumar Menaria

Department of Mathematics and Statistics, Bhupal Nobles' University, Udaipur, India

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http://orcid.org/0009-0007-4561-1031

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APA Style

Menaria, A. K. (2026). Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators. American Journal of Applied Mathematics, 14(4), 244-258. https://doi.org/10.11648/j.ajam.20261404.17

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ACS Style

Menaria, A. K. Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators. Am. J. Appl. Math. 2026, 14(4), 244-258. doi: 10.11648/j.ajam.20261404.17

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AMA Style

Menaria AK. Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators. Am J Appl Math. 2026;14(4):244-258. doi: 10.11648/j.ajam.20261404.17

Copy | Download

@article{10.11648/j.ajam.20261404.17,
  author = {Anil Kumar Menaria},
  title = {Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {4},
  pages = {244-258},
  doi = {10.11648/j.ajam.20261404.17},
  url = {https://doi.org/10.11648/j.ajam.20261404.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261404.17},
  abstract = {Anisotropic cosmological models play an important role in describing the evolution of the early universe, where directional expansion and spatial curvature cannot be neglected. Classical Bianchi models, however, are formulated through local differential equations and therefore do not account for hereditary effects that may influence the interaction between geometry and matter over cosmic time scales. Motivated by the growing use of nonlocal fractional operators in mathematical physics, this work develops a memory-dependent Bianchi type-IV cosmological model by incorporating the Atangana-Baleanu-Caputo fractional derivative into the reduced Einstein evolution equations. The resulting formulation preserves the characteristic curvature contribution through the explicit A-2 geometric potential and leads to a closed nonlinear dynamical system governing the directional scale factors and directional Hubble parameters. The analytical study is carried out by transforming the fractional model into an equivalent Volterra integral equation, from which existence and uniqueness of solutions are established using Banach's fixed-point theorem. Additional theoretical results provide admissibility conditions for the energy density together with a sufficient criterion for the asymptotic decay of anisotropy under suitable expansion assumptions. To investigate the influence of nonlocal memory, a predictor-corrector numerical scheme is implemented for several fractional orders. The computational results demonstrate that decreasing the fractional order strengthens hereditary effects, slows the decay of the shear scalar and anisotropy parameter, delays isotropization, and produces a measurable memory lag in the expansion dynamics while recovering the classical Einstein evolution as the fractional order approaches unity. These findings show that the proposed framework offers a mathematically consistent and computationally reproducible extension of Bianchi type-IV cosmology, providing a useful approach for investigating memory-driven anisotropic evolution within the setting of fractional gravitational dynamics.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Memory-Driven Anisotropic Expansion in Bianchi Type-IV Space-Time via Nonlocal Fractional Operators
AU  - Anil Kumar Menaria
Y1  - 2026/08/14
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261404.17
DO  - 10.11648/j.ajam.20261404.17
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 244
EP  - 258
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261404.17
AB  - Anisotropic cosmological models play an important role in describing the evolution of the early universe, where directional expansion and spatial curvature cannot be neglected. Classical Bianchi models, however, are formulated through local differential equations and therefore do not account for hereditary effects that may influence the interaction between geometry and matter over cosmic time scales. Motivated by the growing use of nonlocal fractional operators in mathematical physics, this work develops a memory-dependent Bianchi type-IV cosmological model by incorporating the Atangana-Baleanu-Caputo fractional derivative into the reduced Einstein evolution equations. The resulting formulation preserves the characteristic curvature contribution through the explicit A-2 geometric potential and leads to a closed nonlinear dynamical system governing the directional scale factors and directional Hubble parameters. The analytical study is carried out by transforming the fractional model into an equivalent Volterra integral equation, from which existence and uniqueness of solutions are established using Banach's fixed-point theorem. Additional theoretical results provide admissibility conditions for the energy density together with a sufficient criterion for the asymptotic decay of anisotropy under suitable expansion assumptions. To investigate the influence of nonlocal memory, a predictor-corrector numerical scheme is implemented for several fractional orders. The computational results demonstrate that decreasing the fractional order strengthens hereditary effects, slows the decay of the shear scalar and anisotropy parameter, delays isotropization, and produces a measurable memory lag in the expansion dynamics while recovering the classical Einstein evolution as the fractional order approaches unity. These findings show that the proposed framework offers a mathematically consistent and computationally reproducible extension of Bianchi type-IV cosmology, providing a useful approach for investigating memory-driven anisotropic evolution within the setting of fractional gravitational dynamics.
VL  - 14
IS  - 4
ER  -

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