Variable Fractional-Order Reaction-Diffusion System for Edge Preservation in Biomedical Imaging

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DOI: https://doi.org/10.28924/2291-8639-24-2026-23
Published: Jan 29, 2026
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Nidal Anakira, Iqbal H. Jebril, Osama Ogilat, Iqbal M. Batiha, Tala Sasa, Abed Al-Rahman M. Malkawi, Variable Fractional-Order Reaction-Diffusion System for Edge Preservation in Biomedical Imaging, Int. J. Anal. Appl., 24 (2026), 23.

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Nidal Anakira, Iqbal H. Jebril, Osama Ogilat, Iqbal M. Batiha, Tala Sasa, Abed Al-Rahman M. Malkawi

Abstract

This paper introduces a novel variable fractional-order reaction-diffusion system (VFO-RDs) to model anisotropic diffusion for edge preservation in biomedical imaging. By leveraging the Caputo nabla variable fractional-order difference operator, the proposed model captures the memory-dependent nature of biological tissues. We establish sufficient conditions for tempered Mittag-Leffler stability (MLS) of the equilibrium point using Lyapunov functions (LFs) and Lipschitz-type bounds on the nonlinear reaction term. Eigenvalue-based constraints on the discrete Laplacian guarantee contractive dynamics. Numerical simulations in both 1D and 2D domains demonstrate the edge-preserving capabilities of the method under various fractional-order (FO) scenarios. The results confirm that the proposed framework effectively maintains critical spatial features and improves stability, providing a viable tool for advanced biomedical image analysis.

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References

  1. M.A. Al-Betar, M.S. Braik, Q.Y. Shambour, G. Al-Naymat, T. Porntaveetus, Ameliorated Elk Herd Optimizer for Global Optimization and Engineering Problems, Artif. Intell. Rev. 58 (2025), 360. https://doi.org/10.1007/s10462-025-11360-1.
  2. A.S. Hussain, M. Qousini, R.N. Abbas, E.A. Az-Zo'bi, M.A. Tashtoush, Flexible Distributional Modeling with the ACT-G Family Using Recurrent Neural Networks and Firefly Optimization, Int. J. Robot. Control Syst. 5 (2025), 3318–3349.
  3. A.S. Al-Bahtiti, B.A. Tayeh, A. Baghdadi, W.S. Alaloul, Y.I. Abu Aisheh, Drivers for Adopting Augmented Reality and Virtual Reality Technologies in the Construction Project Management in Gaza City, J. Asian Arch. Build. Eng. (2025). https://doi.org/10.1080/13467581.2025.2507235.
  4. A.F. Jameel, N.R. Anakira, M.M. Rashidi, A.K. Alomari, A. Saaban, M.A. Shakhatreh, Differential Transformation Method for Solving High Order Fuzzy Initial Value Problems, Ital. J. Pure Appl. Math. 39 (2018), 194–208.
  5. A.F. Jameel, N. Anakira, A.K. Alomari, I. Hashim, S. Momani, A New Approximation Method for Solving Fuzzy Heat Equations, J. Comput. Theor. Nanosci. 13 (2016), 7825–7832. https://doi.org/10.1166/jctn.2016.5784.
  6. N.R. Anakira, A.K. Alomari, I. Hashim, Numerical Scheme for Solving Singular Two-Point Boundary Value Problems, J. Appl. Math. 2013 (2013), 468909. https://doi.org/10.1155/2013/468909.
  7. A. Jameel, N.R. Anakira, A.K. Alomari, I. Hashim, M.A. Shakhatreh, Numerical Solution of Nth Order Fuzzy Initial Value Problems by Six Stages, J. Nonlinear Sci. Appl. 09 (2016), 627–640. https://doi.org/10.22436/jnsa.009.02.26.
  8. N. Anakira, I. H. Jebri, I. M. Batiha, M. S. Hijazi, T. Sasa, Criteria for Finite-Time Convergence in Discrete Variable-Order Fractional Fitzhugh–Nagumo Reaction–Diffusion Systems, Eur. J. Pure Appl. Math. 18 (2025), 7005. https://doi.org/10.29020/nybg.ejpam.v18i4.7005.
  9. O. Ogilat, I.H. Jebril, I.M. Batiha, N. Anakira, T. Sasa, Distributed Control for Mittag–Leffler Synchronization of Variable‐Order Fractional Gierer–Meinhardt Reaction‐Diffusion Systems, J. Math. 2025 (2025), 6554797. https://doi.org/10.1155/jom/6554797.
  10. I.H. Jebril, I.M. Batiha, S. Momani, A. Biswas, Control Strategies for Mittag-Leffler Synchronization in Variable-Order Fractional Fitzhugh-Nagumo Reaction-Diffusion Networks, Contemp. Math. 6 (2025), 6414–6443. https://doi.org/10.37256/cm.6520257613.
  11. S. Momani, I.M. Batiha, N. Djenina, A. Ouannas, Analyzing the Stability of Caputo Fractional Difference Equations with Variable Orders, Prog. Fract. Differ. Appl. 11 (2025), 139–151. https://doi.org/10.18576/pfda/110110.
  12. I.H. Jebril, I. Bendib, A. Ouannas, I.M. Batiha, J. Oudetallah, et al., Finite-Time Synchronization of the Discrete Reaction-Diffusion Fitzhugh–Nagumo Model, AIP Conf. Proc. 3338 (2025), 040025. https://doi.org/10.1063/5.0295065.
  13. O. Aljabari, S. Momani, I.M. Batiha, M. Odeh, Comparative Study of Fractional Heun’s Method and Fractional q-Heun's Method for Solving Fractional Differential Equations, Prog. Fract. Differ. Appl. 12 (2026), 37–56. https://doi.org/10.18576/pfda/120103.
  14. I.M. Batiha, H.O. Al-Khawaldeh, I.H. Jebril, N. Anakira, A. Bataihah, et al., A Novel Numerical Approach for Handling Fractional Quantum Initial-Value Problems, Int. J. Fuzzy Log. Intell. Syst. 25 (2025), 416–430. https://doi.org/10.5391/ijfis.2025.25.4.416.
  15. F. Mainardi, Fractional Calculus: Theory and Applications, Mathematics 6 (2018), 145. https://doi.org/10.3390/math6090145.
  16. S. Kempfle, I. Schäfer, H. Beyer, Fractional Calculus via Functional Calculus: Theory and Applications, Nonlinear Dyn. 29 (2002), 99–127. https://doi.org/10.1023/a:1016595107471.
  17. K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.
  18. D. del-Castillo-Negrete, Fractional Calculus: Basic Theory and Applications, Lecture Notes, Institute of Mathematics, UNAM, Mexico City, 2005.
  19. S.S. Ray, A. Atangana, S.C.O. Noutchie, M. Kurulay, N. Bildik, et al., Fractional Calculus and Its Applications in Applied Mathematics and Other Sciences, Math. Probl. Eng. 2014 (2014), 849395. https://doi.org/10.1155/2014/849395.
  20. J.A. Tenreiro Machado, M.F. Silva, R.S. Barbosa, I.S. Jesus, C.M. Reis, et al., Some Applications of Fractional Calculus in Engineering, Math. Probl. Eng. 2010 (2009), 639801. https://doi.org/10.1155/2010/639801.
  21. D. Kumar, D. Baleanu, Editorial: Fractional Calculus and Its Applications in Physics, Front. Phys. 7 (2019), 81. https://doi.org/10.3389/fphy.2019.00081.
  22. V. Tarasov, On History of Mathematical Economics: Application of Fractional Calculus, Mathematics 7 (2019), 509. https://doi.org/10.3390/math7060509.
  23. I. Petras, Special Issue: Fractional Calculus and Its Applications, Risk Decis. Anal. 7 (2018), 1–3. https://doi.org/10.3233/RDA-180138.
  24. B. Ross, The Development of Fractional Calculus 1695–1900, Hist. Math. 4 (1977), 75–89. https://doi.org/10.1016/0315-0860(77)90039-8.
  25. M.H. Annaby, Z.S. Mansour, q-Fractional Calculus and Equations, Springer, Berlin, 2012. https://doi.org/10.1007/978-3-642-30898-7.
  26. T.M. Atanacković, S. Pilipović, B. Stanković, D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes, Wiley, 2014. https://doi.org/10.1002/9781118577530.
  27. C. Lubich, Discretized Fractional Calculus, SIAM J. Math. Anal. 17 (1986), 704–719. https://doi.org/10.1137/0517050.
  28. R. Gorenflo, Fractional Calculus, in: Fractals and Fractional Calculus in Continuum Mechanics, Springer, Vienna, 1997: pp. 277–290. https://doi.org/10.1007/978-3-7091-2664-6_6.
  29. M. Chen, W. Deng, Discretized Fractional Substantial Calculus, ESAIM: Math. Model. Numer. Anal. 49 (2015), 373–394. https://doi.org/10.1051/m2an/2014037.
  30. S. Pooseh, R. Almeida, D.F. Torres, Discrete Direct Methods in the Fractional Calculus of Variations, Comput. Math. Appl. 66 (2013), 668–676. https://doi.org/10.1016/j.camwa.2013.01.045.
  31. T. Abdeljawad, On Riemann and Caputo Fractional Differences, Comput. Math. Appl. 62 (2011), 1602–1611. https://doi.org/10.1016/j.camwa.2011.03.036.
  32. X. Li, K. Wu, X. Liu, Mittag–Leffler Stabilization for Short Memory Fractional Reaction-Diffusion Systems via Intermittent Boundary Control, Appl. Math. Comput. 449 (2023), 127959. https://doi.org/10.1016/j.amc.2023.127959.
  33. I. Bendib, A. Ouannas, M. Dalah, Mittag–Leffler Synchronization of Fractional‐order Reaction–Diffusion Systems, Asian J. Control. (2025). https://doi.org/10.1002/asjc.3702.
  34. Y. Cao, Y. Kao, J.H. Park, H. Bao, Global Mittag–Leffler Stability of the Delayed Fractional-Coupled Reaction-Diffusion System on Networks Without Strong Connectedness, IEEE Trans. Neural Netw. Learn. Syst. 33 (2022), 6473–6483. https://doi.org/10.1109/tnnls.2021.3080830.
  35. G. Stamov, I. Stamova, C. Spirova, Reaction-Diffusion Impulsive Fractional–Order Bidirectional Neural Networks with Distributed Delays: Mittag–Leffler Stability Along Manifolds, AIP Conf. Proc. 2172 (2019), 050002. https://doi.org/10.1063/1.5133521.
  36. R.A. Askey, R. Roy, Gamma Function, in: NIST Handbook of Mathematical Functions, Cambridge University Press, pp. 135–148, 2010.
  37. C. Goodrich, A.C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-25562-0.
  38. D. Luo, Z. Luo, Uniqueness and Novel Finite-Time Stability of Solutions for a Class of Nonlinear Fractional Delay Difference Systems, Discret. Dyn. Nat. Soc. 2018 (2018), 8476285. https://doi.org/10.1155/2018/8476285.
  39. Y. Wei, Y. Chen, Y. Wei, X. Zhao, Lyapunov Stability Analysis for Nonlinear Nabla Tempered Fractional Order Systems, Asian J. Control. 25 (2022), 3057–3066. https://doi.org/10.1002/asjc.3003.