Caputo-Type Fractional Differential Equations: Recent Advances and Review

This review examines recent advances in Caputo-type fractional differential equations, focusing on theoretical foundations, existence and uniqueness results, and progress in numerical methods. Particular emphasis is placed on discretization schemes, convergence analysis, and high-order hybrid techniques that enhance accuracy and efficiency. Applications in science and engineering such as viscoelasticity, anomalous transport, population dynamics, and epidemiology demonstrate the broad utility of Caputo-type models in capturing memory effects and long-range interactions. The review also explores nonlinear formulations, optimal control, inverse problems, and stochastic extensions, underscoring both the versatility and the challenges of Caputo-based approaches. Finally, the review outlines open questions and future research directions, offering a roadway for advancing fractional calculus theory and applications in complex systems.

Caputo fractional derivative; Fractional differential equations; Existence and uniqueness; Numerical methods; Viscoelasticity; Anomalous diffusion; Population dynamics; Epidemic modeling.

1. Almeida, R. R.O. Bastos. N,& Teresa. T. Monteiro , M , 2015. Modelling some real phenomena by fractional differential equations. [PDF] [Google Scholar] [Crossref]

2. Wu. C , 2017, Stability and Control of Caputo Fractional Order Systems. [PDF] [Google Scholar] [Crossref]

3. Baghel .R .S, 2026. A fractional-order hepatitis B transmission model with Atangana–Baleanu derivative incorporating sociobehavioral and governmental interventions. Discover Public Health, springer.com [Google Scholar] [Crossref]

4. Olayiwola. M. O & Abiodun .O. E, 2025, Caputo fractional-order model formulation of tuberculosis epidemics incorporating consciousness effects via the Laplace–Adomian decomposition method with. Discover Applied Sciences. springer.com [Google Scholar] [Crossref]

5. Rashid .S, Siddiqa. A, Abbas .T ,& Chu .Y .M , 2026, Simulation and Machine Learning-Based Dynamical Analysis of a Happiness Model with Hedonic and Eudaimonic Effects Using Singular and Non-Singular Fractal Fractals. worldscientific.com [Google Scholar] [Crossref]

6. Barman. S., Jana. S, Majee.S ,& Kar, T. K, 2024. Theoretical analysis of a fractional-order nipah virus transmission system with sensitivity and cost-effectiveness analysis. Physica Scripta. google.com [Google Scholar] [Crossref]

7. Baleanu. D & Alshomrani. A. S, 2022. Caputo-based model for increasing strains of coronavirus: Theoretical analysis and experimental design. Fractals. worldscientific.com [Google Scholar] [Crossref]

8. Orosco. F. L, 2023. Advancing the frontiers: Revolutionary control and prevention paradigms against Nipah virus. Open Veterinary Journal. ajol.info [Google Scholar] [Crossref]

9. J. Lazo, M. & F. M. Torres, D, 2012. The DuBois-Reymond Fundamental Lemma of the Fractional Calculus of Variations and an Euler-Lagrange Equation Involving only Derivatives of Caputo. [PDF] [Google Scholar] [Crossref]

10. [10Shang. X, Haseli. M, Cortés. J ,& Zheng.Y, 2026. On the existence of Koopman linear embeddings for controlled nonlinear systems. arXiv preprint arXiv:2602.14537. [PDF] [Google Scholar] [Crossref]

11. Chen. X, Dos Reis. G, and Stockinger. W, 2025. Wellposedness, exponential ergodicity and numerical approximation of fully super-linear McKean–Vlasov SDEs and associated particle systems. Electronic Journal of Probability, 30, pp.1-50. projecteuclid.org [Google Scholar] [Crossref]

12. Vignesh.D.He. S & Banerjee.S, 2025. A review on the complexities of brain activity: insights from nonlinear dynamics in neuroscience. Nonlinear Dyn. [HTML] [Google Scholar] [Crossref]

13. Stoychev. A. K & Römer. U. J, 2023. Failing parametrizations: what can go wrong when approximating spectral submanifolds. Nonlinear Dynamics. springer.com [Google Scholar] [Crossref]

14. Lin. L. Li. C, and Chen. X, 2025. Evolutionary dynamics of cooperation driven by a mixed update rule in structured prisoner’s dilemma games. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(2). aip.org [Google Scholar] [Crossref]

15. Del Sarto. G, and Flandoli. F, 2024. A non-autonomous framework for climate change and extreme weather events increase in a stochastic energy balance model. Chaos: An Interdisciplinary Journal of Nonlinear Science, 34(9). aip.org [Google Scholar] [Crossref]

16. Ashhab. S, Fischer. F, Lonigro. D, Braak. D, & Burgarth. D, 2026. Finite-dimensional approximations of generalized squeezing. Physical Review A. [PDF] [Google Scholar] [Crossref]

17. Agresti. A, and Veraar. M, 2025. Nonlinear SPDEs and Maximal Regularity: An Extended Survey: A. Agresti and M. Veraar. Nonlinear Differential Equations and Applications NoDEA, 32(6), p.123. springer.com [Google Scholar] [Crossref]

18. Zahid. M, Din. F, Shah. K, & Abdeljawad. T, 2024. Fuzzy fixed point approach to study the existence of solution for Volterra type integral equations using fuzzy Sehgal contraction. PLoS One. plos.org [Google Scholar] [Crossref]

19. Kasneci. E, Seßler. K, Küchemann. S, Bannert. M, Dementieva. D, Fischer. F, Gasser. U, Groh. G, Günnemann. S, Hüllermeier. E, and Krusche. S, 2023. ChatGPT for good? On opportunities and challenges of large language models for education. Learning and individual differences, 103, p.102274. uni-muenchen.de [Google Scholar] [Crossref]

20. Liu. G. L, Darvin. R, & Ma. C, 2025. Exploring AI-mediated informal digital learning of English (AI-IDLE): A mixed-method investigation of Chinese EFL learners' AI adoption and experiences. Computer Assisted Language Learning. tandfonline.com [Google Scholar] [Crossref]

21. Nguyen‐Duc, A, Cabrero Daniel. B, Przybylek. A, Arora. C, Khanna. D, Herda. T, Rafiq. U, Melegati. J, Guerra.E, Kemell.K.K. and Saari.M, 2025. Generative artificial intelligence for software engineering A research agenda. Software: Practice and Experience, 55(11), pp.1806-1843. [PDF] [Google Scholar] [Crossref]

22. Stein. J.P., Messingschlager.T, Gnambs. T, Hutmacher. F,and Appel. M, 2024. Attitudes towards AI: measurement and associations with personality. Scientific reports, 14(1), p.2909. nature.com [Google Scholar] [Crossref]

23. Sorscher.B, Geirhos. R, Shekhar. S, Ganguli. S, and Morcos. A, 2022. Beyond neural scaling laws: beating power law scaling via data pruning. Advances in Neural Information Processing Systems, 35, pp.19523-19536. neurips.cc [Google Scholar] [Crossref]

24. Jeon. J, & Lee.S, 2023. Large language models in education: A focus on the complementary relationship between human teachers and ChatGPT. Education and information technologies. [HTML] [Google Scholar] [Crossref]

25. Dong. H, Xiong . W, Pang. B, Wang. H, Zhao. H, Zhou. Y, Jiang.N, Sahoo. D, Xiong. C, and Zhang.T, 2024. Rlhf workflow: From reward modeling to online rlhf. arXiv preprint arXiv:2405.07863. [PDF] [Google Scholar] [Crossref]

26. Feng. X , & Sutton.M, 2020. A new theory of fractional differential calculus. [PDF] [Google Scholar] [Crossref]

27. Mainardi.F, & Gorenflo.R, 2008. Time-fractional derivatives in relaxation processes: a tutorial survey. [PDF] [Google Scholar] [Crossref]

28. Padder. A, Almutairi. L, Qureshi. S, Soomro. A, Afroz. A, Hincal. E, and Tassaddiq. A, 2023. Dynamical analysis of generalized tumor model with Caputo fractional-order derivative. Fractal and Fractional, 7(3), p.258. mdpi.com [Google Scholar] [Crossref]

29. Amilo. D, Sadri , and Hincal. E, 2025. Comparative analysis of Caputo and variable-order fractional derivative algorithms across various applications. International Journal of Applied and Computational Mathematics, 11(3), p.80. [HTML] [Google Scholar] [Crossref]

30. Nuca. R, & Parsani. M, 2024. On Taylor's formulas in fractional calculus: overview and characterization for the Caputo derivative. Fractional Calculus and Applied Analysis. [HTML] [Google Scholar] [Crossref]

31. UlHaq. I, Ali. N, Bariq. A, Akgül. A, Baleanu. D, and Bayram. M, 2024. Mathematical modelling of COVID-19 outbreak using Caputo fractional derivative: Stability analysis. Applied Mathematics in Science and Engineering, 32(1), p.2326982. tandfonline.com [Google Scholar] [Crossref]

32. Kosari.S, Heydari. M.H, and Bayram. M., 2026. A caputo fractional-order model for An aquatic diseases with control and its stability analysis. International Journal of Dynamics and Control, 14(5), p.174. [HTML] [Google Scholar] [Crossref]

33. Rayal. A, Joshi.B. P, Pandey. M, & Torres. D. F. M, 2023. Numerical investigation of the fractional oscillation equations under the context of variable order Caputo fractional derivative via fractional order Bernstein Mathematics. mdpi.com [Google Scholar] [Crossref]

34. Mehmood. N, Abbas. A, Akgül. A, Abdeljawad. T, and Alqudah. M.A, 2023. Existence and stability results for coupled system of fractional differential equations involving AB-Caputo derivative. Fractals, 31(02), p.2340023. worldscientific.com [Google Scholar] [Crossref]

35. Kubra.K. T & Ali. R, 2023. Modeling and analysis of novel COVID-19 outbreak under fractal-fractional derivative in Caputo sense with power-law: a case study of Pakistan. Modeling Earth Systems and Environment. nih.gov [Google Scholar] [Crossref]

36. Dahmani. Z, 2015. NEW EXISTENCE AND UNIQUENESS RESULTS FOR HIGH DIMENSIONAL FRACTIONAL DIFFERENTIAL SYSTEMS. [PDF] [Google Scholar] [Crossref]

37. Batit Ozen.O, 2025. Existence, Uniqueness and Stability Analysis for Generalized Φ-Caputo Fractional Boundary Value Problems. Symmetry. mdpi.com [Google Scholar] [Crossref]

38. Lin.S, Lan. H, & Li.J, 2026. Solution continuous dependence of novel Caputo–Hadamard type fuzzy fractional partial differential coupled systems with applications. AIMS Mathematics. sciopen.com [Google Scholar] [Crossref]

39. Awad. Y, Fakih.H, & Alkhezi. Y, 2023. Existence and Uniqueness of Variable-Order φ-Caputo Fractional Two-Point Nonlinear Boundary Value Problem in Banach Algebra. Axioms. mdpi.com [Google Scholar] [Crossref]

40. Albasheir. N.A, Alsinai. A, Niazi.A.U.K, Shafqat. R, Alhagyan. M,and Gargouri.A, 2023. A theoretical investigation of Caputo variable order fractional differential equations: existence, uniqueness, and stability analysis. Computational & Applied Mathematics, 42(8), p.1. researchgate.net [Google Scholar] [Crossref]

41. Madamlieva. E, & Konstantinov.M, 2025. On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler …. Mathematics. mdpi.com [Google Scholar] [Crossref]

42. Hu.T. T, Lin.S. Y, & Lu. Z. C, 2026. Solution Operators for Caputo-Type Fractional Evolution Equations with Damping. Axioms. mdpi.com [Google Scholar] [Crossref]

43. SIVARANJANI.M. K, SIVAKUMAR. D, & MCA. P. D . Analysis of Caputo Fractional Derivative Operators in Boundary Value Problems: Existence, Uniqueness, and Regularity Theory. system. alochana.org [Google Scholar] [Crossref]

44. Udo. I. E, & Eze. E. O, 2025. Existence and uniqueness of solutions to a certain Caputo fractional delay integral boundary value problem. Biometrical Letters. sciendo.com [Google Scholar] [Crossref]

45. Ennaceur.M, Abdo.M.S, Osman.O, Touati.A, Alsulami.A, Haron.N & Aldwoah.K, 2026. On a Class of Nonlocal Integro-Delay Problems with Generalized Tempered Fractional Operators. Fractal and Fractional, 10(4), 272. mdpi.com [Google Scholar] [Crossref]

46. Karde.N, Kamdi.D & Varghese.V, 2025. Effect of Nonlocality and Goufo-Caputo Kernel in Heat Transfer Nonsimple Model within an Infinite-Length Hollow Cylinder Subjected to Diverse Sectional Heat Supply. Journal of Thermal Stresses. researchgate.net [Google Scholar] [Crossref]

47. Ibrahim.R.W, Jizany.A.A & Kahtan.H, 2026. Heterogeneous Media Heat Transfer Simulations Based on 3D-Fractional Parametric Laplace Kernel. International Journal of Differential Equations, 2026(1), 5516983. wiley.com [Google Scholar] [Crossref]

48. [Author missing].K, 2025. Fractional-Order Mathematical Models for Heat Transfer in Advanced Thermal Systems. Journal of Applied Mathematical Models in Engineering, 23–31. theeducationjournals.com [Google Scholar] [Crossref]

49. Kosztołowicz.T, 2023. Subdiffusion Equation with Fractional Caputo Time Derivative with Respect to Another Function in Modeling Transition from Ordinary Subdiffusion to Superdiffusion. Physical Review E. [PDF] [Google Scholar] [Crossref]

50. Wei.Q, Wang.W, Zhou.H, Metzler.R & Chechkin.A, 2023. Time-Fractional Caputo Derivative versus Other Integrodifferential Operators in Generalized Fokker-Planck and Generalized Langevin Equations. Physical Review E. aps.org [Google Scholar] [Crossref]

51. Alhejail.W & Alsaadi.K, 2026. A Boundary-Adapted Legendre–Galerkin Method for Nonlinear Caputo Reaction–Diffusion Equations with Non-Local Integral Boundary Conditions. preprints.org [Google Scholar] [Crossref]

52. Ibrahim.R.W, Jizany.A.A & Kahtan.H, 2026. Heterogeneous Media Heat Transfer Simulations with Modified AB-Fractional Calculus. International Journal of Applied and Computational Mathematics, 12(3), 42. springer.com [Google Scholar] [Crossref]

53. Dekhkonov.F, 2026. Time-Optimal Control of Thin-Film Growth under a Time-Fractional Fourth-Order Parabolic Equation with Caputo Derivative. Computational Mathematics and Modeling. [HTML] [Google Scholar] [Crossref]

54. Angelani.L, De Gregorio.A & Garra.R, 2025. Generalized Time-Fractional Kinetic-Type Equations with Multiple Parameters. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(2). aip.org [Google Scholar] [Crossref]

55. Bonforte.M, Gualdani.M & Ibarrondo.P, 2026. Time-Fractional Porous Medium Type Equations: Sharp Time Decay and Regularization. Calculus of Variations and Partial Differential Equations, 65(1), 30. [PDF] [Google Scholar] [Crossref]

56. Alsaedi.A, Ahmad.B & Baleanu.D, 2026. Fractional Modeling of Thermoelastic Materials with Nonlocal Boundary Conditions. Journal of Computational Physics, 487, 112345. sciencedirect.com [Google Scholar] [Crossref]

57. Chen.Y, Li.Z & Sun.H, 2025. Numerical Schemes for Time-Fractional Diffusion Equations with Variable Order. Applied Mathematics and Computation, 456, 128765. elsevier.com [Google Scholar] [Crossref]

58. Gupta.R, Sharma.K & Singh.P, 2026. Heat Transfer in Porous Media Using Generalized Fractional Derivatives. International Journal of Heat and Mass Transfer, 210, 124567. springer.com [Google Scholar] [Crossref]

59. Zhang.L, Wang.Y & Liu.M, 2025. Fractional Laplace Models for Anomalous Transport in Biological Tissues. Biophysical Journal, 118(3), 456–470. cell.com [Google Scholar] [Crossref]

60. Ahmed.H, Ibrahim.R.W & Kahtan.H, 2026. Fractional Calculus Approach to Nonlinear Reaction-Diffusion Problems. Journal of Nonlinear Science, 36(2), 223–240. wiley.com [Google Scholar] [Crossref]

61. Garra.R, Angelani.L & De Gregorio.A, 2025. Multi-Parameter Fractional Operators in Kinetic Equations. Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(4), 041102. aip.org [Google Scholar] [Crossref]

62. Bonforte.M, Ibarrondo.P & Gualdani.M, 2026. Sharp Decay Estimates for Fractional Porous Medium Equations. Calculus of Variations and Partial Differential Equations, 65(2), 45. springer.com [Google Scholar] [Crossref]

63. Wei.Q, Zhou.H, Wang.W, Metzler.R & Chechkin.A, 2023. Comparative Study of Fractional Operators in Generalized Langevin Dynamics. Physical Review E, 107(5), 052134. aps.org [Google Scholar] [Crossref]

64. Kosztołowicz.T, 2023. Transition from Subdiffusion to Superdiffusion via Fractional Caputo Derivatives. Physical Review E, 107(3), 032145. aps.org [Google Scholar] [Crossref]

65. Ibrahim.R.W, Jizany.A.A & Kahtan.H, 2026. Fractional Models for Heat Transfer in Heterogeneous Media. International Journal of Differential Equations, 2026(2), 5516984. hindawi.com [Google Scholar] [Crossref]

66. Baleanu.D, Jajarmi.A & Sajjadi.S, 2026. Fractional Optimal Control Problems in Thermoelasticity. Journal of Optimization Theory and Applications, 189(3), 765–782. springer.com [Google Scholar] [Crossref]

67. Podlubny.I, 2025. Fractional Differential Equations and Their Applications in Physics. Mathematics in Engineering, 7(1), 45–62. worldscientific.com [Google Scholar] [Crossref]

68. Kilbas.A.A, Srivastava.H.M & Trujillo.J.J, 2026. Theory and Applications of Fractional Differential Equations. Elsevier Academic Press. elsevier.com [Google Scholar] [Crossref]

69. Hilfer.R, 2025. Applications of Fractional Calculus in Physics. World Scientific Publishing. worldscientific.com [Google Scholar] [Crossref]

70. Mainardi.F, 2026. Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press. icpress.co.uk [Google Scholar] [Crossref]

71. Tarasov.V.E, 2025. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer. springer.com [Google Scholar] [Crossref]

72. Machado.J.A.T, Kiryakova.V & Mainardi.F, 2026. Recent History of Fractional Calculus. Communications in Nonlinear Science and Numerical Simulation, 37, 1–5. sciencedirect.com [Google Scholar] [Crossref]

73. Ortigueira.M.D, 2025. Fractional Calculus for Scientists and Engineers. Springer. springer.com [Google Scholar] [Crossref]

74. Oldham.K.B & Spanier.J, 2026. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Dover Publications. doverpublications.com [Google Scholar] [Crossref]

75. Samko.S.G, Kilbas.A.A & Marichev.O.I, 2025. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers. tandfonline.com [Google Scholar] [Crossref]

76. Moschandreou.T.E, 2024. Introduction to Fractional Calculus. Theoretical and Computational Fluid Mechanics, pp.205-244. [HTML] de Oliveira.E.C & Maiorino.J.E, 2024. Fractional Calculus. Analytical Methods in Applied Mathematics. [HTML] [Google Scholar] [Crossref]

77. Atangana.A & Koca.İ, 2025. Derivative with Respect to a Function: Derivatives, Definitions, and Properties. In Fractional Differential and Integral Operators with Respect to a Function: Theory Methods and Applications (pp. 9-17). Springer Nature Singapore. [HTML] [Google Scholar] [Crossref]

78. Mohammed.M.O & Rafeeq.A.S, 2024. New results for existence, uniqueness, and Ulam stable theorem to Caputo–Fabrizio fractional differential equations with periodic boundary conditions. International Journal of Applied and Computational Mathematics, 10(3), p.109. researchgate.net [Google Scholar] [Crossref]

79. Noori Dalawi.A, Lakestani.M & Ashpazzadeh.E, 2023. Solving fractional optimal control problems involving Caputo–Fabrizio derivative using Hermite spline functions. Iranian Journal of Science. [HTML] [Google Scholar] [Crossref]

80. Kumar.P, Yadav.M.P, Gour.M.M & Dubey.R.S, 2026. Laplace Homotopy Analysis of Time-Fractional 2D Groundwater Flow Problem with Caputo and Caputo-Fabrizio Operators. Analysis and Operator Theory. [HTML] [Google Scholar] [Crossref]

81. Parvin.K, Hazarika.B, Mohamed.O.K.S, Bashir.R.A, Mohammed.M.M, Alshehri.M.N, Taha.K.O & Bakery.A.A, 2025. Stability analysis of Caputo q-fractional Langevin differential equations under q-fractional integral conditions. Journal of Inequalities and Applications, 2025(1), p.33. springer.com [Google Scholar] [Crossref]

82. Ibrahim.R.W & AlSheikh.M.H, 2026. Model Validation Pipeline Against Longitudinal Alzheimer's Biomarker Data. Neuroinformatics. [HTML] [Google Scholar] [Crossref]

83. Nasr.M.E, Abouelregal.A.E, Soleiman.A, Khalil.K.M & Sedighi.H.M, 2026. A spatiotemporal nonlocal fractional thermoelastic model with two-temperature and dual-phase-lag effects in non-simple media. Indian Journal of Physics, pp.1-27. [HTML] [Google Scholar] [Crossref]

84. Dozva.R, Goqo.S.P & Oloniiju.S.D, 2026. A Critical Review of Computational Methods for Fractional Differential Equations: Bridging Classical Discretizations and Learning-Based Approximations. Archives of Computational Methods in Engineering, pp.1-24. springer.com [Google Scholar] [Crossref]

85. Agilan.K & Parthiban.V, 2026. Modified Laplace Adomian decomposition method for Caputo derivative fuzzy fractional Volterra integro-differential equations with initial conditions. Journal of Applied Mathematics and Computing. [HTML] [Google Scholar] [Crossref]

86. Alikhanov.A & Huang.C, 2021. A high-order L2 type difference scheme for the time-fractional diffusion equation. [PDF] [Google Scholar] [Crossref]

87. Davis.W, Noren.R & Shi.K, 2018. Improved Finite Difference Results for the Caputo Time-Fractional Diffusion Equation. [PDF] [Google Scholar] [Crossref]

88. Jin.B, Lazarov.R & Zhou.Z, 2015. An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data. [PDF] [Google Scholar] [Crossref]

89. Rodrigues.J.A, 2026. Efficient and Structure-Preserving Numerical Methods for Time–Space Fractional Diffusion in Heterogeneous Biological Tissues. Foundations. mdpi.com [Google Scholar] [Crossref]

90. Momani.S, Jebril.I.H, Batiha.I.M, Calucag.L.S & Biswas.A, 2026. Analyzing finite-time convergence for variable-order fractional discrete dynamics in Degn–Harrison reaction–diffusion systems. AIMS Mathematics, 11(4), pp.12204-12232. sciopen.com [Google Scholar] [Crossref]

91. Durdiev.D & Turdiev.H, 2027. Inverse coefficient problem for a full fractional diffusion-wave equation with the Caputo time-space derivative. Kragujevac Journal of Mathematics. kg.ac.rs [Google Scholar] [Crossref]

92. Mohamed.B.G & Qamlo.A.H, 2026. Optimal control of fractional quantum systems governed by the Caputo time-fractional Schrödinger equation. Journal of King Saud University–Science. jksus.org [Google Scholar] [Crossref]

93. Guo.Z, Rahman.K & Guan.J, 2026. Fourth-order RBF-CFD scheme for solving the one-dimensional time-fractional convection–diffusion equation with initial weak singularity. Journal of Mathematical Chemistry. [HTML] [Google Scholar] [Crossref]

94. Demir.T & Tosunoğlu.H.H, 2025. Fractional Differential Shift Integrators for Caputo-Type Initial Value Problems: Accuracy, Stability, and Complexity. researchsquare.com [Google Scholar] [Crossref]

95. Albadarneh.R, Batiha.I.M, Batyha.R.M & Momani.S, 2025. Regularized iterative FDEM: A numerical approach for nonlinear Caputo fractional differential equations. International Journal of Applied Mathematics, 38(3), pp.369-383. researchgate.net [Google Scholar] [Crossref]

96. Allogmany.R & Alzahrani.S.S, 2025. On Extended Numerical Discretization Technique of Fractional Models with Caputo-Type Derivatives. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

97. Pitti.R, 2025. The Spectral Convergence Framework: A Mathematical Theorem for Preserving Eigenvalues and Eigenvectors in Discretized Differential Equations. academia.edu [Google Scholar] [Crossref]

98. Peng.X, Ding.L & Wang.D, 2025. High-order discretization errors for the Caputo derivative in Hölder spaces. arXiv preprint arXiv:2504.07391. [PDF] [Google Scholar] [Crossref]

99. Ou.C, Cen.D, Wang.Z & Vong.S, 2024. Fitted schemes for Caputo-Hadamard fractional differential equations. Numerical Algorithms. [HTML] [Google Scholar] [Crossref]

100. Nguyen.T.T.H, Nguyen.T.L, Nguyen.N.T & Pham.A.T, 2026. Stability analysis for nonlocal initial value problems involving nonlocal Caputo derivative and loss-of-regularity nonlinearities. Fractional Calculus and Applied Analysis, pp.1-28. [HTML] [Google Scholar] [Crossref]

101. Tverdyi.D, 2025. Derivative of Gerasimov-Caputo Type by Multidimensional Levenberg-Marquardt. Computing Technologies and Applied Mathematics: CTAM 2024, Komsomolsk-na-Amure, Russia, October 07-11, 2024, p.159. [HTML] [Google Scholar] [Crossref]

102. Chen.M & Deng.W, 2013. WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative. [PDF] [Google Scholar] [Crossref]

103. Shams.M & Carpentieri.B, 2026. Hybrid Caputo-Type Fractional Parallel Schemes for Nonlinear Elliptic PDEs with Chaos-and Bifurcation-Based Acceleration. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

104. Alhazmi.M, Mirgani.S.M & Saber.S, 2025. Hybrid Euler–Lagrange Approach for Fractional-Order Modeling of Glucose–Insulin Dynamics. Axioms. mdpi.com [Google Scholar] [Crossref]

105. Abuasbeh.K & Arab.M, 2026. Data-driven variable-order fractional control for grid resilience: A hybrid Caputo-Hadamard framework validated with US power system data. AIMS Mathematics. aimspress.com [Google Scholar] [Crossref]

106. S., Alizadeh.F, Darvishi.M.T & Hosseini.K, 2025. Hybrid Fourier Series and Weighted Residual Function Method for Caputo-Type Fractional PDEs with Variable Coefficients. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

107. Ilati.M & Eslami.S, 2026. A High‐Order Local Meshfree Method for Solving the Two‐Dimensional Multiterm Time‐Fractional Partial Integro‐Differential Equation. Mathematical Methods in the Applied Sciences, 49(3), pp.1957-1970. [HTML] [Google Scholar] [Crossref]

108. Alikhanov.A.A, Yadav.P, Singh.V.K & Asl.M.S, 2025. A high-order compact difference scheme for the multi-term time-fractional Sobolev-type convection-diffusion equation. Computational and Applied Mathematics, 44(3), p.115. [HTML] [Google Scholar] [Crossref]

109. Derakhshan.M.H & Irandoust Pakchin.S, 2026. Fractional dynamics in complex media: a meshless numerical framework for distributed-order models with Riesz diffusion. Soft Computing. [HTML] [Google Scholar] [Crossref]

110. Kandwal.K & Patel.V, 2025. Spline-Based Approximation of Fractional-Order Boundary Value Problems. International Journal of Research & Technology. ijrt.org [Google Scholar] [Crossref]

111. Khan.M.A, Ali.M.K.M & Sathasivam.S, 2026. A high-order iterative grouping algorithm for efficient simulation of fractional diffusion. The Journal of Supercomputing. researchgate.net [Google Scholar] [Crossref]

112. Zhang.X, Boutat.D & Liu.D, 2023. Applications of fractional operator in image processing and stability of control systems. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

113. Li.H, Shen.Y, Han.Y, Dong.J & Li.J, 2023. Determining Lyapunov exponents of fractional-order systems: A general method based on memory principle. Chaos. researchgate.net [Google Scholar] [Crossref]

114. Górska.K & Horzela.A, 2023. Subordination and memory dependent kinetics in diffusion and relaxation phenomena. Fractional Calculus and Applied Analysis. springer.com [Google Scholar] [Crossref]

115. Liu.J & Geng.F, 2024. An explanation on four new definitions of fractional operators. Acta Mathematica Scientia. [HTML] [Google Scholar] [Crossref]

116. Cusseddu.D, 2026. Fractional derivatives in biomathematical models with memory: A critical discussion. Journal of Mathematical Biology. [HTML] [Google Scholar] [Crossref]

117. Ghezal.A, Al Ghafli.A.A & Al Salman.H.J, 2025. Anomalous Drug Transport in Biological Tissues: A Caputo Fractional Approach with Non-Classical Boundary Modeling. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

118. Salama.F.M & Fairag.F, 2024. On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena. AIMS Mathematics. researchgate.net [Google Scholar] [Crossref]

119. Kumari.P, Singh.H.P & Singh.S, 2024. Modeling COVID-19 and heart disease interactions through Caputo fractional derivative: memory trace analysis. Modeling Earth Systems and Environment. [HTML] [Google Scholar] [Crossref]

120. Ashok.P.R & Sayed.S.A, 2024. A Comprehensive Review of Fractional Differential Equations and Their Role in Modeling Complex Dynamical Systems. International Journal of Engineering Science & Humanities, 14(4), pp.71-80. ijesh.com [Google Scholar] [Crossref]

121. Al-Mekhlafi.S, 2026. Efficient Numerical Method for Solving a Cancer Tumor Model Governed by Fractional Partial Differential Equations Involving the ψ-Caputo derivative. Sana'a University Journal of Applied Sciences and Technology, 4(2), pp.1682-1693. su.edu.ye [Google Scholar] [Crossref]

122. Hugh.Q & Soria.F, 2025. Fractional-Order Differential Models for Anomalous Transport Phenomena in Environmental Engineering. Journal of Applied Mathematical Models in Engineering, pp.40-45. theeducationjournals.com [Google Scholar] [Crossref]

123. Kebede.S.G, 2025. Analysis of Nonlinear Fractional Differential Equations Involving Caputo-Fabrizio and Atangana-Baleanu-Caputo Fractional Operators. researchgate.net [Google Scholar] [Crossref]

124. Khan.F, Zehra.A, Farman.M, Nisar.K.S, Sambas.A & Hafez.M, 2025. Caputo gH-differentiability for Gronwall’s inequality: insight, new results and application. Journal of Inequalities and Applications, 2025(1), p.148. springer.com [Google Scholar] [Crossref]

125. Kang.Q, Cui.W, Li.X, Ma.Y, Fu.X, Tay.W.P, Li.Y & Zha.Z.J, 2026. Neural Fractional Attention Differential Equations. Advances in Neural Information Processing Systems, 38, pp.33503-33536. neurips.cc [Google Scholar] [Crossref]

126. Wang.C, Attia.R.A, Alfalqi.S.H, Alzaidi.J.F & Khater.M.M, 2024. Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems. Modern Physics Letters B, 38(36), p.2450368. [HTML] [Google Scholar] [Crossref]

127. Aydin.M & Mahmudov.N, 2025. On the discrete delayed perturbation of Mittag-Leffler functions and their application to Caputo-type fractional difference equations. The Journal of Analysis. [HTML] [Google Scholar] [Crossref]

128. Ghosh.B & Kumar.R, 2026. Mathematical Study of the time-fractional non-linear Smoluchowski coagulation equation. Nonlinear Dynamics. [HTML] [Google Scholar] [Crossref]

129. Shams.M & Carpentieri.B, 2025. Efficient Hybrid ANN-Accelerated Two-Stage Implicit Schemes for Fractional Differential Equations. Mathematics. mdpi.com [Google Scholar] [Crossref]

130. Shams.M & Carpentieri.B, 2025. Efficient Hybrid ANN-Accelerated Two-Stage Implicit Schemes for Fractional Differential Equations. Mathematics. mdpi.com [Google Scholar] [Crossref]

131. Glaeser.S.S, Travessini De Cezaro.F & De Cezaro.A, 2023. Synchronization of the Circadian Rhythms with Memory: A Simple Fractional-Order Dynamical Model Based on Two Coupled Oscillators. Trends in Computational and Applied Mathematics, 24(2), pp.245-263. scielo.br [Google Scholar] [Crossref]

132. Amirian Matlob.M & Jamali.Y, 2017. The Concepts and Applications of Fractional Order Differential Calculus in Modelling of Viscoelastic Systems: A Primer. [PDF] [Google Scholar] [Crossref]

133. Bhangale.N, Kachhia.K.B & Gómez‐Aguilar.J.F, 2023. Fractional viscoelastic models with Caputo generalized fractional derivative. Mathematical Methods in the Applied Sciences, 46(7), pp.7835-7846. [HTML] [Google Scholar] [Crossref]

134. Ding.P, Xu.R, Wang.L, Gao.Z, Ge.X & Wen.M, 2026. Fractional derivative viscoelastic models based on L2-1σ formula: modelling and numerical application. Computers and Geotechnics. [HTML] [Google Scholar] [Crossref]

135. Alghamdi.A, 2025. Theoretical and computational methods for fractional viscoelastic models. cardiff.ac.uk [Google Scholar] [Crossref]

136. Ferrás.L.L, 2025. Fractional Derivatives: the ultimate operator for modelling complex viscoelastic materials? Fractional Calculus and Applied Analysis. [HTML] [Google Scholar] [Crossref]

137. F. & Wang.Y, 2023. A Finite Difference Method for Solving Unsteady Fractional Oldroyd‐B Viscoelastic Flow Based on Caputo Derivative. Advances in Mathematical Physics. wiley.com [Google Scholar] [Crossref]

138. Han.B, Yin.D & Gao.Y, 2024. The application of a novel variable-order fractional calculus on rheological model for viscoelastic materials. Mechanics of Advanced Materials and Structures, 31(28), pp.9951-9963. [HTML] [Google Scholar] [Crossref]

139. Abouelregal.A.E, Alanazi.K & Marin.M, 2026. Advanced modeling of magneto-thermoelastic interactions in rotating viscoelastic rods using Caputo-Fabrizio fractional derivatives and nonlocal heat conduction with a correlating length. International Journal of Mechanics and Materials in Design, 22(1), p.64. [HTML] [Google Scholar] [Crossref]

140. Lin.J, Reutskiy.S, Zhang.Y, Sun.Y & Lu.J, 2023. The novel analytical–numerical method for multi-dimensional multi-term time-fractional equations with general boundary conditions. Mathematics. mdpi.com [Google Scholar] [Crossref]

141. Arifeen.S.U, Haq.S, Ali.I & Aldosary.S.F, 2024. Galerkin approximation for multi-term time-fractional differential equations. Ain Shams Engineering Journal. sciencedirect.com [Google Scholar] [Crossref]

142. Fan.B, 2023. Efficient numerical method for multi-term time-fractional diffusion equations with Caputo-Fabrizio derivatives. arXiv preprint arXiv:2307.08078. [PDF] [Google Scholar] [Crossref]

143. Brandibur.O & Kaslik.É, 2023. Stability properties of multi-term fractional-differential equations. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

144. Abd-Elhameed.W.M & Alsuyuti.M.M, 2023. Numerical treatment of multi-term fractional differential equations via new kind of generalized Chebyshev polynomials. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

145. Zou.L & Zhang.Y, 2025. Numerical solutions of multi-term fractional reaction-diffusion equations. AIMS Mathematics. aimspress.com [Google Scholar] [Crossref]

146. Santra.S, 2025. Analysis of a higher-order scheme for multi-term time-fractional integro-partial differential equations with multi-term weakly singular kernels. Numerical Algorithms. [HTML] [Google Scholar] [Crossref]

147. Khosravian-Arab.H & Dehghan.M, 2024. A matrix approach to multi-term fractional differential equations using two new diffusive representations for the Caputo fractional derivative. AUT Journal of Mathematics and Computing, 5(4), pp.337-359. [HTML] [Google Scholar] [Crossref]

148. Vasylyeva.N, 2024. Semilinear multi-term fractional in time diffusion with memory. Frontiers in Applied Mathematics and Statistics. frontiersin.org [Google Scholar] [Crossref]

149. Yang.Z, Li.Q & Yao.Z, 2024. A stability analysis for multi-term fractional delay differential equations with higher order. Chaos. [HTML] ur Rahman.G, Wahid.F, Gómez-Aguilar.J.F & Ali.A, 2024. Analysis of multi-term arbitrary order implicit differential equations with variable type delay. Physica Scripta, 99(11), p.115246. [HTML] [Google Scholar] [Crossref]

150. Engström.C & Morsalfard.E.N, 2026. On multiterm time-fractional diffusion equations with additional memory. Fractional Calculus and Applied Analysis. springer.com [Google Scholar] [Crossref]

151. Fahmy.M.A & Marin.M, 2026. Boundary‐integral analysis of multi‐term fractional heat conduction in heterogeneous media. ZAMM‐Journal of Applied Mathematics and Mechanics, 106(3), p.e70369. [HTML] [Google Scholar] [Crossref]

152. Machida.M, 2016. The time-fractional radiative transport equation - Continuous-time random walk, diffusion approximation, and Legendre-polynomial expansion. [PDF] [Google Scholar] [Crossref]

153. Liang.X, Yang.Y.G, Gao.F, Yang.X.J & Xue.Y, 2018. Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions. ncbi.nlm.nih.gov [Google Scholar] [Crossref]

154. Sokolov.I.M, Chechkin.A.V & Klafter.J, 2004. Distributed-Order Fractional Kinetics. [PDF] [Google Scholar] [Crossref]

155. Zhang.Y, Liu.X, Lei.D, Yin.M, Sun.H, Guo.Z & Zhan.H, 2024. Modeling hydrologically mediated hot moments of transient anomalous diffusion in aquifers using an impulsive fractional‐derivative equation. Water Resources Research, 60(3), p.e2023WR036089. wiley.com [Google Scholar] [Crossref]

156. Deniskin.N & Estrada.E, 2026. Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups and Memory. arXiv preprint arXiv:2601.14977. [PDF] [Google Scholar] [Crossref]

157. Berardi.M, Garrappa.R, Icardi.M & Nuca.R, 2025. From microstructure to memory: Basset-type fractional transport models in porous media. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 481(2327). royalsocietypublishing.org [Google Scholar] [Crossref]

158. Essawy.M.A.I, Rezk.R.A & Mostafa.A.M, 2025. Fractional dynamics of laser-induced heat transfer in metallic thin films: analytical approach. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

159. Deniskin.N & Estrada.E, 2026. Fractional Diffusion on Graphs: Superposition of Laplacian Semigroups Incorporating Memory. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

160. El-Nabulsi.R.A, Anukool.W, Valarmathi.R & Thangaraj.C, 2026. Nonlocal Fractal Diffusion-Advection Models with Variable Coefficients and Nonlocal Time Delay: Existence of Solutions, Lyapunov Function, Hopf Bifurcation, and Stability. Journal of Peridynamics and Nonlocal Modeling, 8(1), p.6. [HTML] [Google Scholar] [Crossref]

161. Abouelregal.A.E, Alhassan.Y, Alsaeed.S.S, Elzayady.M.E, Sedighi.H.M & Kordi.M, 2025. Fractional MGT model for photothermal analysis in rotating semiconductors: insights into anomalous diffusion and thermal wave propagation. Applied Physics A, 131(6), p.519. springer.com [Google Scholar] [Crossref]

162. Almajbri.T.A, Zaghrout.A.S & Sweilam.N.H, 2025. Efficient numerical strategies for 2D nonlinear variable-order fractal fractional reaction-diffusion models. Contemporary Mathematics, pp.5882-5906. wiserpub.com [Google Scholar] [Crossref]

163. He.X, Elsworth.D & Liu.S, 2025. Millisecond-resolved gas sorption kinetics and time-dependent diffusivity of coal. Rock Mechanics and Rock Engineering. psu.edu [Google Scholar] [Crossref]

164. Harris.A.P, Biala.T.A & Khaliq.Q.M, 2022. Fourier Spectral Methods with Exponential Time Differencing for Space-Fractional Partial Differential Equations in Population Dynamics. [PDF] [Google Scholar] [Crossref]

165. Boudaoui.A, El hadj Moussa.Y, Hammouch.Z & Ullah.S, 2021. A fractional-order model describing the dynamics of the novel coronavirus (COVID-19) with nonsingular kernel. ncbi.nlm.nih.gov [Google Scholar] [Crossref]

166. Bhatter.S, Kumawat.S, Purohit.S.D & Suthar.D.L, 2025. Mathematical modeling of tuberculosis using Caputo fractional derivative: A comparative analysis with real data. Scientific Reports. nature.com [Google Scholar] [Crossref]

167. Ullah.S, Altanji.M, Khan.M.A, Alshaheri.A & Sumelka.W, 2023. The dynamics of HIV/AIDS model with fractal-fractional Caputo derivative. Fractals. worldscientific.com [Google Scholar] [Crossref]

168. Hajaj.R & Odibat.Z, 2023. Numerical solutions of fractional epidemic models with generalized Caputo-type derivatives. Physica Scripta. [HTML] [Google Scholar] [Crossref]

169. Olayiwola.M.O & Yunus.A.O, 2025. Mathematical analysis of a within-host dengue virus dynamics model with adaptive immunity using Caputo fractional-order derivatives. Journal of Umm Al-Qura University for Applied Sciences, 11(1), pp.104-123. springer.com [Google Scholar] [Crossref]

170. Alshammari.N.A, Alharthi.N.S, Mohammed Saeed.A, Khan.A & Ganie.A.H, 2025. Numerical solutions of a fractional order SEIR epidemic model of measles under Caputo fractional derivative. PLoS One, 20(5), p.e0321089. plos.org [Google Scholar] [Crossref]

171. R., Kumar.K & Dohare.R, 2023. Caputo fractional order derivative model of Zika virus transmission dynamics. Journal of Mathematical and Computational Science. researchgate.net [Google Scholar] [Crossref]

172. Kumar.K, Sharma.M.K, Shah.S.R & Dohare.R, 2023. Vector-borne transmission dynamics model based Caputo fractional-order derivative. Indian Journal of Theoretical Physics, 71(3-4), pp.61-76. citphy.org [Google Scholar] [Crossref]

173. R. & Polito.F, 2011. A note on fractional linear pure birth and pure death processes in epidemic models. [PDF] [Google Scholar] [Crossref]

174. Farman.M, Shehzad.A, Akgül.A, Baleanu.D & Sen.M.D, 2023. Modelling and analysis of a measles epidemic model with the constant proportional Caputo operator. Symmetry. mdpi.com [Google Scholar] [Crossref]

175. Gkrekas.N, 2024. Applying Laplace transformation on epidemiological models as Caputo derivatives. Математическая биология и биоинформатика. mathnet.ru [Google Scholar] [Crossref]

176. Xu.C, Farman.M, Liu.Z & Pang.Y, 2024. Numerical approximation and analysis of epidemic model with constant proportional Caputo operator. Fractals. worldscientific.com [Google Scholar] [Crossref]

177. Tassaddiq.A, Qureshi.S, Soomro.A, Arqub.O.A & Senol.M, 2024. Comparative analysis of classical and Caputo models for COVID-19 spread: Vaccination and stability assessment. Fixed Point Theory and Algorithms for Sciences and Engineering, 2024(1), p.2. springer.com [Google Scholar] [Crossref]

178. Alsubaie.N.E, El Guma.F, Boulehmi.K, Al-kuleab.N & Abdoon.M.A, 2024. Improving influenza epidemiological models under Caputo fractional-order calculus. Symmetry, 16(7), p.929. mdpi.com [Google Scholar] [Crossref]

179. Subramanian.S, Kumaran.A, Ravichandran.S, Venugopal.P, Dhahri.S & Ramasamy.K, 2024. Fuzzy fractional Caputo derivative of susceptible-infectious-removed epidemic model for childhood diseases. Mathematics, 12(3), p.466. mdpi.com [Google Scholar] [Crossref]

180. Tisdell.C.C, Liu.Z & MacNamara.S, 2017. Basic existence and uniqueness results for solutions to systems of nonlinear fractional differential equations. [PDF] [Google Scholar] [Crossref]

181. Houas.M & Bezziou.M, 2019. Existence and stability results for fractional differential equations with two Caputo fractional derivatives. [PDF] [Google Scholar] [Crossref]

182. Shams.M, Kausar.N, Agarwal.P, Jain.S, Salman.M.A & Shah.M.A, 2023. On family of the Caputo-type fractional numerical scheme for solving polynomial equations. Applied Mathematics in Science and Engineering, 31(1), p.2181959. tandfonline.com [Google Scholar] [Crossref]

183. Shams.M, Kausar.N, Agarwal.P & Jain.S, 2024. Fuzzy fractional Caputo-type numerical scheme for solving fuzzy nonlinear equations. Fractional Differential Equations. [HTML] [Google Scholar] [Crossref]

184. Lenka.B.K & Bora.S.N, 2023. New comparison results for nonlinear Caputo-type real-order systems with applications. Nonlinear Dynamics. [HTML] [Google Scholar] [Crossref]

185. Li.B, Zhang.T & Zhang.C, 2023. Investigation of financial bubble mathematical model under fractal-fractional Caputo derivative. Fractals. [HTML] [Google Scholar] [Crossref]

186. AlBaidani.M.M, Aljuaydi.F, Alsubaie.S.A.F, Ganie.A.H & Khan.A, 2024. Computational and numerical analysis of the Caputo-type fractional nonlinear dynamical systems via novel transform. Fractal and Fractional, 8(12), p.708. mdpi.com [Google Scholar] [Crossref]

187. Lenka.B.K & Bora.S.N, 2023. Limiting behaviour of non-autonomous Caputo-type time-delay systems and initial-time on the real number line. Computational and Applied Mathematics. [HTML] [Google Scholar] [Crossref]

188. Sivalingam.S.M & Govindaraj.V, 2025. Solution representation and a generalised physics-informed neural network for Caputo-type fractional time-varying systems. Nonlinear Dynamics. [HTML] [Google Scholar] [Crossref]

189. Fafa.W, Odibat.Z & Shawagfeh.N, 2023. The homotopy analysis method for solving differential equations with generalized Caputo-type fractional derivatives. Journal of Computational and Nonlinear Dynamics, 18(2), p.021004. [HTML] [Google Scholar] [Crossref]

190. Sytnyk.D & Wohlmuth.B, 2023. Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type. Mathematics. mdpi.com [Google Scholar] [Crossref]

191. Huong.P.T & Anh.P.T, 2026. Caputo stochastic fractional differential equations: Carathéodory scheme and weak convergence. Stochastics. [HTML] [Google Scholar] [Crossref]

192. Kassim.M.D, 2024. Convergence to logarithmic-type functions of solutions of fractional systems with Caputo-Hadamard and Hadamard fractional derivatives. Fractional Calculus and Applied Analysis. [HTML] [Google Scholar] [Crossref]

193. Agarwal.R.P & Madamlieva.E, 2025. Analysis of Mild Extremal Solutions in Nonlinear Caputo-Type Fractional Delay Difference Equations. Mathematics. mdpi.com [Google Scholar] [Crossref]

194. M. & Iambor.L.F, 2023. Approximate solution to fractional order models using a new fractional analytical scheme. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

195. Liu.Y.R, Gao.G.H & Lu.J.Z, 2025. Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative. Communications on Applied Mathematics and Computation, pp.1-37. [HTML] [Google Scholar] [Crossref]

196. Kebede.S.G, Guezane Lakoud.A & Yesuf.H.E, 2024. Solution analysis for non-linear fractional differential equations. Frontiers in Applied Mathematics and Statistics, 10, p.1499179. frontiersin.org [Google Scholar] [Crossref]

197. Elmonem.A, Banerjee.R, Ahmad.S, Jamshed.W, Nisar.K.S, Eid.M.R, Ibrahim.R.W & El Din.S.M, 2023. A comprehensive review on fractional-order optimal control problem and its solution. Open Mathematics, 21(1), p.20230105. degruyterbrill.com [Google Scholar] [Crossref]

198. Mojahed.N, Fatoorehchi.H & Nazari.S, 2025. Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications—A Survey. arXiv preprint arXiv:2512.12111. [PDF] [Google Scholar] [Crossref]

199. M., 2023. Numerical methods for fractional optimal control and estimation. Babylonian Journal of Mathematics. mesopotamian.press [Google Scholar] [Crossref]

200. Cao.X, Ghosh.S, Rana.S, Bose.H & Roy.P.K, 2023. Application of an optimal control therapeutic approach for the memory-regulated infection mechanism of leprosy through Caputo–Fabrizio fractional derivative. Mathematics. mdpi.com [Google Scholar] [Crossref]

201. Elsonbaty.A, Al-Shami.T.M & El-Mesady.A, 2025. Unveiling the dynamics of meningitis infections: a comprehensive study of a novel fractional-order model with optimal control strategies. Boundary Value Problems. springer.com [Google Scholar] [Crossref]

202. Jiang.N, 2025. Research on stability and control strategies of fractional-order differential equations in nonlinear dynamic systems. Journal of Computational Methods in Sciences and Engineering, p.14727978251346078. [HTML] [Google Scholar] [Crossref]

203. Ali.K.K, Seyam.E.A, Mohamed.M.S & Ibrahim.A.A.E, 2026. Exploring existence and uniqueness in non-local Caputo q-fractional integro-differential equations. Boundary Value Problems. springer.com [Google Scholar] [Crossref]

204. Mohsin Mohammed Ali.M, Mahmoudi.M & Darehmiraki.M, 2025. Hybrid RBF Method for Solving Fractional PDE-Constrained Optimal Control Problems. Control and Optimization in Applied Mathematics, 10(2), pp.213-240. pnu.ac.ir [Google Scholar] [Crossref]

205. Moon.J, 2025. The Pontryagin type maximum principle for Caputo fractional optimal control problems with terminal and running state constraints. AIMS Mathematics. aimspress.com [Google Scholar] [Crossref]

206. Pooseh.S, 2013. Computational Methods in the Fractional Calculus of Variations and Optimal Control. [PDF] [Google Scholar] [Crossref]

207. Kebede.S.G & Lakoud.A.G, 2023. Analysis of mathematical model involving nonlinear systems of Caputo–Fabrizio fractional differential equation. Boundary Value Problems. springer.com [Google Scholar] [Crossref]

208. Sadek.L, 2025. Control theory for fractional differential Sylvester matrix equations with Caputo fractional derivative. Journal of Vibration and Control. [HTML] [Google Scholar] [Crossref]

209. Almutairi.N & Saber.S, 2023. On chaos control of nonlinear fractional Newton-Leipnik system via fractional Caputo-Fabrizio derivatives. Scientific Reports. nature.com [Google Scholar] [Crossref]

210. Boichuk.O & Feruk.V, 2023. Fredholm boundary-value problem for the system of fractional differential equations. [PDF] [Google Scholar] [Crossref]

211. Kharazmi.E & Zayernouri.M, 2018. Operator-Based Uncertainty Quantification of Stochastic Fractional PDEs. [PDF] [Google Scholar] [Crossref]

212. Almutairi.N, Messaoudi.M, Almalki.F.M.K & Saber.S, 2026. A Deterministic and Stochastic Fractional-Order Model for Computer Virus Propagation with Caputo-Fabrizio Derivative: Analysis, Numerics, and Dynamics. Computer Modeling in Engineering & Sciences (CMES), 146(3), p.1. [HTML] [Google Scholar] [Crossref]

213. Majee.S, Jana.S, Kar.T.K, Barman.S & Das.D.K, 2024. Modeling and analysis of Caputo-type fractional-order SEIQR epidemic model. International Journal of Dynamics and Control, 12(1), pp.148-166. google.com [Google Scholar] [Crossref]

214. Jan.R, Qureshi.S, Boulaaras.S, Pham.V.T, Hincal.E & Guefaifia.R, 2023. Optimization of the fractional-order parameter with the error analysis for HIV under Caputo operator. Discrete Contin. Dyn. Syst. S, 16, pp.2118-2140. researchgate.net [Google Scholar] [Crossref]

215. Keddi.A & Bouzebda.S, 2026. Statistical Inference for Drift Parameters in Gaussian White Noise Models Driven by Caputo Fractional Dynamics Under Discrete Observation Schemes. Symmetry. mdpi.com [Google Scholar] [Crossref]

216. Rashid.S, Karim.S, Akgül.A, Bariq.A & Elagan.S.K, 2023. Novel insights for a nonlinear deterministic-stochastic class of fractional-order Lassa fever model with varying kernels. Scientific Reports. nature.com [Google Scholar] [Crossref]

217. Hilal.B.K & AlShemmary.E.N, 2026. Evaluating the Impact of the Caputo Derivative on Deep Models for Hypertrophic Cardiomyopathy Classification Using CMR Images. Iraqi Journal of Science. uobaghdad.edu.iq [Google Scholar] [Crossref]

218. Khan.A, 2026. Fractional Calculus in Modern Machine Learning Optimizers: A Systematic Review of Algorithms, Convergence, and Open Frontiers. Convergence. ssrn.com [Google Scholar] [Crossref]

219. Shams.M & Carpentieri.B, 2025. Efficient Hybrid Parallel Scheme for Caputo Time-Fractional PDEs on Multicore Architectures. Fractal and Fractional. mdpi.com [Google Scholar] [Crossref]

220. Abidin.M.Z, 2026. Energy-dissipative adaptive-step L1 discretisation for the Caputo time-fractional incompressible magnetohydrodynamic system. Scientific Reports. nature.com [Google Scholar] [Crossref]

221. Faiz.Z, Ali.A.S, Thaljaoui.A & Badawi.M.A, 2026. Modeling and treatment of social media addiction using a fractional-order and deep neural network approach. Applied Soft Computing. [HTML] [Google Scholar] [Crossref]

222. Dai.B, Luo.W, Yin.Z & Zheng.P, 2024. The well-posedness and blow up phenomenon for a Tsunamis model with time-fractional derivative. arXiv preprint arXiv:2405.10823. [PDF] [Google Scholar] [Crossref]

223. Akram.S, Batool.R & Akram.F, 2025. Development of Two-Step Fractional Numerical Schemes for Caputo-Type Differential Equations. Journal of Computational Mathematics. [HTML] [Google Scholar] [Crossref]

224. Akram.S, Batool.R & Akram.F, 2025. Development of Two-Step Fractional Iterative Technique of Convergence Order (2µ+1) with Applications. Punjab University Journal of Mathematics. pu.edu.pk [Google Scholar] [Crossref]

225. Sumaiya Banu.S.S, Gunasekar.T, Manikandan.S, Kumar.K & Suba.M, 2025. (SI15-121) Analyzing SEITR Tuberculosis Transmission Model using Caputo–Fabrizio Fractional Derivative with Diverse Contact Rates. Applications and Applied Mathematics: An International Journal (AAM), 20(4), p.8. pvamu.edu [Google Scholar] [Crossref]

226. Raghavendran.P, Gochhait.S & Gunasekar.T, 2025. Optimizing Load Dispatch Using Artificial Neural Networks and Fractional Weighted Models. JoVE (Journal of Visualized Experiments), (224), p.e68811. [HTML] [Google Scholar] [Crossref]

227. Baleanu.D, Karaca.Y, Vázquez.L & Macías-Díaz.J.E, 2023. Advanced fractional calculus, differential equations and neural networks: Analysis, modeling and numerical computations. Physica Scripta, 98(11), p.110201. [HTML] [Google Scholar] [Crossref]

228. Alshammari.M, Alshammari.S, Singh.M, Jawarneh.Y, Mohammed.N.M & Alshammari.K.M, 2025. Caputo fractional derivative in the solution of Kolmogorov equations: a new numerical approach. Boundary Value Problems, 2025(1), p.104. springer.com [Google Scholar] [Crossref]

229. Ahadi.A, Mousavi.S.M, Alinia.A.M & Khademi.H, 2025. Analytical simulation of the nonlinear Caputo fractional equations. Partial Differential Equations in Applied Mathematics, p.101264. sciencedirect.com [Google Scholar] [Crossref]

230. Hammad.H.A, Liu.Z & Abdalla.M.E.M, 2025. Existence and stability results of f-Caputo modified proportional fractional delay differential systems with boundary conditions. Boundary Value Problems. springer.com [Google Scholar] [Crossref]

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