Analysis of a class of fractal hybrid fractional differential equation with application to a biological model

Almeida, R., Malinowska, A. B. & Monteiro, M. T. T. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Mathe. Methods Appl. Sci. 41(1), 336–352 (2018).Article 
ADS 
MathSciNet 

Google Scholar 
Das, S. & Pan, I. Fractional order signal processing: introductory concepts and applications (Springer Science & Business Media, 2011).
Google Scholar 
Araz, S. İ. Analysis of a Covid-19 model: optimal control, stability and simulations. Alexandria Eng. J. 60(1), 647–658 (2021).Article 

Google Scholar 
Awadalla, M. & Yameni, Y. Modeling exponential growth and exponential decay real phenomena by \(\psi\)-Caputo fractional derivative. J. Adv. Mathe. Comput. Sci. 28(2), 1–13 (2018).Article 

Google Scholar 
Atangana, A. & İğret Araz, S. Mathematical model of COVID-19 spread in Turkey and South Africa: theory, methods, and applications. Adv. Diff. Equ. 2020(1), 1–89 (2020).Article 
MathSciNet 

Google Scholar 
Ahmed, S., Ahmed, A., Mansoor, I., Junejo, F. & Saeed, A. Output feedback adaptive fractional-order super-twisting sliding mode control of robotic manipulator. Iran. J. Sci. Technol. Trans. Elect. Eng. 45, 335–347 (2021).Article 

Google Scholar 
Shah, K., Jarad, F. & Abdeljawad, T. On a nonlinear fractional order model of dengue fever disease under Caputo-Fabrizio derivative. Alex. Eng. J. 59(4), 2305–2313 (2020).Article 

Google Scholar 
Ahmed, S., Wang, H., Aslam, M. S., Ghous, I. & Qaisar, I. Robust adaptive control of robotic manipulator with input time-varying delay. Int. J. Control Autom. Syst. 17(9), 2193–2202 (2019).Article 

Google Scholar 
Shaikh, A., Nisar, K. S., Jadhav, V., Elagan, S. K. & Zakarya, M. Dynamical behaviour of HIV/AIDS model using fractional derivative with Mittag-Leffler kernel. Alex. Eng. J. 61(4), 2601–2610 (2022).Article 

Google Scholar 
Peter, O. J. et al. Analysis and dynamics of fractional order mathematical model of COVID-19 in Nigeria using atangana-baleanu operator. Comput. Mater. Continua 66(2), 1823–1848 (2021).Article 

Google Scholar 
Shaikh, A. S., Shaikh, I. N. & Nisar, K. S. A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control. Adv. Differ. Equ. 2020(1), 373 (2020).Article 
MathSciNet 
PubMed 
PubMed Central 

Google Scholar 
Teodoro, G. S., Machado, J. T. & De Oliveira, E. C. A review of definitions of fractional derivatives and other operators. J. Comput. Phys. 388, 195–208 (2019).Article 
ADS 
MathSciNet 

Google Scholar 
Khalil, R., Al Horani, M., Yousef, A. & Sababheh, M. A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014).Article 
MathSciNet 

Google Scholar 
Hilfer, R. (ed.) Applications of fractional calculus in physics (World scientific, 2000).
Google Scholar 
Kilbas, A. A. Hadamard-type fractional calculus. J. Korean Math. Soc. 38(6), 1191–1204 (2001).MathSciNet 

Google Scholar 
Zhang, T. & Li, Y. Global exponential stability of discrete-time almost automorphic Caputo-Fabrizio BAM fuzzy neural networks via exponential Euler technique. Knowl.-Based Syst. 246, 108675 (2022).Article 

Google Scholar 
Khan, M. et al. Dynamics of two-step reversible enzymatic reaction under fractional derivative with Mittag-Leffler Kernel. PLoS One 18(3), e0277806 (2023).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Caputo, M. & Fabrizio, M. Applications of new time and spatial fractional derivatives with exponential kernels. Progress Fract. Differ. Appl. 2(1), 1–11 (2016).Article 

Google Scholar 
Caputo, M. & Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress Fract. Diff. Appl. 1(2), 73–85 (2015).
Google Scholar 
Losada, J. & Nieto, J. J. Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 87–92 (2015).
Google Scholar 
Gul, R., Sarwar, M., Shah, K., Abdeljawad, T. & Jarad, F. Qualitative analysis of implicit Dirichlet boundary value problem for Caputo-Fabrizio fractional differential equations. J. Function Spaces 2020, 1–9 (2020).Article 
MathSciNet 

Google Scholar 
Atangana, A. & Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. (2016). arXiv preprint arXiv:1602.03408.Xu, C., Liu, Z., Pang, Y., Saifullah, S. & Inc, M. Oscillatory, crossover behavior and chaos analysis of HIV-1 infection model using piece-wise Atangana-Baleanu fractional operator: Real data approach. Chaos, Solitons Fractals 164, 112662 (2022).Article 
MathSciNet 

Google Scholar 
Saifullah, S., Ali, A., Irfan, M. & Shah, K. Time-fractional Klein-Gordon equation with solitary/shock waves solutions. Math. Probl. Eng. 2021, 1–15 (2021).Article 

Google Scholar 
Alomari, A. K., Abdeljawad, T., Baleanu, D., Saad, K. M. & Al-Mdallal, Q. M. Numerical solutions of fractional parabolic equations with generalized Mittag-Leffler kernels. Numer. Methods Partial Differ. Equ. 40(1), e22699 (2024).Article 
MathSciNet 

Google Scholar 
Saad Alshehry, A., Imran, M., Shah, R. & Weera, W. Fractional-view analysis of fokker-planck equations by ZZ transform with mittag-leffler kernel. Symmetry 14(8), 1513 (2022).Article 
ADS 

Google Scholar 
Atangana, A. Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system. Chaos, Solitons Fractals 102, 396–406 (2017).Article 
ADS 
MathSciNet 

Google Scholar 
He, J. H. Fractal calculus and its geometrical explanation. Res. Phys. 10, 272–276 (2018).ADS 

Google Scholar 
Hu, Y. & He, J. H. On fractal space-time and fractional calculus. Therm. Sci. 20(3), 773–777 (2016).Article 

Google Scholar 
Qureshi, S. & Atangana, A. Fractal-fractional differentiation for the modeling and mathematical analysis of nonlinear diarrhea transmission dynamics under the use of real data. Chaos Solitons Fractals 136, 109812 (2020).Article 
MathSciNet 

Google Scholar 
Srivastava, H. M. & Saad, K. M. Numerical simulation of the fractal-fractional Ebola virus. Fractal Fract. 4(4), 49 (2020).Article 

Google Scholar 
Xiao, B., Huang, Q., Chen, H., Chen, X. & Long, G. A fractal model for capillary flow through a single tortuous capillary with roughened surfaces in fibrous porous media. Fractals 29(01), 2150017 (2021).Article 
ADS 

Google Scholar 
Liang, M. et al. An analytical model for the transverse permeability of gas diffusion layer with electrical double layer effects in proton exchange membrane fuel cells. Int. J. Hydrogen Energy 43(37), 17880–17888 (2018).Article 
ADS 
CAS 

Google Scholar 
Yu, X. et al. Characterization of water migration behavior during spontaneous imbibition in coal: From the perspective of fractal theory and NMR. Fuel 355, 129499 (2024).Article 
CAS 

Google Scholar 
Ahmad, I., Ahmad, N., Shah, K. & Abdeljawad, T. Some appropriate results for the existence theory and numerical solutions of fractals-fractional order malaria disease mathematical model. Res. Control Optim. 14, 100386 (2024).
Google Scholar 
Ur Rahman, M. Generalized fractal-fractional order problems under non-singular Mittag-Leffler kernel. Res. Phys. 35, 105346 (2022).
Google Scholar 
Eiman, Shah K., Sarwar, M. & Abdeljawad, T. A comprehensive mathematical analysis of fractal-fractional order nonlinear re-infection model. Alex. Eng. J. 103, 353–365 (2024).Article 

Google Scholar 
Khan, S. Existence theory and stability analysis to a class of hybrid differential equations using confirmable fractal fractional derivative. J. Frac. Calc. Nonlinear Sys. 5(1), 1–11 (2024).Article 

Google Scholar 
El-Dessoky, M. M. & Khan, M. A. Modeling and analysis of an epidemic model with fractal-fractional Atangana-Baleanu derivative. Alex. Eng. J. 61(1), 729–746 (2022).Article 

Google Scholar 
Smith, H. An Introduction to Delay Differential Equations with Applications to the Life Sciences 119–130 (Springer, 2011).Book 

Google Scholar 
Balachandran, B., Kalmár-Nagy, T. & Gilsinn, D. E. Delay differential equations (Springer, 2009).
Google Scholar 
Balachandran, K., Kiruthika, S. & Trujillo, J. Existence of solutions of nonlinear fractional pantograph equations. Acta Mathe. Sci. 33(3), 712–720 (2013).Article 
MathSciNet 

Google Scholar 
Basim, M., Ahmadian, A., Senu, N. & Ibrahim, Z. B. Numerical simulation of variable-order fractal-fractional delay differential equations with nonsingular derivative. Eng. Sci. Technol. Int. J. 42, 101412 (2023).
Google Scholar 
Shafiullah, Shah K., Sarwar, M. & Abdeljawad, T. On theoretical and numerical analysis of fractal-fractional non-linear hybrid differential equations. Nonlinear Eng. 13(1), 20220372 (2024).Article 
ADS 

Google Scholar 
Abbas, M. I. & Ragusa, M. A. On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry 13(2), 264 (2021).Article 
ADS 

Google Scholar 
Guo, C., Hu, J., Hao, J., Celikovsky, S. & Hu, X. Fixed-time safe tracking control of uncertain high-order nonlinear pure-feedback systems via unified transformation functions. Kybernetika 59(3), 342–364 (2023).MathSciNet 

Google Scholar 
Ulam, S. M. Problems in modern mathematics (Courier Corporation, 2004).
Google Scholar 
Hyers, D. H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27(4), 222–224 (1941).Article 
ADS 
MathSciNet 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Rassias, T. M. On the stability of the linear mapping in Banach spaces. Proc. Am. Mathe. Soc. 72(2), 297–300 (1978).Article 
MathSciNet 

Google Scholar 
Rhaima, M., Mchiri, L., Makhlouf, A. B. & Ahmed, H. Ulam type stability for mixed Hadamard and Riemann-Liouville fractional stochastic differential equations. Chaos Solitons Fractals 178, 114356 (2024).Article 
MathSciNet 

Google Scholar 
Huang, J. & Luo, D. Ulam-Hyers stability of fuzzy fractional non-instantaneous impulsive switched differential equations under generalized Hukuhara differentiability. Int. J. Fuzzy Syst. 2024, 1–12 (2024).
Google Scholar 
Guo, C., Hu, J., Wu, Y. & Celikovsky, S. Non-Singular Fixed-Time Tracking Control of Uncertain Nonlinear Pure-Feedback Systems With Practical State Constraints. IEEE Trans. Circuits Syst. I Regul. Pap. 70(9), 3746–3758 (2023).Article 

Google Scholar 
Peng, Y., Zhao, Y. & Hu, J. On The Role of Community Structure in Evolution of Opinion Formation: A New Bounded Confidence Opinion Dynamics. Inf. Sci. 621, 672–690 (2023).Article 

Google Scholar 
Khan, S., Shah, K., Debbouche, A., Zeb, S. & Antonov, V. Solvability and Ulam-Hyers stability analysis for nonlinear piecewise fractional cancer dynamic systems. Phys. Scr. 99(2), 025225 (2024).Article 
ADS 

Google Scholar 
Khan, M. A. & Atangana, A. Numerical Methods for Fractal-fractional Differential Equations and Engineering: Simulations and Modeling (CRC Press, 2023).Book 

Google Scholar 
Chen, X., Shi, C. & Wang, D. Dynamic behaviors for a delay Lasota-Wazewska model with feedback control on time scales. Adv. Differ. Equ. 2020(1), 1–13 (2020).MathSciNet 
CAS 

Google Scholar 
Xu, G., Huang, M., Hu, J., Liu, S. & Yang, M. Bisphenol A and its structural analogues exhibit differential potential to induce mitochondrial dysfunction and apoptosis in human granulosa cells. Food Chem. Toxicol. 188, 114713 (2024).Article 
CAS 
PubMed 

Google Scholar 
Luo, W. et al. Update: Innate lymphoid cells in inflammatory bowel disease. Dig. Dis. Sci. 67(1), 56–66 (2022).Article 
PubMed 

Google Scholar