Oscillatory Behavior of Solution of Hilfer Fractional Differential Equation
DOI:
https://doi.org/10.71107/gfd62y51Keywords:
Oscillation theory, Fractional differential equation, Impulsive, Hilfer derivativeAbstract
In this paper, we study the oscillation of impulsive fractional differential equations.Using the inequality principle and Bihari Lemma,sufficient conditions are found for both the asymptotic and oscillatory phases of the equation. An example is given to illustrate the validity of our main results. The oscillation of an impulsive fractional differential equation with two different Caputo derivatives is being studied for the fist time.
Downloads
References
[1] A. Hermosillo-Arteaga, M. P. Romo, and R. Magana-del Toro, “Response spectra generation using a fractional differential model,” Soil Dynamics and Earthquake Engineering 115, 719–729 (2018).
[2] Y. Jiang, B. Xia, X. Zhao, T. Nguyen, C. Mi, and R. A. de Callafon, “Data-based fractional differential models for nonlinear dynamic modeling of a lithium-ion battery,” Energy 135, 171–181 (2017).
[3] L. Feng and S. Sun, “Oscillation theorems for three classes of conformable fractional differential equations,” Advances in Difference Equations 2019, 1–30 (2019).
[4] Y. Bolat, “On the oscillation of fractional-order delay differential equations with constant coefficients,” Communications in Nonlinear Science and Numerical Simulation 19, 3988–3993 (2014).
[5] Y. Z. Wang, Z. L. Han, Z. H. A. O. Ping, and S. R. Sun, “Oscillation theorems for fractional neutral differential equations,” Hacettepe Journal of Mathematics and Statistics 44, 1477–1488 (2015).
[6] Y. Zhou, B. Ahmad, and A. Alsaedi, “Existence of nonoscillatory solutions for fractional neutral differential equations,” Applied Mathematics Letters 72, 70–74 (2017).
[7] Q. Ma, J. Pecaric, and J. Zhang, “Integral inequalities of systems and the estimate for solutions of certain nonlinear two-dimensional fractional differential systems,” Computers and Mathematics with Applications 61, 3258–3267 (2011).
[8] A. Ortega, J. J. Rosales, J. M. Cruz-Duarte, et al., “Fractional model of the dielectric dispersion,” Optik-International Journal for Light and Electron Optics 180, 754–759 (2019).
[9] A. Raheem and M. Maqbul, “Oscillation criteria for impulsive partial fractional differential equations,” Computers and Mathematics with Applications 73, 1781–1788 (2017).
[10] I. Stamova, “Global stability of impulsive fractional differential equations,” Applied Mathematics and Computation 237, 605–612 (2014).
[11] S. R. Grace, R. P. Agarwal, J. Y. Wong, and A. Zafer, “On the oscillation of fractional differential equations,” Fractional Calculus and Applied Analysis 15, 222–231 (2012).
[12] J. Wang, X. Li, and W. Wei, “On the natural solution of an impulsive fractional differential equation of order q∈(1,2),” Communications in Nonlinear Science and Numerical Simulation 17, 4384–4394 (2012).
[13] P. Prakash, S. Harikrishnan, J. Nieto, and K. Hoon, “Oscillation of a time fractional partial differential equation,” Electronic Journal of Qualitative Theory of Differential Equations 2014, 1–10 (2014).
[14] D. X. Chen, “Oscillatory behavior of a class of fractional differential equations with damping,” UPB Scientific Bulletin, Series A 75, 107–118 (2013).
[15] D. X. Chen, “Oscillation criteria of fractional differential equations,” Advances in Difference Equations 2012, 1–10 (2012).
[16] D. X. Chen, “Oscillation criteria of fractional differential equations,” Advances in Difference Equations 2012, 1–10 (2012).
[17] W. N. Li, “Oscillation of solutions for certain fractional partial differential equations,” Advances in Difference Equations 2016, 1–8 (2016).
[18] W. N. Li, “On the forced oscillation of certain fractional partial differential equations,” Applied Mathematics Letters 50, 5–9 (2015).
[19] W. N. Li, “Forced oscillation criteria for a class of fractional partial differential equations with damping term,” Mathematical Problems in Engineering 2015, Article ID 393624 (2015).
[20] M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals 14, 433–440 (2002).
[21] Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,” Computers and Mathematics with Applications 59, 1063–1077 (2010).
[22] S. Liu, X. Wu, X. F. Zhou, and W. Jiang, “Asymptotical stability of Riemann–Liouville fractional nonlinear systems,” Nonlinear Dynamics 86, 65–71 (2016).
[23] S. Liu, X. F. Zhou, X. Li, and W. Jiang, “Asymptotical stability of Riemann–Liouville fractional singular systems with multiple time-varying delays,” Applied Mathematics Letters 65, 32–39 (2017).
[24] K. Li, J. Peng, and J. Jia, “Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives,” Journal of Functional Analysis 263, 476–510 (2012).
[25] R. P. Agarwal and A. Özbekler, “Lyapunov type inequalities for mixed nonlinear Riemann–Liouville fractional differential equations with a forcing term,” Journal of Computational and Applied Mathematics 314, 69–78 (2017).
[26] R. P. Agarwal, V. Lupulescu, D. O’Regan, and G. ur Rahman, “Fractional calculus and fractional differential equations in nonreflexive Banach spaces,” Communications in Nonlinear Science and Numerical Simulation 20, 59–73 (2015).
[27] H. Liu, Y. Pan, S. Li, and Y. Chen, “Adaptive fuzzy backstepping control of fractional-order nonlinear systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems 47, 2209–2217 (2017).
[28] H. Liu, S. Li, G. Li, and H. Wang, “Adaptive controller design for a class of uncertain fractional-order nonlinear systems: an adaptive fuzzy approach,” International Journal of Fuzzy Systems 20, 366–379 (2018).
[29] R. Hilfer, Y. Luchko, and I. Tomovski, “Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives,” Fractional Calculus and Applied Analysis 12, 289–318 (2009).
[30] X. Ding and J. J. Nieto, “Controllability and optimality of linear time-invariant neutral control systems with different fractional orders,” Acta Mathematica Scientia 35, 1003–1013 (2015).
Downloads
Published
Data Availability Statement
Data will be made available from corresponding author by request.
License
Copyright (c) 2025 Aqsa Balqees, Azmat Ullah Khan Niazi, Naveed Iqbal (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Similar Articles
- Rabbiya Fatima, Azmat Ullah Khan Niazi, Hassan Raza, Novel Insights into Oscillation of Impulsive Fractional Differential Equations with Caputo Derivative , Conclusions in Engineering: Vol. 1 No. 1 (2025): Conclusions in Engineering
- Kamal Bashir, Mohamed Mosadag, A Novel Resampling Technique for Imbalanced Classification in Software Defect Prediction by a re-sampling method with filtering , Conclusions in Engineering: Vol. 1 No. 1 (2025): Conclusions in Engineering
- Irfan Haider, Imtiaz Ahmad Khan, Fatima Kainat, Hassan Ali Akhter, Hassaan Khalid, Nawishta Jabeen, Ahmad Hussain, Theoretical analysis of power-law nanofluid across extended sheet with thermal-concentration slip and Soret/Dufour effect , Conclusions in Engineering: Vol. 1 No. 1 (2025): Conclusions in Engineering
You may also start an advanced similarity search for this article.