A Sequential Differential Problem With Caputo and Riemann Liouville Derivatives Involving Convergent Series

Year 2023, Volume: 7 Issue: 2, 319 - 335, 23.07.2023

Yazid Gouari Mahdi Rakah , Zoubir Dahmani

https://doi.org/10.31197/atnaa.1224234

Abstract

In this paper, we study a new nonlinear sequential differential prob-
lem with nonlocal integral conditions that involve convergent series. The
problem involves two fractional order operators: Riemann-Liouville inte-
gral, Caputo and Riemann-Liouville derivatives. We prove an existence
and uniqueness result. Also, we prove a new existence result. We end our
paper by presenting some illustrative examples.

Keywords

Caputo derivative, fixed point, Riemann-Liouville derivative, existence of solution

References

• [1] S. Abbagari, A. Houwe, Y. Saliou, Douvagaï, Y. Chu, M. Inc, H.Rezazadeh, S. Y. Doka: Analytical survey of the predatorprey model with fractional derivative order, AIP Advances, (2021).
• [2] A. Abdenebi, Z. Dahmani: New Van der Pol-Duffing Jerk Fractional Differential Oscillator of Sequential Type, Mathematics, 10, 3546, (2022).
• [3] J. Abolfazl, F. Hadi: The application of Duffing oscillator in weak signal detection, ECTI Transactions on Electrical Engineering, Electronics and Communication, (2011).
• [4] H. Afshari, D. Baleanu: Applications of some fixed point theorems for fractional differential equations with Mittag-Leffler kernel, Advances in Difference Equations, (2020).
• [5] R. Almeida, B.R.O.Bastos, M.T.T. Monteiro: Modeling some real phenomena by fractional differential equations, Math. Methods Appl. Sci. 39(16), 4846-4855, (2016).
• [6] Y. Bahous , Z. Dahmani: A Lane Emden Type Problem Involving Caputo Derivative and Riemann-Liouville Integral,Indian Journal of Industrial and Applied Mathematics. Vol. 10, No. 1, (2019).
• [7] Z. Bekkouche, Z. Dahmani and G. Zhang: Solutions and Stabilities for a 2D-Non Homogeneous Lane-Emden Fractional System, Int. J. Open Problems Compt. Math., Vol. 11, No. 2, (2018).
• [8] A. Benzidane, Z. Dahmani: A class of nonlinear singular differential equations, Journal of Interdisciplinary Mathematics, Vol. 22, No. 6, (2019).
• [9] A. Carpinteri , F. Mainardi: Fractional Calculus in Continuum Mechanics, Springer, New York, NY, (1997).
• [10] Y.M. Chu, M. Ahmad, M.I. Asjad, D. Baleanu: Fractional Model of Second Grande Fluid Induced by Generalized Thermal and Molecular Fluxes With Constant Proportional Caputo, Thermal Science, (2021).
• [11] Y.M. Chu, M.S. Khan, M. Abbas, S. Ali, W. Nazeer: On characterizing of bifurcation and stability analysis for time fractional glycolysis model, Chaos, Solitons and Fractals, (2022).
• [12] Y.M. Chu, M.D. Ikram, M.I. Asjad, A. Ahmadian, F. Ghaemi: Influence of hybrid nanofluids and heat generation on coupled heat and mass transfer flow of a viscous fluid with novel fractional derivative, J Therm Anal Calorim 144, 20572077,(2021).
• [13] Y.M. Chu, M.F. Khan, S. Ullah, S.A.A. Shah, M. Farooq, M. bin Mamat: Mathematical assessment of a fractional-order vectorhost disease model with the CaputoFabrizio derivative, Math Methods Appl. Sci., (2022).
• [14] Z. Dahmani, Y. Bahous and Z. Bekkouche: A two parameter singular fractional diferential equations of Lane Emden type, Turkish J. Ineq., 3 (1), (2019).
• [15] Z. Dahmani, M.A. Abdellaoui, M. Houas: Coupled Systems of Fractional IntegroDiferential Equations Involving Several Functions, Theory and Applications of Mathematics and Computer Science 5 (1), (2015).
• [16] C. L. Ejikeme, M.O. Oyesanya, D. F. Agbebaku, M. B Okofu: Solution to nonlinear Duffing Oscillator with fractional derivatives using Homotopy Analysis Method(HAM), Global Journal of Pure and Applied Mathematics,(2018).
• [17] R. Emden: Gaskugeln, Teubner, Leipzig and Berlin, (1907).
• [18] Y. Gouari, Z. Dahmani, I. Jebril: Application of fractional calculus on a new differential problem of duffing type, Adv. Math. Sci. J. (2020).
• [19] Y. Gouari, Z. Dahmani, M.M. Belhamiti, M.Z. Sarikaya : Uniqueness of Solutions, Stability and Simulations for a Differential Problem Involving Convergent Series and Time Variable Singularities, Rocky Mountain Journal of Mathematics, (2022).
• [20] Y. Gouari, Z. Dahmani, A. Ndiaye: A generalized sequential problem of Lane-Emden type via fractional calculus. Moroccan J. of Pure and Appl. Anal, Vol. 6, Issu. 2, (2020).
• [21] M. Houas, M.E. Samei: Existence and Mittag-Leffler-Ulam-Stability Results for Duffing Type Problem Involving Sequential Fractional Derivatives, International Journal of Applied and Computational Mathematics, (2022).
• [22] R.W. Ibrahim: Stability of A Fractional Differential Equation, International Journal of Mathematical, Computational, Physical and Quantum Engineering., Vol. 7, No. 3, (2013).
• [23] R.W. Ibrahim and H.A. Jalab: Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy 17, (2015).
• [24] M.D. Ikram, M.A. Imran, Y. Chu and A. Akgül: MHD Flow of a Newtonian Fluid in Symmetric Channel with ABC Fractional Model Containing Hybrid Nanoparticles, Combinatorial Chemistry and High Throughput Screening, (2022).
• [25] D. Khan, G. Ali, A. Khan, I. Khan, Y. Chu, K. S. Nisar: A New Idea of FractalFractional Derivative with Power Law Kernel for Free Convection Heat Transfer in a Channel Flow between Two Static Upright Parallel Plates, Computers, Materials and Continua, (2020).
• [26] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo: Theory and Applications of Fractional Differential Equations, Elsevier B.V., Amsterdam, The Netherlands, (2006).
• [27] S.M. Mechee and N. Senu: Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation, Applied Mathematics., 3, (2012).
• [28] K.S. Miller and B. Ross: An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York., (1993).
• [29] J. Niu, R. Liu, Y. Shen, S. Yang: Chaos detection of Duffing system with fractional order derivative by Melnikov method, Chaos 29, 123106, (2019).
• [30] P. Pirmohabbati, A. H. Refahi Sheikhani, H. Saberi Najafi, A. Abdolahzadeh Ziabari: Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities, Journal of AIMS Mathematics, (2020).
• [31] M. Rakah, A. Anber, Z. Dahmani, I. Jebril: An Analytic and Numerical study for two classes of differential equation of fractional order involving Caputo and Khalil derivative. An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), (2022).
• [32] M. Rakah, Z. Dahmani, A. Senouci: New Uniqueness Results for Fractional Differential Equations with a Caputo and Khalil Derivatives. Appl. Math. Inf. Sci. 16, No. 6, 943-952 (2022).
• [33] M. Saqib, S. Shafie, I. Khan, Y. Chu, K. S. Nisar: Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles, Symmetry, (2020).
• [34] J. Sunday: The Duffing oscillator: Applications and computational simulations. Asian Research Journal of Mathematics, (2017).