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

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

References

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  • [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).
Year 2023, Volume: 7 Issue: 2, 319 - 335, 23.07.2023
https://doi.org/10.31197/atnaa.1224234

Abstract

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).
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Yazid Gouari This is me

Mahdi Rakah 0000-0002-5214-815X

Zoubir Dahmani

Early Pub Date August 3, 2023
Publication Date July 23, 2023
Published in Issue Year 2023 Volume: 7 Issue: 2

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