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On the Study of Pantograph Differential Equations with Proportional Fractional Derivative

Year 2023, Volume: 11 Issue: 2, 97 - 103, 30.06.2023
https://doi.org/10.36753/mathenot.1057344

Abstract

This manuscript is devoted to investigate the existence, uniqueness and stability of pantograph equations with Hilfer generalized proportional fractional derivative. The concerned results are obtained using standard theorems.

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References

  • [1] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear. Sci. Numer. Simulat., 44 (2017) 460–481.
  • [2] S. Abbas, M. Benchohra, S. Sivasundaram, Dynamics and Ulam stability for Hilfer type fractional di?erential equations, Nonlinear Stud., 4 (2016) 627–637.
  • [3] I. Ahmed, P. Kumam, F. Jarad, P. Borisut, W. Jirakitpuwapat, On Hilfer generalized proportional fractional derivative, Adv. Di?er. Equ. 2020:329.
  • [4] K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence of solutions of Nonlinear fractional pantograph equations, Acta Math. Sci., 33B (2013) 1-9.
  • [5] K. M. Furati, M. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving HFD, Compur. Math. Appl., 64 (2012) 1616–1626.
  • [6] K. Guan, Q. Wang, X. He, Oscillation of a pantograph di?erential equation with impulsive perturbations, Appl. Math. Comput. , 219 (2012) 3147-3153.
  • [7] R. Hilfer, Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.
  • [8] A. Iserles, On the generalized pantograph functional di?erential equation, Eur. J. Appl. Math., 4 (1993) 1-38.
  • [9] A. A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Di?erential Equations, in: Mathematics Studies, vol.204, Elsevier, 2006.
  • [10] R. Kamocki, C. Obcznnski, On fractional Cauchy-type problems containing Hilfer derivative, Electron. J. Qual. Theory. Di?er. Equ., 50 (2016) 1–12.
  • [11] I. Podlubny, Fractional Di?erential Equations: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999.
  • [12] D. R. Smart, Fixed Point Theorems, Cambridge University Press, 1980.
  • [13] J.VanterlerdaC.Sousa, E.CapelasdeOliveira, Onthe?-Hilferfractionalderivative, Commun.Nonlinear. Sci. Numer. Simulat., 60 2018, 72-91.
  • [14] J. Vanterlerda C. Sousa, E. Capelas de Oliveira, On the Ulam-Hyers-Rassias satibility for nonlinear fractional di?erential equations using the ?-Hilfer operator, arXiv: 1711.07339, (2017).
  • [15] D. Vivek, K. Kanagarajan, S. Sivasundaram, Dynamics and stability of pantograph equationsvia Hilfer fractional derivative, Nonlinear Stud., 23 (2016) 685-698.
Year 2023, Volume: 11 Issue: 2, 97 - 103, 30.06.2023
https://doi.org/10.36753/mathenot.1057344

Abstract

Project Number

-

References

  • [1] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear. Sci. Numer. Simulat., 44 (2017) 460–481.
  • [2] S. Abbas, M. Benchohra, S. Sivasundaram, Dynamics and Ulam stability for Hilfer type fractional di?erential equations, Nonlinear Stud., 4 (2016) 627–637.
  • [3] I. Ahmed, P. Kumam, F. Jarad, P. Borisut, W. Jirakitpuwapat, On Hilfer generalized proportional fractional derivative, Adv. Di?er. Equ. 2020:329.
  • [4] K. Balachandran, S. Kiruthika, J.J. Trujillo, Existence of solutions of Nonlinear fractional pantograph equations, Acta Math. Sci., 33B (2013) 1-9.
  • [5] K. M. Furati, M. D. Kassim, N. E. Tatar, Existence and uniqueness for a problem involving HFD, Compur. Math. Appl., 64 (2012) 1616–1626.
  • [6] K. Guan, Q. Wang, X. He, Oscillation of a pantograph di?erential equation with impulsive perturbations, Appl. Math. Comput. , 219 (2012) 3147-3153.
  • [7] R. Hilfer, Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.
  • [8] A. Iserles, On the generalized pantograph functional di?erential equation, Eur. J. Appl. Math., 4 (1993) 1-38.
  • [9] A. A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Di?erential Equations, in: Mathematics Studies, vol.204, Elsevier, 2006.
  • [10] R. Kamocki, C. Obcznnski, On fractional Cauchy-type problems containing Hilfer derivative, Electron. J. Qual. Theory. Di?er. Equ., 50 (2016) 1–12.
  • [11] I. Podlubny, Fractional Di?erential Equations: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999.
  • [12] D. R. Smart, Fixed Point Theorems, Cambridge University Press, 1980.
  • [13] J.VanterlerdaC.Sousa, E.CapelasdeOliveira, Onthe?-Hilferfractionalderivative, Commun.Nonlinear. Sci. Numer. Simulat., 60 2018, 72-91.
  • [14] J. Vanterlerda C. Sousa, E. Capelas de Oliveira, On the Ulam-Hyers-Rassias satibility for nonlinear fractional di?erential equations using the ?-Hilfer operator, arXiv: 1711.07339, (2017).
  • [15] D. Vivek, K. Kanagarajan, S. Sivasundaram, Dynamics and stability of pantograph equationsvia Hilfer fractional derivative, Nonlinear Stud., 23 (2016) 685-698.
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Harikrishnan Sugumaran

Dvivek Vivek

Elsayed Elsayed 0000-0003-0894-8472

Project Number -
Publication Date June 30, 2023
Submission Date January 13, 2022
Acceptance Date December 31, 2022
Published in Issue Year 2023 Volume: 11 Issue: 2

Cite

APA Sugumaran, H., Vivek, D., & Elsayed, E. (2023). On the Study of Pantograph Differential Equations with Proportional Fractional Derivative. Mathematical Sciences and Applications E-Notes, 11(2), 97-103. https://doi.org/10.36753/mathenot.1057344
AMA Sugumaran H, Vivek D, Elsayed E. On the Study of Pantograph Differential Equations with Proportional Fractional Derivative. Math. Sci. Appl. E-Notes. June 2023;11(2):97-103. doi:10.36753/mathenot.1057344
Chicago Sugumaran, Harikrishnan, Dvivek Vivek, and Elsayed Elsayed. “On the Study of Pantograph Differential Equations With Proportional Fractional Derivative”. Mathematical Sciences and Applications E-Notes 11, no. 2 (June 2023): 97-103. https://doi.org/10.36753/mathenot.1057344.
EndNote Sugumaran H, Vivek D, Elsayed E (June 1, 2023) On the Study of Pantograph Differential Equations with Proportional Fractional Derivative. Mathematical Sciences and Applications E-Notes 11 2 97–103.
IEEE H. Sugumaran, D. Vivek, and E. Elsayed, “On the Study of Pantograph Differential Equations with Proportional Fractional Derivative”, Math. Sci. Appl. E-Notes, vol. 11, no. 2, pp. 97–103, 2023, doi: 10.36753/mathenot.1057344.
ISNAD Sugumaran, Harikrishnan et al. “On the Study of Pantograph Differential Equations With Proportional Fractional Derivative”. Mathematical Sciences and Applications E-Notes 11/2 (June 2023), 97-103. https://doi.org/10.36753/mathenot.1057344.
JAMA Sugumaran H, Vivek D, Elsayed E. On the Study of Pantograph Differential Equations with Proportional Fractional Derivative. Math. Sci. Appl. E-Notes. 2023;11:97–103.
MLA Sugumaran, Harikrishnan et al. “On the Study of Pantograph Differential Equations With Proportional Fractional Derivative”. Mathematical Sciences and Applications E-Notes, vol. 11, no. 2, 2023, pp. 97-103, doi:10.36753/mathenot.1057344.
Vancouver Sugumaran H, Vivek D, Elsayed E. On the Study of Pantograph Differential Equations with Proportional Fractional Derivative. Math. Sci. Appl. E-Notes. 2023;11(2):97-103.

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