On the Study of Pantograph Differential Equations with Proportional Fractional Derivative
Year 2023,
Volume: 11 Issue: 2, 97 - 103, 30.06.2023
Harikrishnan Sugumaran
,
Dvivek Vivek
,
Elsayed Elsayed
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.
References
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- [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
Harikrishnan Sugumaran
,
Dvivek Vivek
,
Elsayed Elsayed
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.