Differential equations with tempered Ψ-Caputo fractional derivative
Abstract
In this paper we define a new type of the fractional derivative, which we call tempered Ψ−Caputo fractional derivative. It is a generalization of the tempered Caputo fractional derivative and of the Ψ−Caputo fractional derivative. The Cauchy problem for fractional differential equations with this type of derivative is discussed and some existence and uniqueness results are proved. We present a Henry-Gronwall type inequality for an integral inequality with the tempered Ψ−fractional integral. This inequality is applied in the proof of an existence theorem. A result on a representation of solutions of linear systems of Ψ−Caputo fractional differential equations is proved and in the last section an example is presented.
Keyword : tempered Riemann-Liouville fractional derivative, tempered Ψ−Caputo fractional derivative
How to Cite
Medveď, M., & Brestovanská, E. (2021). Differential equations with tempered Ψ-Caputo fractional derivative. Mathematical Modelling and Analysis, 26(4), 631-650. https://doi.org/10.3846/mma.2021.13252
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M.S. Abdo, S.K. Panchal and A.K. Saeed. Fractional boundary value problem with ψ−Caputo fractional derivative. Proc. Indian Acad. Sci., 129(65):64–78, 2019. https://doi.org/10.1007/s12044-019-0514-8
M.S. Abdo, S.T. Thabet and B. Ahmad. The existence and Ulam-Hyers stability results for ψ−Hilfer integrodifferential equations. J. Pseudo-Differ. Oper. Appl., 11(131):1757–1780, 2020. https://doi.org/10.1007/s11868-020-00355-x
R. Almeida. A Caputo fractional derivative of a function with respect to another function. Commun. in Nonlinear Sci. and Numer. Simulation, 44:460–481, 2017. https://doi.org/10.1016/j.cnsns.2016.09.006
R. Almeida. A Gronwall inequality for general Caputo fractional operator. Math. Ineq. and Appl., 20(4):1089–1105, 2017. https://doi.org/10.7153/mia-2017-2070
R. Almeida, A.B. Malinowska and M.T.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci., 41:336–352, 2018. https://doi.org/10.1002/mma.4617
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M. Fečkan, J.R. Wang and M. Pospíšil. Fractional-Order Equations and Inclusions. De Gruyter, Berlin, Boston, 2017. https://doi.org/10.1515/9783110522075
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R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. https://doi.org/10.1142/9789812817747
M. Jamaki, K. Kanagarajan and E.M. Elsayed. Katugampola-type fractional differential equations with delay and impulses. Mathematical and Theoretical Physics, 1(3):73–77, 2018. https://doi.org/10.15406/oajmtp.2018.01.00012
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A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204. Amsterdam, Boston, Heidelberg, London, 2006.
K.D. Kucche and J.P. Kharade. Analysis of impulsive φ−Hilfer fractional differential equations. Mediterr. J. Math., 17(5), 2020. https://doi.org/10.1007/s00009-020-01575-7
K.D. Kucche, A.D. Mali and V.C. Da Sousa. On the nonlinear ψ−Hilfer fractional differential equations. Computational and Appl.Math., 38(2), 2019. https://doi.org/10.1007/s40314-019-0833-5
V. Lakshmikantham, L. Leela and J. Vasundra Devi. Theory of Fractional Dynamic Systems. Cambridge Scientific Publ., Cambridge, 2009.
C. Li, W. Deng and L. Zhao. Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations. Discrete and Continuous Dynamical Systems, Series B, 24(4):1989–2015, 2019. https://doi.org/10.3934/dcdsb.2019026
J. Liouville. Memoire sur quelques question de geometrie et de mecanique et sur un nouveau genre de calcul pour resudre ces question. J. Ecole Polytech., 13(21):1–69, 1832.
K. Liu, J.R. Wang and D. O’Regan. Ulam-Hyers-Mittag-Leffler stability for ψ−Hilfer fractional-order delay differential equations. Adv. Differ. Equ., 50:1– 12, 2019. https://doi.org/10.1186/s13662-019-1997-4
M. Medveď. A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl., 214(2):349–366, 1997. https://doi.org/10.1006/jmaa.1997.5532
M. Medveď. Integral inequalities and global solutions of semilinear evolution equations. J. Math. Anal. Appl., 267(2):634–650, 2002. https://doi.org/10.1006/jmaa.2001.7798
M. Medveď and E. Brestovanská. Corrigendum to the paper: New conditions for the exponential stability of fractionally perturbed ODEs. Electron. J. Qual. Theory Differ. Equ., 102:1–2, 2018. https://doi.org/10.14232/ejqtde.2018.1.102
M. Medveď and E. Brestovanská. New conditions for the exponential stability of fractionally perturbed ODEs. Electron. J. Qual. Theory Differ. Equ., 84:1–14, 2018. https://doi.org/10.14232/ejqtde.2018.1.84
M. Medveď and E. Brestovanská. Sufficient conditions for the exponential stability of nonlinear fractionally perturbed ODEs with multiple delays. Fract. Differ. Calc., 9(2):263–278, 2019. https://doi.org/10.7153/fdc-2019-09-17
M. Medveď and M. Pospíšil. On the existence and exponential stability for differential equations with multiple constant delays and nonlinearity depending on fractional substantial integrals. Electron. J. Qual. Theory Differ. Equ., 43:1– 17, 2019. https://doi.org/10.14232/ejqtde.2019.1.43
M.L. Morgado and M. Rebelo. Well-posedness and numerical approximation of tempered fractional terminal value problems. Fract. Calc. Appl. Anal., 20(5):1239–1262, 2017. https://doi.org/10.1515/fca-2017-0065
I. Petráš. Fractional-Order Nonlinear Systems, Modeling, Analysis and Simulation. Springer, Heidelberg, Dortrecht, London, New York, 2011. https://doi.org/10.1007/978-3-642-18101-6
E. Picard. Mémoire sur la theórie des equations aux dérivés partielles et la méthode des approximations successives. J. de Math. Pures Appl., 231(6):145– 210, 1890.
I. Podlubný. Fractional Differential Equations. Academic Press, San Diego, 1999.
B. Riemann. Versuch einer allgemainen auffasung der integration und differentiation. Gesammelte Werke, 1876.
F. Sabzibar, M.M. Meerschaert and J. Chen. Tempered fractional calculus. J. Comput. Physics, 293:14–28, 2015. https://doi.org/10.1016/j.jcp.2014.04.024
H. Schaefer. Über die Methode der a Priori-Schranken. Math. Ann., 129:415– 416, 1955. https://doi.org/10.1007/BF01362380
J.C. Sousa and E.C. De Oliveira. On the ψ−Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul., 60:72–91, 2018. https://doi.org/10.1016/j.cnsns.2018.01.005
J.C. Sousa and E.C. De Oliveira. On the Ulam-Hyers-Rassias stability of nonlinear differential equations using the ψ−Hilfer operator. J. Fixed Point Theory and Appl., 20(3):1–21, 2018. https://doi.org/10.1007/s11784-018-0587-5
J.C. Sousa and E.C. De Oliveira. A Gronwall inequality and the Cauchy-type problems by means of ψ−Hilfer operator. Differ. Equ. Appl., 11(1):87–106, 2019. https://doi.org/10.7153/dea-2019-11-02
J.C. Sousa and E.C. De Oliveira. On the ψ− fractional integral and applications. Comp. Appl. Math., 38(4), 2019. https://doi.org/10.1007/s40314-019-0774-z
J.C. Sousa, K.D. Kucche and E.C. De Oliveira. Stability of ψ−Hilfer impulsive fractional equations. Applied Mathematics Letters, 88:73–80, 2019. https://doi.org/10.1016/j.aml.2018.08.013
J.C. Sousa, F.G. Rodrigues and E.C. De Oliveira. Stability of the Volterra integro-differential equations by means of ψ−Hilfer operator. Mathematical Methods in the Applied Sciences, 12, 2019. https://doi.org/10.1002/mma.5563
V.E. Tarasov. Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles. Fields and Media. Springer, New York, 2010.
D. Vivek, E.M. Elsayed and K. Kanagarajan. Theory and analysis of ψ−fractional differential equations with boundary conditions. Commun. Appl. Anal., 22(3):401–414, 2018.
H. Ye, J. Gao and Y. Ding. A generalized Gronwall inequality and its application to a fractional differential equations. J. Math. Anal. Appl., 328(2):1075–1081, 2007. https://doi.org/10.1016/j.jmaa.2006.05.061
Y. Zhou. Basic Theory of Fractional Differential Equations. World Scientific, New Jersey, London, Singapore, 2014. https://doi.org/10.1142/9069
M.S. Abdo, A.G. Ibrahim and S.K. Panchal. Nonlinear implicit fractional differential equation involving ψ−Caputo fractional derivative. Proceedings of the Jangjeon Mathematical Society, 22(3):387–400, 2019.
M.S. Abdo, A.G. Ibrahim and H.A. Wahash. Ulam–Hyers–Mittag-Leffler stability for ψ−Hilfer problem with fractional order and infinite delay. Results in Appl. Math., 7:100115, 2020. https://doi.org/10.1016/j.rinam.2020.100115
M.S. Abdo, K.Shah, S.K. Panchal and H.A. Wahash. Existence and Ulam stability results of a coupled system for terminal value problems involving ψ−Hilfer fractional operator. Adv. Differ. Equ., 2020(316), 2020. https://doi.org/10.1186/s13662-020-02775-x
M.S. Abdo, S.K. Panchal and H.S. Hussein. Fractional integro-differential equations with nonlocal conditions and ψ−Hilfer fractional derivative. Proc. Indian Acad. Sci., 24(4):564–584, 2019. https://doi.org/10.3846/mma.2019.034
M.S. Abdo, S.K. Panchal and A.K. Saeed. Fractional boundary value problem with ψ−Caputo fractional derivative. Proc. Indian Acad. Sci., 129(65):64–78, 2019. https://doi.org/10.1007/s12044-019-0514-8
M.S. Abdo, S.T. Thabet and B. Ahmad. The existence and Ulam-Hyers stability results for ψ−Hilfer integrodifferential equations. J. Pseudo-Differ. Oper. Appl., 11(131):1757–1780, 2020. https://doi.org/10.1007/s11868-020-00355-x
R. Almeida. A Caputo fractional derivative of a function with respect to another function. Commun. in Nonlinear Sci. and Numer. Simulation, 44:460–481, 2017. https://doi.org/10.1016/j.cnsns.2016.09.006
R. Almeida. A Gronwall inequality for general Caputo fractional operator. Math. Ineq. and Appl., 20(4):1089–1105, 2017. https://doi.org/10.7153/mia-2017-2070
R. Almeida, A.B. Malinowska and M.T.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci., 41:336–352, 2018. https://doi.org/10.1002/mma.4617
G. Butler and T. Rogers. A generalization of a lemma of Bihari and applications to piecewise estimates for integral equations. J. Math. Anal. Appl., 33(1):77–81, 1971. https://doi.org/10.1016/0022-247X(71)90183-1
M. Chen and W. Deng. Discrete fractional substantial calculus. ESAIM: Math. Modell. Numer., 49(2):373–394, 2015. https://doi.org/10.1051/m2an/2014037
M. Fečkan, J.R. Wang and M. Pospíšil. Fractional-Order Equations and Inclusions. De Gruyter, Berlin, Boston, 2017. https://doi.org/10.1515/9783110522075
S. Harikrishnan, K. Kanagarajan and D. Vivek. Existence and stability results for boundary value problem for differential equation with ψ−Hilfer fractional derivative. J. Appl. Nonlinear Dynamics, 8(2):251–259, 2019. https://doi.org/10.5890/JAND.2019.06.008
D. Henry. Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, Berlin, Heidelberg, New York, 1981. https://doi.org/10.1007/BFb0089647
R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. https://doi.org/10.1142/9789812817747
M. Jamaki, K. Kanagarajan and E.M. Elsayed. Katugampola-type fractional differential equations with delay and impulses. Mathematical and Theoretical Physics, 1(3):73–77, 2018. https://doi.org/10.15406/oajmtp.2018.01.00012
U.N. Katugampola. New approach to a generalized fractional integral. Appl. Math. Comput., 218(3):860–865, 2011. https://doi.org/10.1016/j.amc.2011.03.062
A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204. Amsterdam, Boston, Heidelberg, London, 2006.
K.D. Kucche and J.P. Kharade. Analysis of impulsive φ−Hilfer fractional differential equations. Mediterr. J. Math., 17(5), 2020. https://doi.org/10.1007/s00009-020-01575-7
K.D. Kucche, A.D. Mali and V.C. Da Sousa. On the nonlinear ψ−Hilfer fractional differential equations. Computational and Appl.Math., 38(2), 2019. https://doi.org/10.1007/s40314-019-0833-5
V. Lakshmikantham, L. Leela and J. Vasundra Devi. Theory of Fractional Dynamic Systems. Cambridge Scientific Publ., Cambridge, 2009.
C. Li, W. Deng and L. Zhao. Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations. Discrete and Continuous Dynamical Systems, Series B, 24(4):1989–2015, 2019. https://doi.org/10.3934/dcdsb.2019026
J. Liouville. Memoire sur quelques question de geometrie et de mecanique et sur un nouveau genre de calcul pour resudre ces question. J. Ecole Polytech., 13(21):1–69, 1832.
K. Liu, J.R. Wang and D. O’Regan. Ulam-Hyers-Mittag-Leffler stability for ψ−Hilfer fractional-order delay differential equations. Adv. Differ. Equ., 50:1– 12, 2019. https://doi.org/10.1186/s13662-019-1997-4
M. Medveď. A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl., 214(2):349–366, 1997. https://doi.org/10.1006/jmaa.1997.5532
M. Medveď. Integral inequalities and global solutions of semilinear evolution equations. J. Math. Anal. Appl., 267(2):634–650, 2002. https://doi.org/10.1006/jmaa.2001.7798
M. Medveď and E. Brestovanská. Corrigendum to the paper: New conditions for the exponential stability of fractionally perturbed ODEs. Electron. J. Qual. Theory Differ. Equ., 102:1–2, 2018. https://doi.org/10.14232/ejqtde.2018.1.102
M. Medveď and E. Brestovanská. New conditions for the exponential stability of fractionally perturbed ODEs. Electron. J. Qual. Theory Differ. Equ., 84:1–14, 2018. https://doi.org/10.14232/ejqtde.2018.1.84
M. Medveď and E. Brestovanská. Sufficient conditions for the exponential stability of nonlinear fractionally perturbed ODEs with multiple delays. Fract. Differ. Calc., 9(2):263–278, 2019. https://doi.org/10.7153/fdc-2019-09-17
M. Medveď and M. Pospíšil. On the existence and exponential stability for differential equations with multiple constant delays and nonlinearity depending on fractional substantial integrals. Electron. J. Qual. Theory Differ. Equ., 43:1– 17, 2019. https://doi.org/10.14232/ejqtde.2019.1.43
M.L. Morgado and M. Rebelo. Well-posedness and numerical approximation of tempered fractional terminal value problems. Fract. Calc. Appl. Anal., 20(5):1239–1262, 2017. https://doi.org/10.1515/fca-2017-0065
I. Petráš. Fractional-Order Nonlinear Systems, Modeling, Analysis and Simulation. Springer, Heidelberg, Dortrecht, London, New York, 2011. https://doi.org/10.1007/978-3-642-18101-6
E. Picard. Mémoire sur la theórie des equations aux dérivés partielles et la méthode des approximations successives. J. de Math. Pures Appl., 231(6):145– 210, 1890.
I. Podlubný. Fractional Differential Equations. Academic Press, San Diego, 1999.
B. Riemann. Versuch einer allgemainen auffasung der integration und differentiation. Gesammelte Werke, 1876.
F. Sabzibar, M.M. Meerschaert and J. Chen. Tempered fractional calculus. J. Comput. Physics, 293:14–28, 2015. https://doi.org/10.1016/j.jcp.2014.04.024
H. Schaefer. Über die Methode der a Priori-Schranken. Math. Ann., 129:415– 416, 1955. https://doi.org/10.1007/BF01362380
J.C. Sousa and E.C. De Oliveira. On the ψ−Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul., 60:72–91, 2018. https://doi.org/10.1016/j.cnsns.2018.01.005
J.C. Sousa and E.C. De Oliveira. On the Ulam-Hyers-Rassias stability of nonlinear differential equations using the ψ−Hilfer operator. J. Fixed Point Theory and Appl., 20(3):1–21, 2018. https://doi.org/10.1007/s11784-018-0587-5
J.C. Sousa and E.C. De Oliveira. A Gronwall inequality and the Cauchy-type problems by means of ψ−Hilfer operator. Differ. Equ. Appl., 11(1):87–106, 2019. https://doi.org/10.7153/dea-2019-11-02
J.C. Sousa and E.C. De Oliveira. On the ψ− fractional integral and applications. Comp. Appl. Math., 38(4), 2019. https://doi.org/10.1007/s40314-019-0774-z
J.C. Sousa, K.D. Kucche and E.C. De Oliveira. Stability of ψ−Hilfer impulsive fractional equations. Applied Mathematics Letters, 88:73–80, 2019. https://doi.org/10.1016/j.aml.2018.08.013
J.C. Sousa, F.G. Rodrigues and E.C. De Oliveira. Stability of the Volterra integro-differential equations by means of ψ−Hilfer operator. Mathematical Methods in the Applied Sciences, 12, 2019. https://doi.org/10.1002/mma.5563
V.E. Tarasov. Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles. Fields and Media. Springer, New York, 2010.
D. Vivek, E.M. Elsayed and K. Kanagarajan. Theory and analysis of ψ−fractional differential equations with boundary conditions. Commun. Appl. Anal., 22(3):401–414, 2018.
H. Ye, J. Gao and Y. Ding. A generalized Gronwall inequality and its application to a fractional differential equations. J. Math. Anal. Appl., 328(2):1075–1081, 2007. https://doi.org/10.1016/j.jmaa.2006.05.061
Y. Zhou. Basic Theory of Fractional Differential Equations. World Scientific, New Jersey, London, Singapore, 2014. https://doi.org/10.1142/9069