Fractional integro-differential equations with nonlocal conditions and ψ–Hilfer fractional derivative
Abstract
Considering a fractional integro-differential equation with nonlocal conditions involving a general form of Hilfer fractional derivative with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation, we also prove the existence, uniqueness of solutions and Ulam-Hyers stability of this problem by employing a variety of tools of fractional calculus including Banach fixed point theorem and Krasnoselskii's fixed point theorem. An example is provided to illustrate our main results.
Keyword : fractional integro-differential equations, ψ -Hilfer fractional derivative, ψ-fractional integral, existence and and Ulam-Hyers stability, fixed point theorem, Mittag-Leffler function, least squares method
How to Cite
Abdo, M. S., Panchal, S. K., & Shafei Hussien, H. (2019). Fractional integro-differential equations with nonlocal conditions and ψ–Hilfer fractional derivative. Mathematical Modelling and Analysis, 24(4), 564-584. https://doi.org/10.3846/mma.2019.034
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M.S. Abdo and S.K.Panchal. Fractional integro-differential equations involving ψ-Hilfer fractional derivative. Advances in Applied Mathematics and Mechanics, 11(2):338–359, 2019. https://doi.org/10.4208/aamm.OA-2018-0143
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J. Vanterler da C. Sousa and E. Capelas 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
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K.M. Furati and M.D. Kassim. Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Applic., 64(6):1616–1626, 2012. https://doi.org/10.1016/j.camwa.2012.01.009
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R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. https://doi.org/10.1142/9789812817747
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K. Liu, J. Wang and D. O’Regan. Ulam-Hyers-Mittag-Leffler stability for ψHilfer fractional-order delay differential equations. Advances in Difference Equations, 2019(1):50, 2019. https://doi.org/10.1186/s13662-019-1997-4
D.S. Oliveira and E. Capelas de Oliveira. Hilfer-Katugampola fractional derivatives. Computat. Appl. Math., pp. 1–19, 2017.
I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, volume 198. Elsevier, 1998.
S.Z. Rida and H.S. Hussien. Efficient Mittag-Leffler collocation method for solving linear and nonlinear fractional differential equations. Mediterr. J. Math., 15(130), 2018. https://doi.org/10.1007/s00009-018-1174-0
S.G. Samko, A.A. Kilbas and O.I. Marichev. Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon, 1993.
D. Vivek, E. Elsayed and K. Kanagarajan. Theory and analysis of ψ-fractional differential equations with boundary conditions. Communications in Applied Analysis, 22:401–414, 2018.
M.S. Abdo, S.K. Panchal and A.M. Saeed. Fractional boundary value problem with ψ−Caputo fractional derivative. Proc. Indian Acad. Sci. (Math. Sci.), 129(5):65, 2019. https://doi.org/10.1007/s12044-019-0514-8
M.S. Abdo and S.K.Panchal. Fractional integro-differential equations involving ψ-Hilfer fractional derivative. Advances in Applied Mathematics and Mechanics, 11(2):338–359, 2019. https://doi.org/10.4208/aamm.OA-2018-0143
O.P. Agrawal. Some generalized fractional calculus operators and their applications in integral equations. Fract. Calc. Appl. Anal., 15(4):700–711, 2012. https://doi.org/10.2478/s13540-012-0047-7
R. Almeida. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul., 44:460–481, 2017. https://doi.org/10.1016/j.cnsns.2016.09.006
R. Almeida, A.B. Malinowska and M.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Method Appl. Sci., 41(1):336–352, 2018. https://doi.org/10.1002/mma.4617
T.A. Burton and C. Kirk. A fixed point theorem of KrasnoselskiiSchaefer type. Mathematische Nachrichten, 189(1):23–31, 2006. https://doi.org/10.1002/mana.19981890103
J. Vanterler da C. Sousa and E. Capelas de Oliveira. A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator. arXiv preprint arXiv:1709.03634, pp. 1–19, 2017.
J. Vanterler da C. Sousa and E. Capelas 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. Vanterler da C. Sousa and E. Capelas de Oliveira. Stability of the fractional Volterra integro-differential equation by means of ψ-Hilfer operator. arXiv preprint arXiv:1804.02601, pp. 1–15., 2018.
J. Vanterler da C. Sousa and E. Capelas de Oliveira. Ulam-Hyers-Rassias stability for a class of fractional integro-differential equations. Results in Mathematics, 73:111, 2018. https://doi.org/10.1007/s00025-018-0872-z
J. Vanterler da C. Sousa and E. Capelas de Oliveira. Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation. Applied Mathematics Letters, 81:50–56, 2018. https://doi.org/10.1016/j.aml.2018.01.016
J. Vanterler da C. Sousa, D. dos S. Oliveira and E. Capelas de Oliveira. On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation. Mathematical Methods in the Applied Sciences, pp. 1–13, 2018. https://doi.org/10.1002/mma.5430
J. Vanterler da C. Sousa, K.D. Kucche and E. Capelas de Oliveira. Stability of ψ-Hilfer impulsive fractional differential equations. Applied Mathematics Letters, 88:73–80, 2019. https://doi.org/10.1016/j.aml.2018.08.013
K.M. Furati and M.D. Kassim. Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Applic., 64(6):1616–1626, 2012. https://doi.org/10.1016/j.camwa.2012.01.009
S. Harikrishnan, E.M. Elsayed and K. Kanagarajan. Existence and uniqueness results for fractional pantograph equations involving ψ-Hilfer fractional derivative. Dynamics of Continuous, Discrete and Impulsive Systems, 25:319–328, 2018.
R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. https://doi.org/10.1142/9789812817747
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations. North-Holland Math Stud, 204 Elsevier, Amsterdam, 2006.
K.D. Kucche, A.D. Mali and J. Vanterler da C. Sousa. Theory of nonlinear ψHilfer fractional differential equations. arXiv preprint arXiv:1808.01608, 2018.
K. Liu, J. Wang and D. O’Regan. Ulam-Hyers-Mittag-Leffler stability for ψHilfer fractional-order delay differential equations. Advances in Difference Equations, 2019(1):50, 2019. https://doi.org/10.1186/s13662-019-1997-4
D.S. Oliveira and E. Capelas de Oliveira. Hilfer-Katugampola fractional derivatives. Computat. Appl. Math., pp. 1–19, 2017.
I. Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, volume 198. Elsevier, 1998.
S.Z. Rida and H.S. Hussien. Efficient Mittag-Leffler collocation method for solving linear and nonlinear fractional differential equations. Mediterr. J. Math., 15(130), 2018. https://doi.org/10.1007/s00009-018-1174-0
S.G. Samko, A.A. Kilbas and O.I. Marichev. Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon, 1993.
D. Vivek, E. Elsayed and K. Kanagarajan. Theory and analysis of ψ-fractional differential equations with boundary conditions. Communications in Applied Analysis, 22:401–414, 2018.