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Fractional Order Pseudoparabolic Partial Differential Equation: Ulam–Hyers Stability

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Abstract

Using Gronwall inequality we will investigate the Ulam-Hyers and generalized Ulam–Hyers–Rassias stabilities for the solution of a fractional order pseudoparabolic partial differential equation.

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References

  • Abbas, S., Benchohra, M.: Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order. Nonlinear Anal. Hybrid Syst. 4(3), 406–413 (2010a)

    Article  MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M.: Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay. Frac. Cal. Appl. Anal. 13(3), 225–244 (2010b)

    MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M.: Fractional order Riemann–Liouville integral inclusions with two independent variables and multiple delay. Opuscula Math. 33(2), 209–222 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M.: Ulam–Hyers stability for the Darboux problem for partial fractional differential and integro-differential equations via Picard operators. Results Math. 65(1–2), 67–79 (2014a)

    Article  MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M.: On the generalized Ulam–Hyers–Rassias stability for Darboux problem for partial fractional implicit differential equations. Appl. Math. E Notes 14, 20–28 (2014b)

    MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M.: Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses. Appl. Math. Comput. 257, 190–198 (2015)

    MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Gorniewicz, L.: Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative. Sci. Math. Jpn. 72, 271–282 (2010)

    MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Nieto, J.J.: Functional implicit hyperbolic fractional order differential equations with delay. Afr. Diaspora J. Math. New Ser. 15(1), 74–96 (2013a)

    MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Sivasundaram, S.: Ulam stability for partial fractional differential inclusions with multiple delays and impulses via Picard operators. Nonlinear Stud. 20(4), 623–641 (2013b)

    MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Nieto, J.J.: Ulam stabilities for partial impulsive fractional differential equations, Acta Universitatis Palackianae Olomucensis. Fac. Rerum Nat. Math. 53(1), 5–17 (2014a)

    MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Petrusel, A.: Ulam stability for partial fractional differential inclusions via Picard operators theory. Electron. J. Qual. Theor. Differ. Equations 2014(51), 1–13 (2014b)

    Article  MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Darwish, M.A.: New stability results for partial fractional differential inclusions with not instantaneous impulses. Frac. Cal. Appl. Anal. 18(1), 172–191 (2015a)

    MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Trujillo, J.J.: Upper and lower solutions method for partial fractional differential inclusions with not instantaneous impulses. Prog. Frac. Differ. Appl. 1(1), 11–22 (2015b)

    Google Scholar 

  • Abbas, S., Albarakati, W., Benchohra, M., Trujillo, J.J.: Ulam stabilities for partial Hadamard fractional integral equations. Arab. J. Math. 5(1), 1–7 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Lagreg, J.E., Alsaedi, A., Zhou, Y.: Existence and Ulam stability for fractional differential equations of Hilfer–Hadamard type. Adv. Differ. Equations 2017(1), 180 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Abbas, S., Benchohra, M., Graef, J.R., Henderson, J.: Implicit Fractional Differential and Integral Equations: Existence and Stability, vol. 26. Walter de Gruyter GmbH & Co KG, Berlin, Germany (2018)

    Book  MATH  Google Scholar 

  • Ahmad, M.Z., Hanan, I.K., Fadhel, F.S.: Stability of fractional order parabolic partial differential equations using discretized back stepping method. Malays. J. Fundam. Appl. Sci. 13(4), 612–618 (2017)

    Article  Google Scholar 

  • Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)

    Article  MathSciNet  Google Scholar 

  • Chen, P., Zhang, X., Li, Y.: Existence of mild solutions to partial differential equations with non-instantaneous impulses. Electron. J. Differ. Equations 241, 1–11 (2016)

    MathSciNet  MATH  Google Scholar 

  • Chuong, N.M., Ke, T.D., Quan, N.N.: Stability for a class of fractional partial integro-differential equations. J. Integr. Equations Appl. 26(2), 145–170 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • de Oliveira, E.C., Sousa, J.V.C.: Ulam–Hyers–Rassias stability for a class of fractional integro-differential equations. Results Math. 73, 111 (2018). https://doi.org/10.1007/s00025-018-0872-z

    Article  MathSciNet  MATH  Google Scholar 

  • Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific Publ. Comp, New Jersey (2014)

    Book  MATH  Google Scholar 

  • Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27(4), 222–224 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  • Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6(4), 1–15 (2014)

    MathSciNet  MATH  Google Scholar 

  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, Amsterdam (2006)

    Book  MATH  Google Scholar 

  • Long, H.V., Thao, H.T.P.: Hyers–Ulam stability for nonlocal fractional partial integro-differential equation with uncertainty. J. Intell. Fuzzy Syst. 24(1), 233–244 (2018)

    Article  Google Scholar 

  • Long, H.V., Son, N.T.K., Tam, H.T.T., Yao, J.-C.: Ulam stability for fractional partial integro-differential equation with uncertainty. Acta Math. Vietnam. 42(4), 675–700 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Lungu, N., Ciplea, S.A.: Ulam-Hyers-Rassias stability of pseudoparabolic partial differential equations. Carpath. J. Math. 31(2), 233–240 (2015)

    MathSciNet  MATH  Google Scholar 

  • Oliveira, D.S., Oliveira, E.: Capelas de: On a Caputo-type fractional derivative. Adv. Pure Appl. Math (2018) (accepted)

  • Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering, vol. 198. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  • Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications, vol. 1993. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  • Sousa, J.V.C., de Oliveira, E.C: A Gronwall inequality and the Cauchy-type problem by means of \(\psi \)-Hilfer operator (2017). arXiv:1709.03634 [math.CA]

  • Sousa, J.V.C., de Oliveira, E.C.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018a)

    Article  MathSciNet  Google Scholar 

  • Sousa, J. V.C., de Oliveira, E.C.: On a new operator in fractional calculus and applications (2018b). arXiv:1710.03712

  • Sousa, J.V.C., de Oliveira, E.C.: On the stability of a hyperbolic fractional partial differential equation (2018c). arXiv:1805.05546

  • Sousa, J.V.C., de Oliveira, E.C.: Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett. 81, 50–56 (2018d)

    Article  MathSciNet  MATH  Google Scholar 

  • Sousa, J.V.C., de Oliveira, E.C.: On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the \(\psi \)-Hilfer operator. Fixed Point Theory Appl. 20, 96 (2018e). https://doi.org/10.1007/s11784-018-0587-5

    Article  MathSciNet  MATH  Google Scholar 

  • Sousa, J.V.C., de Oliveira, E.C., Rodrigues, F.G.: Stability of the fractional Volterra integro-differential equation by means of \(\psi \)-Hilfer operator (2018). arXiv:1804.02601

  • Ulam, S.M.: A Collection of Mathematical Problems Interscience Publishers, pp. 665–666. Interscience, New York (1968)

  • Vivek, D., Kanagarajan, K., Elsayed, E.M.: Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions. Mediterr. J. Math. 15(1), 15 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • Yan, Z., Zhang, H.: Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with state-dependent delay. Electron. J. Differ. Equations 2013(206), 1–29 (2013)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We remain indebted to the anonymous referee. Your comments and suggestions have improved the paper unequivocally.

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Authors and Affiliations

  1. Department of Applied Mathematics, Imecc-Unicamp, Campinas, SP, 13083-859, Brazil

    J. Vanterler da C. Sousa & E. Capelas de Oliveira

Authors
  1. J. Vanterler da C. Sousa
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  2. E. Capelas de Oliveira
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Correspondence to J. Vanterler da C. Sousa.

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Sousa, J.V.d.C., Oliveira, E.C.d. Fractional Order Pseudoparabolic Partial Differential Equation: Ulam–Hyers Stability . Bull Braz Math Soc, New Series 50, 481–496 (2019). https://doi.org/10.1007/s00574-018-0112-x

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  • DOI: https://doi.org/10.1007/s00574-018-0112-x

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