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Fractional Differential Equations with Mixed Boundary Conditions

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Abstract

In this paper, we discuss the existence and uniqueness of solutions of a boundary value problem for a fractional differential equation of order \(\alpha \in (2,3)\), involving a general form of fractional derivative. First, we prove an equivalence between the Cauchy problem and the Volterra equation. Then, two results on the existence of solutions are proven, and we end with some illustrative examples.

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References

  1. Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agarwal, R., Hristova, S., O’Regan, D.: A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19, 290–318 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahmad, B., Ntouyas, S.K.: A new kind of nonlocal-integral fractional boundary value problems. Bull. Malays. Math. Sci. Soc. 39, 1343–1361 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Almeida, R., Malinowska, A. B., Monteiro, M. T. T.: Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci. (in press)

  5. 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 

  6. Al-Saqabi, B., Kiryakova, V.S.: Explicit solutions of fractional integral and differential equations involving Erdelyi–Kober operators. Appl. Math. Comput 95, 1–13 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Didgar, M., Ahmadi, N.: An efficient method for solving systems of linear ordinary and fractional differential equations. Bull. Malays. Math. Sci. Soc. 38, 1723–1740 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Douglas, J.F.: Some Applications of Fractional Calculus to Polymer Science. Advances in Chemical Physics. Wiley, Hoboken (2007)

    Google Scholar 

  10. Engheta, N.: On fractional calculus and fractional multipoles in electromagnetism. IEEE Trans. Antennas Propag. 44(4), 554–566 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fellah, Z.E.A., Depollier, C.: Application of fractional calculus to the sound waves propagation in rigid porous materials: validation via ultrasonic measurement. Acta Acust. 88, 34–39 (2002)

    Google Scholar 

  12. Kempfle, S., Schäfer, I., Beyer, H.: Fractional differential equations and viscoelastic damping. In: Porto, J.L. Martins de Carvalho, F.A., Fontes, C. C., de Pinho M.D.R. (eds.) Proceedings of the European Control Conference 2001, pp. 1744–1751 (2001)

  13. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)

  14. Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, M., Wang, J.: Existence of local and global solutions for Hadamard fractional differential equations. Electron. J. Differ. Equ. 2015, 1–8 (2015)

    Article  MathSciNet  Google Scholar 

  16. Lokenath, D.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003(54), 3413–3442 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Malinowska, A.B., Odzijewicz, T., Torres, D.F.M.: Advanced Methods in the Fractional Calculus of Variations. Springer Briefs in Applied Sciences and Technology. Springer, Cham (2015)

    Google Scholar 

  18. Qarout, D., Ahmad, B., Alsaedi, A.: Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions. Fract. Calc. Appl. Anal. 19, 463–479 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, translated from the 1987 Russian original. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  20. Wang, J., Dong, X., Zhou, Y.: Analysis of nonlinear integral equations with Erdelyi-Kober fractional operator. Commun. Nonlinear Sci. Numer. Simul. 17, 3129–3139 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wang, J., Zhou, Y., Medved, M.: Existence and stability of fractional differential equations with Hadamard derivative. Topol. Methods Nonlinear Anal. 41, 113–133 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Xu, Y.: Fractional boundary value problems with integral and anti-periodic boundary conditions. Bull. Malays. Math. Sci. Soc. 39, 571–587 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, X., Zhong, Q.: Multiple positive solutions for nonlocal boundary value problems of singular fractional differential equations. Bound. Value Probl. 21 Article Number 65 (2016). https://doi.org/10.1186/s13661-016-0572-0

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Acknowledgements

Work supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013. The author is very grateful to two anonymous referees, for valuable remarks and comments that improved this paper.

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Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193, Aveiro, Portugal

    Ricardo Almeida

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  1. Ricardo Almeida
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Correspondence to Ricardo Almeida.

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Communicated by See Keong Lee.

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Almeida, R. Fractional Differential Equations with Mixed Boundary Conditions. Bull. Malays. Math. Sci. Soc. 42, 1687–1697 (2019). https://doi.org/10.1007/s40840-017-0569-6

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  • DOI: https://doi.org/10.1007/s40840-017-0569-6

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