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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Existence and Uniqueness Solutions of a BVP for Nonlinear Caputo-Hadamard Fractional Differential Equation

Journal of Applied Nonlinear Dynamics 11(2) (2022) 359--374 | DOI:10.5890/JAND.2022.06.007

Abdelatif Boutiara$^1$, Maamar Benbachir$^2$, Kaddour Guerbati$^1$

$^1$ Laboratoire de Math'{e}matiques et Sciences appliqu'{e}es, University of Ghardaia, Algeria

$^2$ Faculty of Sciences, Saad Dahlab University, Blida 1, Algeria

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Abstract

In this paper, we study the existence and uniqueness of solutions for a nonlinear fractional boundary value problem with Caputo-Hadamard derivative. New existence and uniqueness results are established using Banach contraction principle. Other existence results are obtained using scheafer's and krasnoselskii's fixed point theorems. At the end, some illustrative examples are presented.

References

  1. [1]  Ahmad, B. and Ntouyas, S.K. (2015), Initial value problems of fractional order Hadamard-type functional differential equations, Electron, J. Differ. Equ., 77, Zbl 1320.34109.
  2. [2]  Ahmad, B. and Ntouyas, S.K. (2014), A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal., 17, 348-360. MR3181059.
  3. [3]  Alsaedi, A., Ntouyas, S.K., Ahmad, B., and Hobiny, A. (2015), Nonlinear Hadamard fractional differential equations with Hadamard type nonlocal non-conserved conditions, Advances in Difference Equations, 285, Zbl 1351.34003.
  4. [4]  Benchohra, M., Hamani, S., and Ntouyas, S.K. (2008), Boundary value problems for differential equations with fractional order, Surveys in Mathematics and its Applications, 3, 1-12. MR2532767. Zbl 1157.26301.
  5. [5]  Benhamida, W., Hamani, S., and Henderson, J. (2016), A boundary value problem for fractional differential equations with Hadamard derivative and nonlocal conditions, PanAmerican Math. J., 26, 1-11.
  6. [6]  Cao, Y., Ma, W.G., and Ma, L.C. (2014), Local fractional functional method for solving diffusion equations on Cantor sets, Abstract and Applied Analysis, (Vol. 2014).
  7. [7]  Gao, F. (2017), General fractional calculus in non-singular power-law kernel applied to model anomalous diffusion phenomena in heat transfer problems, Thermal Science, 21(Suppl. 1), pp. S11-S18.
  8. [8]  Yang, X.J., Ragulskis, M., and Taha, T. (2019), A new general fractional-order derivative with Rabotnov fractional-exponential kernel, Thermal Science, 23(6B), pp. 3711-3718
  9. [9]  Yang, X.J., Srivastava, H.M., and Tenreiro Marchado, J.A. (2015), A new fractional derivative without singular kernel: application to the modelling of the steady heat flow, Thermal Science, 20(2), 753-756.
  10. [10]  Yang, X.J. (2019), New non-conventional methods for quantitative concepts of anomalous rheology, Thermal Science, 23(6B), 4117-4127.
  11. [11]  Yang, X.J. (2019), New general calculi with respect to another functions applied to describe the newton-like dashpot models in anomalous viscoelasticity, Thermal Science, 23(6B), 3751-3757.
  12. [12]  Yang, X.J. (2019), General Fractional Derivatives: Theory, Methods and Applications, New York: CRC Press.
  13. [13]  Yang, X.J. and Jose, T.M. (2019), A new fractal nonlinear Burgers' equation arising in the acoustic signals propagation. Mathematical Methods in the Applied Sciences, 42(18), 7539-7544.
  14. [14]  Yang, X.J., Feng, G., Yang, J., and Zhou, H.W. (2018), Fundamental solutions of the general fractional-order diffusion equations, Mathematical Methods in the Applied Sciences, 41(18), pp. 9312-9320.
  15. [15]  Yang, A.M., Han, Y., Li, J., and Liu, W.X. (2016), On steady heat flow problem involving Yang-Srivastava-Machado fractional derivative without singular kernel, Thermal Science, 20(suppl 3), S719-S723.
  16. [16]  Yang, X.J., Gao, F., and Jing, H.W. (2019), New mathematical models in anomalous viscoelasticity from the derivative with respect to another function view point, Thermal Science, 23(3A), 1555-1561.
  17. [17]  Yang, X.J., Gao, F., and Yang, J. General fractional derivatives with applications in viscoelasticity. Academic Press.
  18. [18]  Ahmad, B. and Ntouyas, S.K. (2015), On Hadamard fractional integro-differential boundary value problems, J. Appl. Math. Comput, Zbl 1328.34006.
  19. [19]  Arioua, Y. and Benhamidouche, N. (2017), Boundary value problem for Caputo-Hadamard fractional differential equations, Surveys in Mathematics and its Applications, 12, 103-115.
  20. [20]  Su, C.M., Sun, J.P., and Zhao, Y.H. (2017), Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation, Hindawi International Journal of Differential Equations, Volume 2017, Article ID 4683581, 7 pages.
  21. [21]  Butzer, P.L., Kilbas, A.A., and Trujillo, J.J. (2002), Composition of Hadamard-type fractional integration operators and the semi-group property, J. Math. Anal. Appl., 269, 387-400.
  22. [22]  Gambo, Y.Y., Jarad, F., Baleanu, D., and Abdeljawad, T. (2014), On Caputo modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2014, Paper No. 10, 12 p. Zbl 1343.26002.
  23. [23]  Kilbas, A.A., Srivastava, H.H., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier Science B.V. Amsterdam, MR2218073(2007a:34002). Zbl 1092.45003.
  24. [24]  Belbali, H. and Benbachir, M. (2021), Stability for coupled systems on networks with Caputo-Hadamard fractional derivative, Journal of Mathematical Modeling, 9(1), pp. 107–118.
  25. [25]  Boutiara, A., Benbachir, M., and Guerbati, K. (2020), Caputo type fractional differential equation with non local Erdelyi-Kober type integral boundary conditions in Banach spaces, Surveys in Mathematics and its Applications, 15, 99–418.
  26. [26]  Boutiara, A., Benbachir, M., and Guerbati, K. (2019), Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Mathematics, 5(1), 259–272.
  27. [27]  Thiramanus, P., Ntouyas, S.K., and Tariboon, J. (2014), Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal. Art., ID 902054, 9 pp.
  28. [28]  Wafaa, B., Samira, H., and Johnny, H. (2018), Boundary Value Problems For Caputo-Hadamard Fractional Differential Equations, Advances in the Theory of Nonlinear Analysis and its Applications, 2(3), 138-145.
  29. [29]  Wang, J. and Zhang, Y. (2015), On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett., 3985-90.
  30. [30]  Yukunthorn, W., Ahmad, B., Ntouyas, S.K., and Tariboon, J. (2016), On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Analysis: Hybrid Systems, 19, 77-92
  31. [31]  Jarad, F., Abdeljawad, T., and Baleanu, D. (2012), Caputo-type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2012(142). Zbl1346.26002.
  32. [32]  Krasnoselskii, M.A. (1955), Two remarks on the method of successive approximations, Uspekhi Mat. Nauk, 10, 123-127. %
  33. [33]  Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993), Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach Science Publishers, Switzerland. MR1347689(96d:26012). Zbl 0818.26003. % % %
  34. [34]  Yang, X.J., Yi-Ying, F., Carlo, C., and Mustafa, I. (2019), Fundamental solutions of anomalous diffusion equations with the decay exponential kernel, Mathematical Methods in the Applied Sciences, 42(11), 4054-4060.

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