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Year 2020, , 121 - 128, 29.09.2020
https://doi.org/10.32323/ujma.756304

Abstract

References

  • [1] M. D. Qassim, K. M. Furati, N-e. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative. Abstract Appl. Anal. 2012, 2012:17, Article ID 391062.
  • [2] Ahmad Y. A. Salamooni, D. D. Pawar, Existence and uniqueness of boundary value problems for Hilfer-Hadamard-type fractional differential equations. arXiv:1801.10400v1[math.AP] 31 Jan (2018).
  • [3] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, (1999).
  • [4] S. G. Samko, A. A. Kilbas; O. I. Marichev; Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). [Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)]
  • [5] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Editor: Jan Van Mill, Elsevier, Amsterdam, The Netherlands, (2006).
  • [6] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific Publ. Co., Singapore, (2000).
  • [7] R. Hilfer, Threefold introduction to fractional derivatives. In: Anomalous transport: foundations and applications, 2008, 17-73.
  • [8] J. A. Nanware, D. B. Dhaigude, Existence and uniqueness of solution of Riemann-Liouville fractional differential equations with integral boundary conditions. Int. J. Nonlinear Sci., 14, (2012), 410-415.
  • [9] M. D. Kassim and N. E. Tatar, Well-Posedness and Stability for a Differential Problem with Hilfer-Hadamard Fractional Derivative. Abst. Appl. Anal., 2013, (2013), 1-12, Article ID 605029.
  • [10] R. Hilfer, Y. Luchko, ˇZ: Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Cal. Appl. Anal., 12, (2009), 299-318.
  • [11] K. Diethelm, N. J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl., 265, (2002), 229-248.
  • [12] K. M. Furati, M. D. Kassim and N.e.Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative. Computers Math. Appl., (2012), 1616-1626.
  • [13] C. Kou, J. Liu, and Y. Ye, Existence and uniqueness of solutions for the Cachy-type problems of fractional differential equaitions. Discrete Dyn. Nat. Soc., Article ID 142175, (2010), 1-15.
  • [14] H.M. Srivastava, Z: Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput., 211, (2009), 198-210.
  • [15] Zivorad Tomovski, R. Hilfer, H. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transforms Spec. Funct., 21, (11), (2010), 797-814.
  • [16] J. A. Nanware, D. B. Dhaigude, Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions. J. Nonlinear Sci. Appl., 7, (2014), 246-254.
  • [17] Ahmad Y. A. Salamooni, D. D. Pawar, Existence and uniqueness of nonlocal boundary conditions for Hilfer-Hadamard-type fractional differential equations. arXiv:1802.04262v1[math.AP] 12 Feb (2018).
  • [18] D. Vivek, K. Kanagarajan and E. M. Elsayed, Some Existence and Stability Results for Hilfer-fractional Implicit Differential Equations with Nonlocal Conditions. Mediterr. J. Math. 2018 15:15, https://doi.org/10.1007/s00009-017-1061-0, ©Springer International Publishing AG, part of Springer Nature 2018.
  • [19] S. Abbas, M. Benchohra, , M. Bohner, Weak solutions for implicit differential equations with Hilfer–Hadamard fractional derivative. Adv. Dyn. Syst. Appl., 12, (2017), 1-16.
  • [20] S. Abbas, M. Benchohra, J.-E. Lazreg, Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: analysis and stability. Chaos Solitons Fractals, 102, (2017), 47-71.
  • [21] A. Y. A. Salamooni, D. D. Pawar, Unique positive solution for nonlinear Caputo-type fractional q􀀀difference equations with nonlocal and Stieltjes integral boundary conditions. Fractional Differential Calculus, 9(2), (2019), 295-307.

Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem

Year 2020, , 121 - 128, 29.09.2020
https://doi.org/10.32323/ujma.756304

Abstract

In this paper, we study the existence and uniqueness of Generalized Fractional Cauchy-type problem involving Hilfer-Hadamard-type fractional derivative for a nonlinear fractional differential equation. Also, we prove an equivalence between the Cauchy-type problem and Volterra integral equation(VIE).                                                                                                                                                                                                                                                                               

References

  • [1] M. D. Qassim, K. M. Furati, N-e. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative. Abstract Appl. Anal. 2012, 2012:17, Article ID 391062.
  • [2] Ahmad Y. A. Salamooni, D. D. Pawar, Existence and uniqueness of boundary value problems for Hilfer-Hadamard-type fractional differential equations. arXiv:1801.10400v1[math.AP] 31 Jan (2018).
  • [3] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, (1999).
  • [4] S. G. Samko, A. A. Kilbas; O. I. Marichev; Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993). [Translation from the Russian edition, Nauka i Tekhnika, Minsk (1987)]
  • [5] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Editor: Jan Van Mill, Elsevier, Amsterdam, The Netherlands, (2006).
  • [6] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific Publ. Co., Singapore, (2000).
  • [7] R. Hilfer, Threefold introduction to fractional derivatives. In: Anomalous transport: foundations and applications, 2008, 17-73.
  • [8] J. A. Nanware, D. B. Dhaigude, Existence and uniqueness of solution of Riemann-Liouville fractional differential equations with integral boundary conditions. Int. J. Nonlinear Sci., 14, (2012), 410-415.
  • [9] M. D. Kassim and N. E. Tatar, Well-Posedness and Stability for a Differential Problem with Hilfer-Hadamard Fractional Derivative. Abst. Appl. Anal., 2013, (2013), 1-12, Article ID 605029.
  • [10] R. Hilfer, Y. Luchko, ˇZ: Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Cal. Appl. Anal., 12, (2009), 299-318.
  • [11] K. Diethelm, N. J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl., 265, (2002), 229-248.
  • [12] K. M. Furati, M. D. Kassim and N.e.Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative. Computers Math. Appl., (2012), 1616-1626.
  • [13] C. Kou, J. Liu, and Y. Ye, Existence and uniqueness of solutions for the Cachy-type problems of fractional differential equaitions. Discrete Dyn. Nat. Soc., Article ID 142175, (2010), 1-15.
  • [14] H.M. Srivastava, Z: Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput., 211, (2009), 198-210.
  • [15] Zivorad Tomovski, R. Hilfer, H. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transforms Spec. Funct., 21, (11), (2010), 797-814.
  • [16] J. A. Nanware, D. B. Dhaigude, Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions. J. Nonlinear Sci. Appl., 7, (2014), 246-254.
  • [17] Ahmad Y. A. Salamooni, D. D. Pawar, Existence and uniqueness of nonlocal boundary conditions for Hilfer-Hadamard-type fractional differential equations. arXiv:1802.04262v1[math.AP] 12 Feb (2018).
  • [18] D. Vivek, K. Kanagarajan and E. M. Elsayed, Some Existence and Stability Results for Hilfer-fractional Implicit Differential Equations with Nonlocal Conditions. Mediterr. J. Math. 2018 15:15, https://doi.org/10.1007/s00009-017-1061-0, ©Springer International Publishing AG, part of Springer Nature 2018.
  • [19] S. Abbas, M. Benchohra, , M. Bohner, Weak solutions for implicit differential equations with Hilfer–Hadamard fractional derivative. Adv. Dyn. Syst. Appl., 12, (2017), 1-16.
  • [20] S. Abbas, M. Benchohra, J.-E. Lazreg, Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: analysis and stability. Chaos Solitons Fractals, 102, (2017), 47-71.
  • [21] A. Y. A. Salamooni, D. D. Pawar, Unique positive solution for nonlinear Caputo-type fractional q􀀀difference equations with nonlocal and Stieltjes integral boundary conditions. Fractional Differential Calculus, 9(2), (2019), 295-307.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ahmad Y. A. Salamoonı

D.d. Pawar

Publication Date September 29, 2020
Submission Date June 22, 2020
Acceptance Date September 22, 2020
Published in Issue Year 2020

Cite

APA Salamoonı, A. Y. A., & Pawar, D. (2020). Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem. Universal Journal of Mathematics and Applications, 3(3), 121-128. https://doi.org/10.32323/ujma.756304
AMA Salamoonı AYA, Pawar D. Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem. Univ. J. Math. Appl. September 2020;3(3):121-128. doi:10.32323/ujma.756304
Chicago Salamoonı, Ahmad Y. A., and D.d. Pawar. “Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem”. Universal Journal of Mathematics and Applications 3, no. 3 (September 2020): 121-28. https://doi.org/10.32323/ujma.756304.
EndNote Salamoonı AYA, Pawar D (September 1, 2020) Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem. Universal Journal of Mathematics and Applications 3 3 121–128.
IEEE A. Y. A. Salamoonı and D. Pawar, “Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem”, Univ. J. Math. Appl., vol. 3, no. 3, pp. 121–128, 2020, doi: 10.32323/ujma.756304.
ISNAD Salamoonı, Ahmad Y. A. - Pawar, D.d. “Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem”. Universal Journal of Mathematics and Applications 3/3 (September 2020), 121-128. https://doi.org/10.32323/ujma.756304.
JAMA Salamoonı AYA, Pawar D. Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem. Univ. J. Math. Appl. 2020;3:121–128.
MLA Salamoonı, Ahmad Y. A. and D.d. Pawar. “Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem”. Universal Journal of Mathematics and Applications, vol. 3, no. 3, 2020, pp. 121-8, doi:10.32323/ujma.756304.
Vancouver Salamoonı AYA, Pawar D. Existence and Uniqueness of Generalised Fractional Cauchy-Type Problem. Univ. J. Math. Appl. 2020;3(3):121-8.

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