DOI QR코드

DOI QR Code

EXISTENCE AND UNIQUENESS RESULTS FOR CAPUTO FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

  • HAMOUD, AHMED A. (DEPARTMENT OF MATHEMATICS, TAIZ UNIVERSITY) ;
  • ABDO, MOHAMMED S. (DEPARTMENT OF MATHEMATICS, DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY) ;
  • GHADLE, KIRTIWANT P. (DEPARTMENT OF MATHEMATICS, DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY)
  • 투고 : 2018.02.09
  • 심사 : 2018.08.23
  • 발행 : 2018.09.25

초록

This paper successfully applies the modified Adomian decomposition method to find the approximate solutions of the Caputo fractional integro-differential equations. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. Also, the behavior of the solution can be formally determined by analytical approximation. Moreover, we proved the existence and uniqueness results and convergence of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

키워드

참고문헌

  1. K. Abbaoui and Y. Cherruault, Convergence of Adomian's method applied to nonlinear equations, Math. Comput. Modelling, 20, 9 (1994), 69-73. https://doi.org/10.1016/0895-7177(94)00163-4
  2. G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 2 (1988), 501-544. https://doi.org/10.1016/0022-247X(88)90170-9
  3. S. Alkan and V. Hatipoglu, Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order, Tbilisi Mathematical Journal, 10, 2 (2017), 1-13.
  4. M. AL-Smadi and G. Gumah, On the homotopy analysis method for fractional SEIR epidemic model, Research J. Appl. Sci. Engrg. Technol., 7, 18 (2014), 3809-3820.
  5. M. Bani Issa, A. Hamoud, K. Ghadle and Giniswamy, Hybrid method for solving nonlinear Volterra-Fredholm integro-differential equations, J. Math. Comput. Sci. 7, 4 (2017), 625-641.
  6. A. Hamoud and K. Ghadle, The reliable modified of Laplace Adomian decomposition method to solve nonlinear interval Volterra-Fredholm integral equations, Korean J. Math., 25, 3 (2017), 323-334. https://doi.org/10.11568/KJM.2017.25.3.323
  7. A. Hamoud and K. Ghadle, On the numerical solution of nonlinear Volterra-Fredholm integral equations by variational iteration method, Int. J. Adv. Sci. Tech. Research, 3 (2016), 45-51.
  8. A. Hamoud and K. Ghadle, The combined modified Laplace with Adomian decomposition method for solving the nonlinear Volterra-Fredholm integro-differential equations, J. Korean Soc. Ind. Appl. Math., 21 (2017), 17-28.
  9. A. Hamoud and K. Ghadle, Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equations, J. Indian Math. Soc., 85, (1-2) (2018), 52-69.
  10. A. Hamoud and K. Ghadle, Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations, J. Math. Model., 6, 1 (2018), 91-104.
  11. A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. Elsevier, Amsterdam, 204, 2006.
  12. V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods and Appl. 69, 10 (2008), 3337-3343.
  13. X. Ma and C. Huang, Numerical solution of fractional integro-differential equations by a hybrid collocation method, Appl. Math. Comput., 219, 12 (2013), 6750-6760. https://doi.org/10.1016/j.amc.2012.12.072
  14. R. Mittal and R. Nigam, Solution of fractional integro-differential equations by Adomian decomposition method, Int. J. Appl. Math. Mech., 4, 2 (2008), 87-94.
  15. S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993.
  16. A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1999), 77-86.
  17. C. Yang and J. Hou, Numerical solution of integro-differential equations of fractional order by Laplace decomposition method, Wseas Trans. Math., 12, 12 (2013), 1173-2880.
  18. X. Zhang, B. Tang, and Y. He, Homotopy analysis method for higher-order fractional integro-differential equations, Comput. Math. Appl., 62, 8 (2011), 3194-3203. https://doi.org/10.1016/j.camwa.2011.08.032
  19. Y. Zhou, Basic Theory of Fractional Differential Equations, Singapore: World Scientific, 6, 2014.
  20. M. Zurigat, S. Momani and A. Alawneh, Homotopy analysis method for systems of fractional integro-differential equations, Neur. Parallel Sci. Comput., 17, (2009), 169-186.