In this article, the homotopy perturbation method has been successfully applied to find the approximate solution of a Caputo fractional Volterra-Fredholm integro-differential equation. 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 the analytical approximate. Moreover, we proved the existence and uniqueness results of the solution. Finally, an example is included to demonstrate the validity and applicability of the proposed technique.

Citation details of the article

Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 3
Year: 2018

DOI: 10.12732/ijam.v31i3.3

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  1. [1] N. Abel, Solution de quelques problemes a laide dintegrales definites, Christiania Grondahl (Norway) (1881), 16-18.
  2. [2] S. Alkan, V. Hatipoglu, Approximate solutions of Volterra-Fredholm integro-differential equations of fractional order, Tbilisi Mathematical Journal, 10, No 2 (2017), 1-13.
  3. [3] M. AL-Smadi, G. Gumah, On the homotopy analysis method for fractional SEIR epidemic model, Research J. Appl. Sci. Engrg. Technol., 7, No 18 (2014), 3809-3820.
  4. [4] M. Bani Issa, A. Hamoud, K. Ghadle, Giniswamy, Hybrid method for solving nonlinear Volterra-Fredholm integro-differential equations, J. Math. Comput. Sci., 7, No 4 (2017), 625-641.
  5. [5] A. Hamoud, K. Ghadle, The reliable modified of Laplace Adomian decomposition method to solve nonlinear interval Volterra-Fredholm integral equations, Korean J. Math., 25, No 3 (2017), 323-334.
  6. [6] A. Hamoud, K. Ghadle, On the numerical solution of nonlinear VolterraFredholm integral equations by variational iteration method, Int. J. Adv. Sci. Tech. Research, 3 (2016), 45-51.
  7. [7] A. Hamoud, K. Ghadle, The combined modified Laplace with Adomian decomposition method for solving the nonlinear Volterra-Fredholm integrodifferential equations, J. Korean Soc. Ind. Appl. Math., 21 (2017), 17-28.
  8. [8] A. Hamoud, K. Ghadle, Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equations, J. Indian Math. Soc., 85, No (1-2) (2018), 52-69.
  9. [9] A. Hamoud, K. Ghadle, Recent advances on reliable methods for solving Volterra-Fredholm integral and integro-differential equations, Asian J. of Mathematics and Computer Research, 24, No 3 (2018), 128-157.
  10. [10] J.H. He, Homotopy perturbation method: a new nonlinear analytic technique, Appl. Math. Comput., 135 (2003), 73-79.
  11. [11] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178 (1999), 257-262. 348 A.A. Hamoud, K.P. Ghadle, M.Sh.B. Issa, Giniswamy
  12. [12] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Nonlinear Mechanics, 35 (2000), 37-43.
  13. [13] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. Elsevier, Amsterdam, 204 (2006).
  14. [14] V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods and Appl., 69, No 10 (2008), 33373343.
  15. [15] S. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University (1992).
  16. [16] X.Ma, C. Huang, Numerical solution of fractional integro-differential equations by a hybrid collocation method, Appl. Math. Comput., 219, No 12 (2013), 6750-6760.
  17. [17] R. Mittal, R. Nigam, Solution of fractional integro-differential equations by Adomian decomposition method, Int. J. Appl. Math. Mech., 4, No 2 (2008), 87-94.
  18. [18] I. Podlubny, Fractional Differential Equations, Academic Publications, Boston etc. (1999).
  19. [19] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon (1993).
  20. [20] C. Yang, J. Hou, Numerical solution of integro-differential equations of fractional order by Laplace decomposition method, WSEAS Trans. Math., 12, No 12 (2013), 1173-1183.
  21. [21] Y. Zhou, Basic Theory of Fractional Differential Equations, Singapore: World Scientific, 6 (2014).
  22. [22] M. Zurigat, S. Momani, A. Alawneh, Homotopy analysis method for systems of fractional integro-differential equations, Neur. Parallel Sci. Comput., 17 (2009), 169-186.