Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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THE RELIABLE MODIFIED OF ADOMIAN DECOMPOSITION METHOD FOR SOLVING INTEGRO-DIFFERENTIAL EQUATIONS

  • Hamoud, Ahmed A. (Department of Mathematics Dr. Babasaheb Ambedkar Marathwada University) ;
  • Ghadle, Kirtiwant P. (Department of Mathematics Dr. Babasaheb Ambedkar Marathwada University)
  • Received : 2019.05.05
  • Accepted : 2019.08.21
  • Published : 2019.11.15
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Abstract

In this article, we discussed semi-analytical approximated methods for solving mixed Volterra-Fredholm integro-differential equations, namely: Adomian decomposition method and modified Adomian decomposition method. Moreover, we prove the uniqueness results and convergence of the techniques. Finally, an example is included to demonstrate the validity and applicability of the proposed techniques.

Keywords

References

  1. M.A. Araghi and S.S. Behzadi, Solving nonlinear Volterra-Fredholm integro-differential equations using the modified Adomian decomposition method, Comput. Methods in Appl. Math., 9 (2009), 321-331. https://doi.org/10.2478/cmam-2009-0020
  2. S.H. Behiry and S.I. Mohamed, Solving high-order nonlinear Volterra-Fredholm integro-differential equations by differential transform method, Natural Science, 4 (2012), no. 8, 581-587. https://doi.org/10.4236/ns.2012.48077
  3. E. Babolian, Z. Masouri, and S. Hatamzadeh, New direct method to solve nonlinear Volterra-Fredholm integral and integro differential equation using operational matrix with Block-Pulse functions, Progress in Electromagnetic Research, B 8 (2008), 59-76.
  4. S.M. El-Sayed, D. Kaya, and S. Zarea, The decomposition method applied to solve high-order linear Volterra-Fredholm integro-differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 5 (2004), no. 2, 105-112.
  5. F.S. Fadhel, A.O. Mezaal, and S.H. Salih, Approximate solution of the linear mixed Volterra-Fredholm integro-differential equations of second kind by using variational iteration method, Al- Mustansiriyah, J. Sci., 24 (2013), no. 5, 137-146.
  6. M. Ghasemi, M. kajani, and E. Babolian, Application of He's homotopy perturbation method to nonlinear integro differential equations, Appl. Math. Comput., 188 (2007), 538-548. https://doi.org/10.1016/j.amc.2006.10.016
  7. J.H. He and S.Q. Wang, Variational iteration method for solving integro-differential equations, Phys. Lett., A 367 (2007), 188-191. https://doi.org/10.1016/j.physleta.2007.02.049
  8. A.A. Hamoud and K.P. Ghadle, Existence and uniqueness of the solution for Volterra-Fredholm integro-differential equations, Journal of Siberian Federal University. Mathematics & Physics, 11 (2018), no. 6, 692-701. https://doi.org/10.17516/1997-1397-2018-11-6-692-701
  9. A.A. Hamoud and K.P. Ghadle, Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations, Journal of Mathematical Modeling, 6 (2018), no. 1, 91-104.
  10. A.A. Hamoud and K.P. Ghadle, Homotopy analysis method for the first order fuzzy Volterra-Fredholm integro-differential equations, Indonesian J. Elec. Eng. & Comp. Sci., 11 (2018), no. 3, 857-867. https://doi.org/10.11591/ijeecs.v11.i3.pp857-867
  11. A.A. Hamoud and K.P. Ghadle, Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, J. Appl. Comp. Mech., 5 (2019), no. 1, 58-69.
  12. A.A. Hamoud and K.P. Ghadle, Existence and uniqueness of solutions for fractional mixed Volterra-Fredholm integro-differential equations, Indian J. Math., 60 (2018), no. 3, 375-395.
  13. A.A. Hamoud and K.P. Ghadle, The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, Probl. Anal. Issues Anal., 7(25) (2018), no. 1, 41-58.
  14. A.A. Hamoud, K.P. Ghadle, and S.M. Atshan, The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method, Khayyam Journal of Mathematics, 5 (2019), no. 1, 21-39.
  15. A.A. Hamoud and K.P. Ghadle, The reliable modified of Laplace Adomian decomposition method to solve nonlinear interval Volterra-Fredholm integral equations, Korean J. Math., 25 (2017), no. 3, 323-334. https://doi.org/10.11568/kjm.2017.25.3.323
  16. A.A. Hamoud, L.A. Dawood, K.P. Ghadle, and S.M. Atshan, Usage of the modified variational iteration technique for solving Fredholm integro-differential equations, International Journal of Mechanical and Production Engineering Research and Development, 9 (2019), no. 2, 895-902. https://doi.org/10.24247/ijmperdapr201987
  17. A.A. Hamoud, K.H. Hussain, and K.P. Ghadle, The reliable modified Laplace Adomian decomposition method to solve fractional Volterra-Fredholm integro-differential equations, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 26 (2019), 171-184.
  18. A.A. Hamoud and K.P. Ghadle, Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equations, J. Indian Math. Soc., 85 (2018), no. 1-2, 52-69.
  19. A.A. Hamoud and K.P. Ghadle, M. Bani Issa, Giniswamy, Existence and uniqueness theorems for fractional Volterra-Fredholm integro-differential equations, Int. J. Appl. Math., 31 (2018), no. 3, 333-348.
  20. A.A. Hamoud, A.D. Azeez, and K.P. Ghadle, A study of some iterative methods for solving fuzzy Volterra-Fredholm integral equations, Indonesian J. Elec. Eng. & Comp. Sci., 11 (2018), no. 3, 1228-1235. https://doi.org/10.11591/ijeecs.v11.i3.pp1228-1235
  21. A.A. Hamoud and K.P. Ghadle, Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind, Tamkang Journal of Mathematics, 49 (2018), no. 4, 301-315. https://doi.org/10.5556/j.tkjm.49.2018.2718
  22. A.A. Hamoud, M. Bani Issa, and K.P. Ghadle, M. Abdulghani, Existence and convergence results for caputo fractional Volterra integro-differential equations, Journal of Mathematics and Applications, 41 (2018), 109-122.
  23. A.A. Hamoud, N.M. Mohammed, K.P. Ghadle, and S.L. Dhondge, Solving integro-differential equations by using numerical techniques, International Journal of Applied Engineering Research, 14 (2019), no. 14, 3219-3225.
  24. A.A. Hamoud, M. Bani Issa, and K.P. Ghadle, Existence and uniqueness results for nonlinear Volterra-Fredholm integro-differential equations, Nonlinear Functional Analysis and Applications, 23 (2018), no. 4, 797-805. https://doi.org/10.22771/NFAA.2018.23.04.15
  25. A.M. Jerri, Introduction to Integral Equations with Applications, New York, Wiley, 1999.
  26. J. Manafianheris, Solving the integro-differential equations using the modified Laplace Adomian decomposition method, Journal of Mathematical Extension, 6 (2012), no. 1, 41-55.
  27. H.R. Marzban and S.M. Hoseini, Solution of nonlinear Volterra-Fredholm integro-differential equations via hybrid of Block-Pulse functions and Lagrange interpolating polynomials, Hindawi Publishing Corporation, Advances in Numerical Analysis, 868 (2012), no. 279, 1-14.
  28. S.B. Shadan, The use of iterative method to solve two-dimensional nonlinear Volterra-Fredholm integro-differential equations, J. of Communication in Numerical Analysis, (2012), 1-20.
  29. Y. Salih and S. Mehmet, The approximate solution of higher order linear Volterra-Fredholm integro-differential equations in term of Taylor polynomials, Appl. Math. Comput., 112 (2000), 291-308. https://doi.org/10.1016/S0096-3003(99)00059-4
  30. A.M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, Springer Heidelberg Dordrecht London New York, 2011.
  31. A.M. Wazwaz, The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Appl. Math. Comput., 216 (2010), 1304-1309. https://doi.org/10.1016/j.amc.2010.02.023
  32. A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102 (1999), 77-86. https://doi.org/10.1016/S0096-3003(98)10024-3