On a method for solving nonlinear integro differential equation of order $n$

Volume 25, Issue 4, pp 322--340 http://dx.doi.org/10.22436/jmcs.025.04.03

Publication Date: August 09, 2021 Submission Date: March 13, 2021 Revision Date: May 06, 2021 Accteptance Date: July 01, 2021

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Authors

M. A. Abdou - Department of Mathematics, Faculty of Education , Alexandria University, Alexandria, 21526, Egypt. M. I. Youssef - Department of Mathematics, College of Science, Jouf University, P. O. Box 2014, Sakaka, Saudi Arabia. - Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, 21526, Egypt.

Abstract

This work is concerned with the study of a general class of nonlinear integro-differential equations of order n. Using a suitable transformation, we derive an equivalent nonlinear Fredholm-Volterra integral equation (NF-VIE) to this class of equations. The existence of continuous solutions for the NF-VIE is investigated subject to the verification of some sufficient conditions. We apply the modified Adomian's decomposition method (MADM) and homotopy analysis method (HAM) to solve this NF-VIE. The convergence and error estimate of the approximate solution are also studied. The numerical results in this article show that the HAM technique may give an approximate solution with high accuracy and convergence rate faster than the one obtained using the MADM technique provided the convergence control parameter \(\hbar\) is properly chosen when applying the HAM.

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Keywords

• Integro-differential equations
• existence
• uniqueness
• modified Adomian's decomposition method
• homotopy analysis method

MSC

•  45G99
•  34A12
•  45L05
•  45G10

References

• [1] G. Adomian, Stochastic Systems, Academic Press, Orlando (1983)
  • View Article
  • MathSciNet
  • Google Scholar


• [2] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, Orlando (1986)
  • View Article
  • MathSciNet


• [3] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers Group, Dordrecht (1994)
  • View Article
  • MathSciNet
  • Google Scholar


• [4] R. P. Agarwal, M. Meehan, D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, Cambridge (2001)
  • View Article
  • MathSciNet
  • Google Scholar


• [5] A. Alidema, A. Georgieva, Adomian decomposition method for solving two-dimensional nonlinear Volterra Fuzzy integral equations, AIP Conference Proceedings, 2018 (2018), 11 pages
  • View Article
  • Google Scholar


• [6] H. O. Bakodah, M. Al-Mazmumy, S. O. Almuhalbedi, Solving system of integro differential equations using discrete Adomian decomposition method, J. Taibah Univ. Sci., 13 (2019), 805--812
  • View Article
  • Google Scholar


• [7] M. M. Elborai, M. A. Abdou, M. I. Youssef, On Adomian's decomposition method for solving nonlocal perturbed stochastic fractional integro-differential equations, Life Sci. J., 10 (2013), 550--555
  • View Article
  • Google Scholar


• [8] I. L. El-Kalla, Error analysis of Adomian series solution to a class of nonlinear differential equations, Appl. Math. E-Notes, 7 (2007), 214--221
  • View Article
  • MathSciNet
  • Google Scholar


• [9] A. Hamoud, A. Azeez, K. Ghadle, A study of some iterative methods for solving Fuzzy Volterra-Fredholm integral equations, Indones. J. Ele. Eng. Comput. Sci., 11 (2018), 1228--1235
  • View Article
  • Google Scholar


• [10] A. A. Hamoud, K. P. Ghadle, The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, Probl. Anal. Issues Anal., 7 (2018), 41--58
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] A. Hamoud, K. Ghadle, Modified Adomian decomposition method for solving Fuzzy Volterra-Fredholm integral equations, J. Indian Math. Soc., 85 (2018), 53--69
  • View Article
  • Google Scholar


• [12] A. A. Hamoud, K. P. Ghadle, Modified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations, J. Math. Model., 6 (2018), 91--104
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] A. A. Hamoud, K. P. Ghadle, Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind, Tamkang J. Math., 49 (2018), 301--315
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] A. A. Hamoud, K. P. Ghadle, S. M. Atshan, The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method, Khayyam J. Math., 5 (2019), 21--39
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] J. H. He, A short review on analytical methods for to a fully fourth-order nonlinear integral boundary value problem with fractal derivatives, Inter. J. Numer. Methods Heat Fluid Flow, 39 (2020), 4933--4943
  • View Article
  • Google Scholar


• [16] E. Hetmaniok, D. Slota, T. Trawinski, R. Witula, Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind, Numer. Algorithms, 67 (2014), 163--185
  • View Article
  • MathSciNet
  • Google Scholar


• [17] M. H. Holmes, Introduction to Perturbation Methods, Springer, New York (2013)
  • View Article
  • MathSciNet
  • Google Scholar


• [18] M. Issa, A. Hamoud, K. Ghadle, Numerical solutions of Fuzzy integro-differential equations of the second kind, J. Math. Computer Sci., 23 (2021), 67--74
  • View Article
  • Google Scholar


• [19] A. Kurt, O. Tasbozan, Approximate analytical solutions to conformable modified Burgers equation using homotopy analysis method}, Ann. Math. Sil., 33 (2019), 159--167
  • View Article
  • MathSciNet
  • Google Scholar


• [20] S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University (1992)
  • Google Scholar


• [21] S. J. Liao, Beyond Perturbation Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC, U.S.A. (2003)
  • View Article
  • Google Scholar


• [22] S. J. Liao, Homotopy Analysis Method in Nonlinear Differential Equation, Springer, New York (2012)
  • View Article
  • Google Scholar


• [23] S. J. Liao, Advances In The Homotopy Analysis Method, World Scientific Publishing Co., Hackensack (2014)
  • View Article
  • MathSciNet
  • Google Scholar


• [24] S. Maitama, W. D. Zhao, Local fractional homotopy analysis method for solving non-differentiable problems on Cantor sets, Adv. Difference Equ., 2019 (2019), 22 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [25] M. Marino, Instantons and large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory, Cambridge University Press, Cambridge (2015)
  • View Article
  • MathSciNet
  • Google Scholar


• [26] T. Muta, Foundations of Quantum Chromodynamics: An Introduction to Perturbative Methods in Gauge Theories, Third ed., World Scientific Pub. Co., Hackensack (2010)
  • View Article
  • MathSciNet
  • Google Scholar


• [27] A. H. Nayfeh, Perturbation Methods, Wiley-Interscience, New York (2000)
  • View Article
  • MathSciNet
  • Google Scholar


• [28] J. Singh, D. Kumar, D. Baleanu, S. Rathore, An efficient numerical algorithm for the fractional drinfeld-sokolov-wilson equation, Appl. Math. Comput., 335 (2018), 12--24
  • View Article
  • MathSciNet
  • Google Scholar


• [29] R. Singh, G. Nelakanti, J. Kumar, Approximate solution of Urysohn integral equations using the Adomian decomposition method, Sci. World J., 2014 (2014), 6 pages
  • View Article
  • Google Scholar


• [30] A. M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102 (1999), 77--86
  • View Article
  • Google Scholar


• [31] L.-J. Xie, A new modification of Adomian decomposition method for Volterra integral equations of the second kind, J. Appl. Math., 2013 (2013), 7 pages
  • View Article
  • MathSciNet
  • Google Scholar