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

Volume 25, Issue 4, pp 322--340
Publication Date: August 09, 2021 Submission Date: March 13, 2021 Revision Date: May 06, 2021 Accteptance Date: July 01, 2021
• 215 Views

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.

Share and Cite

ISRP Style

M. A. Abdou, M. I. Youssef, On a method for solving nonlinear integro differential equation of order $n$, Journal of Mathematics and Computer Science, 25 (2022), no. 4, 322--340

AMA Style

Abdou M. A., Youssef M. I., On a method for solving nonlinear integro differential equation of order $n$. J Math Comput SCI-JM. (2022); 25(4):322--340

Chicago/Turabian Style

Abdou, M. A., Youssef, M. I.. "On a method for solving nonlinear integro differential equation of order $n$." Journal of Mathematics and Computer Science, 25, no. 4 (2022): 322--340

Keywords

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

•  45G99
•  34A12
•  45L05
•  45G10

References

• [1] G. Adomian, Stochastic Systems, Academic Press, Orlando (1983)

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

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

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

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

• [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

• [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

• [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

• [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

• [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

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

• [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

• [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

• [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

• [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

• [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

• [17] M. H. Holmes, Introduction to Perturbation Methods, Springer, New York (2013)

• [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

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

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

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

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

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

• [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

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

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

• [27] A. H. Nayfeh, Perturbation Methods, Wiley-Interscience, New York (2000)

• [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

• [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

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

• [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