[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Some Powerful Techniques for Solving Nonlinear Volterra-Fredholm Integral Equations

Journal of Applied Nonlinear Dynamics 10(3) (2021) 461--469 | DOI:10.5890/JAND.2021.09.007

Ahmed A. Hamoud$^{1}$ , Nedal M. Mohammed$^{2}$, Kirtiwant P. Ghadle$^{3}$

$^{1}$ Department of Mathematics, Taiz University, Taiz-380 015, Yemen

$^{2}$ Department of Computer Science, Taiz University, Taiz-380 015, Yemen

$^{3}$ Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India

Download Full Text PDF

 

Abstract

The main object of the present paper is to study the behavior of the approximated solutions of the nonlinear mixed Volterra-Fredholm integral equations by using Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM), Variational Iteration Method (VIM) and Homotopy Analysis Method (HAM). Moreover, we discuss some new uniqueness results. Finally, some examples are included to demonstrate the validity and applicability of the proposed techniques.

References

  1. [1]  Hosseini, S., Shahmorad, S., and Talati, F. (2014), {A matrix based method for two dimensional nonlinear Volterra-Fredholm integral equations}, { Numer. Algor.}, 1-19.
  2. [2]  Shekarabi, F., Maleknejad, K., and Ezzati, R. (2014), {Application of two-dimensional Bernstein polynomials for solving mixed Volterra-Fredholm integral equations}, { Afr. Math.}, 1-15.
  3. [3]  Brunner, H. (1990), On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, { Siam J. Numer. Anal.}, 27(4), 987-1000.
  4. [4]  Bhrawy, A., Abdelkawy, M., Tenreiro Machado, J., and Amin, A. (2016), {Legendre-Gauss-Lobatto collocation method for solving multi-dimensional Fredholm integral equations}, { Comput. Math. Appl.}, 4, 1-13.
  5. [5]  Bani Issa, M. and Hamoud, A. (2020), Some approximate methods for solving system of nonlinear integral equations, Technology Reports of Kansai University, {\bf62}(3), 388-398.
  6. [6]  Bazrafshan, F., Mahbobi, A., Neyrameh, A., Sousaraie, A., and Ebrahim, M. (2011), Solving two-dimensional integral equations, { Journal of King Saud University (Science)}, 23, 111-114.
  7. [7]  Ghasemi, M., Kajani, M., and Davari, A. (2007), { Numerical solution of two-dimensional nonlinear differential equation by homotopy perturbation method}, { Appl. Math. Comput.}, 189(1), 341-345.
  8. [8]  Maleknejad, K. and Sohrabi, S. (2008), {Legendre polynomial solution of nonlinear Volterra-Fredholm integral equations}, { Internat. J. Engrg. Sci.}, 19(2-5), 49-52.
  9. [9]  Behzadi, Sh. (2014), Homotopy approximation technique for solving nonlinear Volterra-Fredholm integral equations of the first kind, { Int. J. Ind. Math.}, 6(4), 315-320.
  10. [10]  Dastjerdi, H., Ghaini, F., and Hadizadeh, M. (2013), {A meshless approximate solution of mixed Volterra-Fredholm integral equations}, { Inter. J. Comput. Math.}, 90(3), 527-538.
  11. [11]  Hussain, K., Hamoud, A., and Mohammed, N. (2019), { Some new uniqueness results for fractional integro-differential equations}, { Nonlinear Funct. Anal. Appl.}, {\bf 24}(4), 827-836.
  12. [12]  Paripour, M. and Kamyar, M. (2013), {Numerical solution of nonlinear Volterra-Fredholm integral equations by using new basis functions,} { Commun. Numer. Anal.}, 1(17), 1-12.
  13. [13]  Wazwaz, A.M. (2011), Linear and Nonlinear Integral Equations Methods and Applications, Springer Heidelberg Dordrecht London New York.
  14. [14]  Dong, C., Chen, Z., and Jiang, W. (2013), {A modified homotopy perturbation method for solving the nonlinear mixed Volterra-Fredholm integral equation}, { J. Comput. Appl. Math.}, 239(1), 359-366.
  15. [15]  Wang, K., Wang, Q., and Guan, K. (2013), {Iterative method and convergence analysis for a kind of mixed nonlinear Volterra-Fredholm integral equation}, { Appl. Math. Comput.}, 225(1), 631-637.
  16. [16]  Wang, K. and Wang, Q. (2014), {Taylor polynomial method and error estimation for a kind of mixed Volterra-Fredholm integral equations}, { Appl. Math. Comput.}, 229(25), 53-59.
  17. [17]  Hamoud, A. and Ghadle, K. (2019), Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, { J. Appl. Comput. Mech.}, {\bf 5}(1), 58-69.
  18. [18]  Hamoud, A. and Ghadle, K. (2018), { The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques}, { Probl. Anal. Issues Anal.}, {\bf 7}(25), 41-58.
  19. [19]  Dawood, L., Hamoud, A., and Mohammed, N. (2020), Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations, { Journal of Mathematics and Computer Science}, {\bf21}(2), 158-163.
  20. [20]  Hamoud, A. and Ghadle, K. (2018), Usage of the homotopy analysis method for solving fractional Volterra-Fredholm integro-differential equation of the second kind, { Tamkang J. Math.}, {\bf 49}(4), 301-315.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates