Solving non-linear functional equations by relaxed new iterative method

: pp. 421–429
Received: December 18, 2023
Revised: May 16, 2024
Accepted: June 20, 2024

Rhofir K., Radid A.  Solving non-linear functional equations by relaxed new iterative method.  Mathematical Modeling and Computing. Vol. 11, No. 2, pp. 421–429 (2024)

LASTI-ENSA Khouribga, Sultan Moulay Slimane University, Morocco
LMFA-FSAC Casablanca, Hassan II University, Morocco

For solving various equations of the form $v=f+N (v)$, the new iterative method and the new algorithm proposed by V. Daftardar–Gejji et al. [Daftardar–Gejji V., Jafari H. J. Math. Anal. Appl. 316 (2), 753–763 (2006); Kumar M., Jhinga A., Daftardar–Gejji V. Int. J. Appl. Comp. Math. 6 (2), 26 (2020)] are been employed successfully and accurately.  Our aim in this paper is to present a relaxed new iterative method by introducing a controlled parameter $\omega$ in order to extend these methods.  According to the values of the parameter $\omega$, we discuss and provide the convergence analysis.  The proposed algorithm is fast, effective and simple to implement as compared to the existing one.  Numerous non-linear equations are solved to show the applicability and efficiency of the algorithm compared to the other methods.

  1. Adomian G.  Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston (1994).
  2. He J.-H.  Homotopy perturbation technique.  Computer Methods in Applied Mechanics and Engineering.  178 (3), 257–262 (1999).
  3. He J.-H.  Variational iteration method – a kind of nonlinear analytical technique: some examples.  International Journal of Non-Linear Mechanics.  34 (4), 699–708 (1999).
  4. Daftardar–Gejji V., Jafari H.  An iterative method for solving nonlinear functional equations.  Journal of Mathematical Analysis and Applications.  316 (2), 753–763 (2006).
  5. Daftardar–Gejji V., Kumar M.  New Iterative Method: A Review.  Frontiers in Fractional Calculus.  1, 233–268 (2018).
  6. Bhalekar S., Daftardar–Gejji V.  New iterative method: Application to partial differential equations.  Applied Mathematics and Computation.  203 (2), 778–783 (2008).
  7. Daftardar–Gejji V., Bhalekar S.  An iterative method for solving fractional differential equations.  Proceedings in Applied Mathematics and Mechanics.  7 (1), 2050017–2050018 (2007).
  8. Daftardar–Gejji V., Bhalekar S.  Solving fractional boundary value problems with Dirichlet boundary conditions using a new iterative method.  Computers & Mathematics with Applications.  59 (5), 1801–1809 (2010).
  9. Sari M., Gunay G., Gurarslan G.  A solution to the telegraph equation by using DGJ Method.  International Journal of Nonlinear Science.  17 (1), 57–66 (2014).
  10. Hemeda A.-A.  New iterative method: application to nth-order integro-differential equations.  International Mathematical Forum.  7 (47), 2317–2332 (2012).
  11. Bhalekar S., Daftardar–Gejji V.  Solving fractional-order logistic equation using new iterative method.  Internationa Journal of Differential Equations.  2012, 975829 (2012).
  12. Jhinga A., Daftardar–Gejji V.  A new finite-diference predictor–corrector method for fractional differential equations.  Applied Mathematics and Computations.  336, 418–432 (2018).
  13. Bhalekar S., Daftardar–Gejji V.  Convergence of the New Iterative Method.  International Journal of Differential Equations.  2011, 989065 (2011).
  14. Radid A., Rhofir K.  SOR-Like New Iterative Method for Solving the Epidemic Model and the Prey and Predator Problem.  Discrete Dynamics in Nature and Society.  2020, 9053754 (2020).
  15. Kumar M., Jhinga A., Daftardar–Gejji V.  New Algorithm for Solving Non-linear Functional Equations.  International Journal of Applied and Computational Mathematics.  6 (2), 26 (2020).
  16. Joshi M. C., Bose R. K.  Some Topics in Nonlinear Functional Analysis. Wiley, Hoboken (1985).