Further, the dynamical behavior of the proposed method is revealed.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:Newton’s iteration method is widely used in numerical methods, but its convergence is low.
📊
Temel Bulgular & Veri:Here, we develop a two-step optimal fourth-order iterative method based on linear combination of two iterative schemes for nonlinear equations, and we explore the convergence criteria of the proposed method and also demonstrate its validity and efficiency by considering some test problems.
💡
Sonuç & Katkı:Further, the dynamical behavior of the proposed method is revealed.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Iterative method · nonlinear equation · order of convergence
Abstract
Newton’s iteration method is widely used in numerical methods, but its convergence is low. Though a higher order iteration algorithm leads to a fast convergence, it is always complex. An optimal iteration formulation is much needed for both fast convergence and simple calculation. Here, we develop a two-step optimal fourth-order iterative method based on linear combination of two iterative schemes for nonlinear equations, and we explore the convergence criteria of the proposed method and also demonstrate its validity and efficiency by considering some test problems. We present both numerical as well as graphical comparisons. Further, the dynamical behavior of the proposed method is revealed.
Nadeem, G. A., Ali, F., He, J. H. (2022). New optimal fourth-order iterative method based on linear combination technique. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 6, 1692–1708. https://ajindex.com/articles/529445
S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by mod- ified Adomian decomposition method, Appl. Math. Comput. 145, 887–893, 2003
F. Ali, W. Aslam and A. Rafiq, Some new iterative techniques for the problems in- volving nonlinear equations, Int. J. Comput. Methods, 16, 1950037 (18 pages), 2019
F. Ali, W. Aslam, K. Ali, M. A. Anwar and A. Nadeem, New family of iterative methods for solving nonlinear models, Discrete Dyn. Nat. Soc. 9619680, 12 pages, 2018
M. Ali, N. Anjum and Q.T. Ain, Homotopy Perturbation Method for the Attachment Oscillator Arising in Nanotechnology, Fibers Polym. 22, 1601-1606, 2021
N. Anjum and J.H. He, Analysis of nonlinear vibration of nano / microelectrome- chanical system switch induced by electromagnetic force under zero initial conditions, Alex. Eng. J. 59 (6), 4343–4352, 2020
N. Anjum and J.H. He, Higher-order homotopy perturbation method for conserva- tive nonlinear oscillators generally and microelectromechanical systems’ oscillators particularly, Int. J. Mod. Phys. B. 34 (32), 2050313, 2020
N. Anjum and J.H. He, Homotopy perturbation method for N/MEMS oscillators, Math. Methods Appl. Sci. DOI: 10.1002/mma.6583
N. Anjum and J.H. He, Nonlinear dynamic analysis of vibratory behavior of a graphene nano/microelectromechanical system, Math. Meth. App. Sci. DOI: 10.1002/mma.6699
N. Anjum and J.H. He, Two Modifications of the Homotopy Perturbation Method for Nonlinear Oscillators”, J. App. Compu. Mech. 6, Special Issue, 1420–1425, 2020
N. Anjum, J.H. He, Q.T. Ain and D. Tian, Li-He’s modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectrome- chanical system, Facta Univ. Ser.: Mech. Eng. DOI:10.22190/FUME210112025A
N. Anjum, M. Suleman and D. Lu, Numerical iteration for nonlinear oscillators by Elzaki transform, J. Low Freq. Noise Vib. Act. Control, 39 (4), 879–884, 2020
E. Babolian and J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 132, 167–172, 2002
S. Bhalekar and V. Daftardar-Gejji, Convergence of new iterative method, Int. J. Differ. Equ. 2011, 989065, 10 pages, 2011
C. Chun, Some fourth-order iterative methods for solving nonlinear equations, Appl. Math. Comput. 195 (2), 454–459, 2018
V. Daftardar-Gejji and H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2), 753–76