An extension of high-order Kou's method for solving nonlinear systems and its stability analysis

Yantong Guo; Quansheng Wu; Xiaofeng Wang; Yantong Guo; Quansheng Wu; Xiaofeng Wang

doi:10.3934/era.2025074

Electronic Research Archive

2025, Volume 33, Issue 3: 1566-1588. doi: 10.3934/era.2025074

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An extension of high-order Kou's method for solving nonlinear systems and its stability analysis

1.
School of Mathematical Sciences, Bohai University, Jinzhou 121000, China
2.
School of Mathematics and Computer Science, Chaoyang Normal University, Chaoyang 122000, China

Received: 10 January 2025 Revised: 12 March 2025 Accepted: 14 March 2025 Published: 19 March 2025

In this paper, Kou's method is extended to solve nonlinear systems. The convergence order of the iterative method is proved. Using fractal theory, we study the theoretical operators related to the iterative method, and analyze the stability of the iterative method. Properties related to strange fixed points and critical points are explored. The fractal results indicate that the iterative method is most stable when the parameter $ \gamma $ equals zero. The extended iterative method is applied to solve the Hammerstein equation and some nonlinear systems. The dynamic plane and numerical experiments show that the extended iterative method can solve the nonlinear system of equations with good convergence and stability.
- nonlinear systems,
- iterative method,
- stability
Citation: Yantong Guo, Quansheng Wu, Xiaofeng Wang. An extension of high-order Kou's method for solving nonlinear systems and its stability analysis[J]. Electronic Research Archive, 2025, 33(3): 1566-1588. doi: 10.3934/era.2025074

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Abstract

In this paper, Kou's method is extended to solve nonlinear systems. The convergence order of the iterative method is proved. Using fractal theory, we study the theoretical operators related to the iterative method, and analyze the stability of the iterative method. Properties related to strange fixed points and critical points are explored. The fractal results indicate that the iterative method is most stable when the parameter $ \gamma $ equals zero. The extended iterative method is applied to solve the Hammerstein equation and some nonlinear systems. The dynamic plane and numerical experiments show that the extended iterative method can solve the nonlinear system of equations with good convergence and stability.

References

[1]	C. Li, H. Zhang, X. Yang, A new linearized ADI compact difference method on graded meshes for a nonlinear 2D and 3D PIDE with a WSK, Comput. Math. Appl., 176 (2024), 349–370. https://doi.org/10.1016/j.camwa.2024.11.006 doi: 10.1016/j.camwa.2024.11.006
[2]	H. Zhang, X. Jiang, F. Wang, X. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation, J. Appl. Math. Comput., 70 (2024), 1127–1151. https://doi.org/10.1007/s12190-024-02000-y doi: 10.1007/s12190-024-02000-y
[3]	X. Yang, Z. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972
[4]	Z. Chen, H. Zhang, H. Chen, Adi compact difference scheme for the two-dimensional integro-differential equation with two fractional Riemann–Liouville integral kernels, Fractal Fract., 8 (2024), 707. https://doi.org/10.3390/fractalfract8120707 doi: 10.3390/fractalfract8120707
[5]	J. Wang, X. Jiang, X. Yang, H. Zhang, A new robust compact difference scheme on graded meshes for the time-fractional nonlinear Kuramoto-Sivashinsky equation, Comput. Appl. Math., 43 (2024), 381. https://doi.org/10.1007/s40314-024-02883-4 doi: 10.1007/s40314-024-02883-4
[6]	X. Shen, X. Yang, H. Zhang, The high-order ADI difference method and extrapolation method for solving the two-dimensional nonlinear parabolic evolution equations, Mathematics, 12 (2024), 3469. https://doi.org/10.3390/math12223469 doi: 10.3390/math12223469
[7]	X. Yang, W. Wang, Z. Zhou, H. Zhang, An efficient compact difference method for the fourth-order nonlocal subdiffusion problem, Taiwan. J. Math., 29 (2025), 35–66. https://doi.org/10.11650/tjm/240906 doi: 10.11650/tjm/240906
[8]	X. Yang, Z. Zhang, Superconvergence analysis of a robust orthogonal Gauss collocation method for 2D fourth-order subdiffusion equations, J. Sci. Comput., 100 (2024). https://doi.org/10.1007/s10915-024-02616-z
[9]	N. Sidorov, Solvability of Nonlinear equations with parameters: Branching, regularization, group symmetry and solutions blow-up, Symmetry, 14 (2022), 226. https://doi.org/10.3390/sym14020226 doi: 10.3390/sym14020226
[10]	Q. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Levy processes, IEEE Trans. Autom. Control, 70 (2025), 1176–1183. http://doi.org/10.1109/TAC.2024.3448128 doi: 10.1109/TAC.2024.3448128
[11]	X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7 doi: 10.1007/s10915-024-02511-7
[12]	J. Wang, X. Jiang, X. Yang, H. Zhang, A compact difference scheme for mixed-type time-fractional black-scholes equation in European option pricing, Math. Methods Appl. Sci., 2025. https://doi.org/10.1002/mma.10717
[13]	K. R. Nilay, A new semi-explicit atomistic molecular dynamics simulation method for membrane proteins, J. Comput. Methods Sci. Eng., 19 (2019), 259–286. https://doi.org/10.3233/jcm-180851 doi: 10.3233/jcm-180851
[14]	X. Wang, N. Shang, Local convergence analysis of a novel derivative-free method with and without memory for solving nonlinear systems, Int. J. Comput. Math., (2025), 1–18. http://doi.org/10.1080/00207160.2025.2464701
[15]	D. Ruan, X. Wang, Y. Wang, Local convergence of seventh-order iterative method under weak conditions and its applications, Eng. Comput., 2025. http://doi.org/10.1108/EC-08-2024-0775
[16]	D. Ruan, X. Wang, A high-order Chebyshev-type method for solving nonlinear equations: local convergence and applications, Electron. Res. Arch., 33 (2025), 1398–1413. https://doi.org/10.3934/era.2025065 doi: 10.3934/era.2025065
[17]	X. Wang, W. Li, Fractal behavior of King's optimal eighth-order iterative method and its numerical application, Math. Comm., 29 (2024), 217–236.
[18]	X. Wang, T. Zhang, Y. Qin, Efficient two-step derivative-free iterative methods with memory and their dynamics, Int. J. Comput. Math., 93 (2015), 1423–1446. http://doi.org/10.1080/00207160.2015.1056168 doi: 10.1080/00207160.2015.1056168
[19]	A. Cordero, J. R. Torregrosa, M. P. Vassileva, Three-step iterative methods with optimal eighth-order convergence, J. Comput. Appl. Math., 235 (2011), 3189–3194. http://doi.org/10.1016/j.cam.2011.01.004 doi: 10.1016/j.cam.2011.01.004
[20]	A. Cordero, M. Moscoso-Martinez, J. R. Torregrosa, Chaos and stability in a new iterative family for solving nonlinear equations, Algorithms, 14 (2021), 101. http://doi.org/10.3390/a14040101 doi: 10.3390/a14040101
[21]	J. M. Ortega, W. C, Rheinboldt, Iterative solution of nonlinear equations in several variables, SIAM, 52 (2000), 521–572. https://doi.org/10.1137/1.9780898719468.bm doi: 10.1137/1.9780898719468.bm
[22]	F. A. Potra, V. Pta'k, Nondiscrete induction and iterative processes. SIAM Rev., 1984. https://doi.org/10.1137/1029105
[23]	A. Cordero, J. R. Torregrosa, On interpolation variants of Newton's method for functions of several variables, J. Comput. Appl. Math., 234 (2010), 34–43. https://doi.org/10.1016/j.cam.2009.12.002 doi: 10.1016/j.cam.2009.12.002
[24]	J. Kou, Y. Li, X. Wang, A composite fourth-order iterative method for solving non-linear equations, Appl. Math. Comput., 184 (2007), 471–475. http://doi.org/10.1016/j.amc.2006.05.181 doi: 10.1016/j.amc.2006.05.181
[25]	K. Falconer, Fractal geometry: mathematical foundations and applications, Wiley, 2013.
[26]	B. B. Mandelbrot, Fractal geometry: what is it, and what does it do?, Proc. R. Soc. Lond. A Math. Phys. Sci., 423 (1989), 3–16. https://doi.org/10.1098/rspa.1989.0038
[27]	M. Y. Lee, Y. I. Kim, The dynamical analysis of a uniparametric family of three-point optimal eighth-order multiple-root finders under the Möbius conjugacy map on the Riemann sphere, Numer. Algorithms, 83 (2020), 1063–1090. https://doi.org/10.1007/s11075-019-00716-8 doi: 10.1007/s11075-019-00716-8
[28]	X. Wang, W. Li, Choosing the best members of the optimal eighth-order Petković's family by its fractal behavior, Fractal Fract., 6 (2022), 749. http://doi.org/10.3390/fractalfract6120749 doi: 10.3390/fractalfract6120749
[29]	S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Appl., 366 (2010), 24–32. https://doi.org/10.1016/j.jmaa.2010.01.047 doi: 10.1016/j.jmaa.2010.01.047
[30]	A. Cordero, T. Lotfi, K. Mahdiani, J. R. Torregrosa, A stable family with high order of convergence for solving nonlinear equations, Appl. Numer. Math., 254 (2015), 240–251. https://doi.org/10.1016/j.amc.2014.12.141 doi: 10.1016/j.amc.2014.12.141
[31]	H. H. H. Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math., 176 (2005), 425–432. https://doi.org/10.1016/j.cam.2004.07.027 doi: 10.1016/j.cam.2004.07.027
[32]	C. Chun, Some third-order families of iterative methods for solving nonlinear equations, Appl. Math. Comput., 188 (2007), 924–933. https://doi.org/10.1016/j.amc.2006.10.043 doi: 10.1016/j.amc.2006.10.043
[33]	A. Cordero, J. R. Torregrosa, Variants of Newton's method using fifth-order quadrature formulas, Appl. Math. Comput., 190 (2007), 686–698. https://doi.org/10.1016/j.amc.2007.01.062 doi: 10.1016/j.amc.2007.01.062
[34]	J. Kou, On Chebyshev-Halley methods with sixth-order convergence for solving non-linear equations, Appl. Math. Comput., 190 (2007), 126–131. https://doi.org/10.1016/j.amc.2007.01.011 doi: 10.1016/j.amc.2007.01.011
[35]	Y. I. Kim, R. Behl, S. S. Motsa, Higher-order efficient class of Chebyshev-Halley-type methods, Appl. Math. Comput., 273 (2016), 1148–1159. https://doi.org/10.1016/j.amc.2015.09.013 doi: 10.1016/j.amc.2015.09.013
[36]	S. Qureshi, A. Soomro, A. A. Shaikh, E. Hincal, N. Gokbulut, A novel multistep iterative technique for models in medical sciences with complex dynamics, Comput. Math. Methods Med., 2022 (2022), 7656451. https://doi.org/10.1155/2022%2F7656451 doi: 10.1155/2022%2F7656451
[37]	J. R. Sharma, S. Kumar, H. Singh, A new class of derivative-free root solvers with increasing optimal convergence order and their complex dynamics, SeMA J., 80 (2023), 333–352. https://doi.org/10.1007/s40324-022-00288-z doi: 10.1007/s40324-022-00288-z
[38]	O. S. Solaiman, I. Hashim, Two new efficient sixth order iterative methods for solving nonlinear equations, J. King Saud Univ. Sci., 31 (2019), 701–705. https://doi.org/10.1016/j.jksus.2018.03.021 doi: 10.1016/j.jksus.2018.03.021
[39]	J. Kou, Y. Li, X. Wang, A modification of Newton method with third-order convergence, Appl. Numer. Math., 181 (2019), 1106–1111. https://doi.org/10.1016/j.amc.2006.01.076 doi: 10.1016/j.amc.2006.01.076

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