On the Convergence, Dynamics and Applications of a New Class of Nonlinear System Solvers
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On the Convergence, Dynamics and Applications of a New Class of Nonlinear System Solvers Ioannis K. Argyros1 · Debasis Sharma2 Shanta Kumari Sunanda2
· Sanjaya Kumar Parhi3 ·
Accepted: 31 August 2020 © Springer Nature India Private Limited 2020
Abstract In this paper, a uni-parametric class of third-order iterative algorithms for solving systems of nonlinear equations is proposed. The local convergence of the suggested schemes is studied using generalized Lipschitz-type condition on the first-order Fréchet derivative. Furthermore, we analyze the numerical stability of the new methods applying complex dynamics tool. The nonlinear systems related to the equation of molecular interaction, a boundary value problem, the integral equation from Chandrasekhar’s work, etc. are discussed. The most interesting fact about the proposed third-order family is that it generates a super convergent scheme (for γ = 2) for solving quadratic nonlinear systems (QNS). This particular method produces much better results for QNS in comparison with other third-order schemes. Also, it is observed that the approximate computational order of convergence (ACOC) of this scheme (for γ = 2) is approximately four (3.99–4.00) while solving QNS. Keywords Iterative methods · Nonlinear systems · Local convergence · Complex dynamics · Stability · Parameter space Mathematics Subject Classification 49M15 · 47H99 · 65D99 · 65G99 · 65H10 · 65J15
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Debasis Sharma [email protected] Ioannis K. Argyros [email protected] Sanjaya Kumar Parhi [email protected] Shanta Kumari Sunanda [email protected]
1
Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA
2
Department of Mathematics, IIIT Bhubaneswar, Bhubaneswar, Odisha 751003, India
3
Department of Mathematics, Fakir Mohan University, Balasore, Odisha 756020, India 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
(2020) 6:142
Introduction In numerical analysis and applied mathematics, constructing efficient iterative algorithms to obtain approximate solutions of nonlinear systems is a challenging and important task. These equations find many applications to a large number of phenomena in the domain of engineering and applied sciences. Various application problems like transfer of mass and heat within porous catalyst particles, Hammerstein integral equations and several boundary value problems related to integro-differential equations can be solved by finding approximate but accurate solutions of systems of nonlinear equations. These problems can be transformed into systems of nonlinear equations [1–4] of the form G(s) = 0,
(1)
where G : Ω ⊂ Rn → Rn , n ≥ 1, by applying different techniques such as discretization, Adomian decomposition, etc. Further to this, nonlinear reaction-diffusion equations appear in some chemical problems like autocatalytic chemical reactions [5] or the electronic structure analysis of the hydrogen atom can be treated numerically [6] by applying iterative algorithms. The numerical treatment of various chemical
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