On Computational Efficiency and Dynamical Analysis for a Class of Novel Multi-step Iterative Schemes
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On Computational Efficiency and Dynamical Analysis for a Class of Novel Multi-step Iterative Schemes K. Sayevand1 · R. Erfanifar1 · H. Esmaeili2 Accepted: 17 October 2020 © Springer Nature India Private Limited 2020
Abstract This paper presents a novel multi-step iterative scheme to solve system of nonlinear equations. Because the cost of calculating the Fréchet derivative evaluation and its inversion is significant, in order to achieve its high computational efficiency, we have tried to calculate Fréchet derivative and its inverse less per cycle by using the proposed multi-step iterative schemes. The basic iterative scheme has a convergence order of four; therefore, repetition of the second step can achieve higher convergence order. In fact, adding a new step to the base iterative scheme each time increases the convergence order by two units. The multi-step iterative schemes have convergence-order 2m, where m is the number of steps of the multistep iterative schemes. Also, the computational efficiency of the iterative scheme is compared with other available methods. The numerical results presented confirm the theoretical results. A number of nonlinear system of equations associated with the numerical approximation of ordinary, partial, and fractional differential equations are made up and solved. Keywords Iterative schemes · Multi-step scheme · Computational efficiency · Systems of nonlinear equations · Fractional calculus Mathematics Subject Classification 65F05 · 65L03 · 65M99
Introduction Analysing the systems of nonlinear equations (SNEs) is an important matter in the field of numerical analysis. Many engineering and physical processes could be expressed in the form of SNEs, and most practical problems lead to finding the roots of these equations. Also,
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K. Sayevand [email protected] R. Erfanifar [email protected] H. Esmaeili [email protected]
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Faculty of Mathematics and Statistics, Malayer University, Malayer, Iran
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Department of Mathematics, Bu-Ali Sina University, Hamedan, Iran 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
(2020) 6:163
there are a few categories for SNEs, the solutions of which could be presented in closed form. Therefore, solving SNEs becomes of particular importance. In fact, solving nonlinear problems are very difficult analytically and, as a result, a numerical method has always been sought to solve problems in the desired form. Iterative schemes are one of the most important ways to get a better approximation of SNEs in applied mathematics. The structure of these schemes for SNEs is one of the interesting and challenging tasks in numerical analysis. In recent decades, iterative schemes have been used in different fields such as dynamic models, economics, physics, engineering, etc [1–12]. In this regard, it is of paramount importance to provide new iterative schemes to solve nonlinear ordinary differential equations (ODEs), partial differential equations (PDEs) and fractional differential equations (FDEs). Consider th
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