New technique for the approximation of the zeros of nonlinear scientific models
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Faisal Ali
Abstract
Most of the problems in mathematical and engineering sciences can be studied in the context of nonlinear equations. In this paper, we develop a new family of iterative methods for the approximation of the zeros of mathematical models whose governing equations are nonlinear in nature. The proposed methods are based on decomposition technique due to Daftardar-Gejji and Jaffri [1]. The new family gives several iterative schemes as special cases. The convergence analysis of proposed methods is also presented. In order to determine the performance of newly developed methods, numerical as well as graphical analysis of four complex mathematical models from diverse fields of science and engineering are considered. We also consider some complex polynomials to visualize the roots through polynomiography in the context of proposed methods.
Acknowledgement
The authors are grateful to the referees for their suggestions to improve the quality of the paper.
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Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This research is not being funded by any agency.
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Conflict of interest statement: The authors declare that there is no conflicts of interest.
References
[1] V. Daftardar-Gejji and H. Jafari, “An iterative method for solving nonlinear functional equations,” J. Math. Anal. Appl., vol. 316, pp. 753–763, 2006, https://doi.org/10.1016/j.jmaa.2005.05.009.Search in Google Scholar
[2] M. A. Ostrowski, Solution of Equations in Euclidean and Banach Space, 3rd ed., New York, Academic Press, 1960.Search in Google Scholar
[3] S. Abbasbandy, “Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method,” Appl. Math. Comput., vol. 145, pp. 887–893, 2003, https://doi.org/10.1016/s0096-3003(03)00282-0.Search in Google Scholar
[4] G. Adomian, Nonlinear Stochastic System and Applications to Physics, Dordrecht, Kluwer Academic Publishers, 1989.10.1007/978-94-009-2569-4Search in Google Scholar
[5] E. Babolian and J. Biazar, “Solution of nonlinear equations by modified Adomian decomposition method,” Appl. Math. Comput., vol. 132, pp. 167–172, 2002, https://doi.org/10.1016/s0096-3003(01)00184-9.Search in Google Scholar
[6] S. Bhalekar and V. Daftardar-Gejji, “Convergence of new iterative method,” Int. J. Differ. Eqs., vol. 2011, pp. 1–10, 2011, https://doi.org/10.1155/2011/989065.Search in Google Scholar
[7] M. Dehghan and M. Hajarian, “New iterative method for solving non-linear equations with fourth-order convergence,” Int. J. Comput. Math., vol. 87, no. 4, pp. 834–839, 2010, https://doi.org/10.1080/00207160802217201.Search in Google Scholar
[8] M. Javidi, “Fourth-order and fifth-order iterative methods for nonlinear algebraic equations,” Math. Comput. Model., vol. 50, pp. 66–71, 2009, https://doi.org/10.1016/j.mcm.2009.02.004.Search in Google Scholar
[9] M. S. Kang, F. Ali, and A. Rafiq, “Iterative methods for solving scalar equations,” J. Nonlinear Sci. Appl., vol. 9, pp. 1035–1042, 2016, https://doi.org/10.22436/jnsa.009.03.31.Search in Google Scholar
[10] A. M. Noor, M. Waseem, I. K. Noor, and A. M. Ali, “New iterative technique for solving nonlinear equations,” Appl. Math. Comput., vol. 265, pp. 1115–1129, 2015, https://doi.org/10.1016/j.amc.2015.05.129.Search in Google Scholar
[11] A. Rafiq and M. Rafiullah, “Some multi-step iterative methods for solving nonlinear equations,” Comput. Math. Appl., vol. 58, pp. 1589–1597, 2009, https://doi.org/10.1016/j.camwa.2009.07.031.Search in Google Scholar
[12] S. Saba, A. Naseem, and M. I. Saleem, “Modified abbasbanday’s method free from second derivative for solving nonlinear equations,” Open J. Math. Sci, vol. 3, pp. 109–114, 2019, https://doi.org/10.30538/oms2019.0053.Search in Google Scholar
[13] J. F. Traub, Iterative Methods for the Solution of Equations, New Jersey, USA, Prentice-Hall Englewood Cliffs, 1964.Search in Google Scholar
[14] S. Weerakon and I. G. T. Fernando, “A variant of Newton’s method with accelerated third-order convergence,” Appl. Math. Lett., vol. 13, pp. 87–93, 2000, https://doi.org/10.1016/S0893-9659(00)00100-2.Search in Google Scholar
[15] M. Zingil and F. S. Topal, “Oscillation criteria for nonlinear dynamic equations on time scales,” Open J. Math. Sci., vol. 2, no. 1, pp. 307–322, 2018, https://doi.org/10.30538/oms2018.0037.Search in Google Scholar
[16] D. K. R. Babajee, Analysis of Higher Order Variants of Newton’s Method and their Applications to Differential and Integral Equations and in Ocean Acidification, December, Ph.D. Thesis, University of Mauritius, Mauritius, 2010.Search in Google Scholar
[17] R. F. King, “A family of fourth-order methods for solving nonlinear equations,” SIAM J. Numer. Anal., vol. 10, no. 5, pp. 876–879, 1973, https://doi.org/10.1137/0710072.Search in Google Scholar
[18] F. Ali, W. Aslam, M. A. Anwar, and A. Nadeem, “New family of iterative methods for solving nonlinear models,” Discrete Dynam Nat. Soc., vol. 12, 2018, Art no. 9619680. https://doi.org/10.1155/2018/9619680.Search in Google Scholar
[19] F. Ali, W. Aslam, and A. Rafiq, “Some new iterative techniques for the problems involving nonlinear equations,” Int. J. Comput. Methods, vol. 17, no. 7, pp. 18, 2020, https://doi.org/10.1142/S0219876219500373.Search in Google Scholar
[20] R. Behl, P. Maroju, and S. S. Motsa, “Efficient family of sixth-order methods for nonlinear models with its dynamics,” Int. J. Comput. Methods, vol. 15, no. 1, pp. 26, 2018.10.1142/S021987621840008XSearch in Google Scholar
[21] J. M. Douglas, Process Dynamics and Control, vol. 2, Englewood Cliffs, Prentice-Hall, 1972.Search in Google Scholar
[22] B. Mandelbrot, The Fractal Geometry of Nature, New York, W. H. Freeman and Co., 1982.Search in Google Scholar
[23] K. Gdawiec, W. Kotarski, and A. Lisowska, “Polynomiography based on the nonstandard Newton like root finding methods,” Abstr. Appl. Anal., vol. 2015, pp. 19, 2015, Art no. 797594, https://doi.org/10.1155/2015/797594.Search in Google Scholar
[24] B. Kalantari, Polynomial Root-Finding and Polynomiography, Hackensack, World Sci. Publishing Co., 2009.10.1142/6265Search in Google Scholar
[25] W. Kotarski, K. Gdawiec, and A. Lisowska, “Polynomiography via Ishikawa and Mann iterations, ISVCLNCS 2012, Part I,” vol. 7431, pp. 305–313, https://doi.org/10.1007/978-3-642-33179-4_30.Search in Google Scholar
[26] W. Nazeer, M. Tanveer, S. M. Kang, and A. Naseem, “A new Householder’s method free from second derivatives for solving nonlinear equations and polynomiography,” J. Nonlinear Sci. Appl., vol. 9, pp. 998–1007, 2016, https://doi.org/10.22436/jnsa.009.03.28.Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
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Articles in the same Issue
- Frontmatter
- Original Research Articles
- Research on prediction model of thermal and moisture comfort of underwear based on principal component analysis and Genetic Algorithm–Back Propagation neural network
- Design of the state feedback-based feed-forward controller asymptotically stabilizing the double-pendulum-type overhead cranes with time-varying hoisting rope length
- Blowup and global existence of mild solutions for fractional extended Fisher–Kolmogorov equations
- Lipschitz stability of nonlinear ordinary differential equations with non-instantaneous impulses in ordered Banach spaces
- Exploring the effects of awareness and time delay in controlling malaria disease propagation
- Algebro-geometric constructions of the Heisenberg hierarchy
- New technique for the approximation of the zeros of nonlinear scientific models
- Norm inequalities on variable exponent vanishing Morrey type spaces for the rough singular type integral operators
- Existence, stability and controllability results of fractional dynamic system on time scales with application to population dynamics
- Numerical solution for the fractional-order one-dimensional telegraph equation via wavelet technique
- Bell-shaped soliton solutions and travelling wave solutions of the fifth-order nonlinear modified Kawahara equation
- Existence and uniqueness of solutions of nonlinear fractional order problems via a fixed point theorem
