New optimal fourth-order iterative method based on linear combination technique

Akbar Nadeem; Faisal Ali; Ji-huan He

doi:10.15672/hujms.909721

Research Article

Year 2021, , 1692 - 1708, 14.12.2021

Akbar Nadeem , Faisal Ali , Ji-huan He

https://doi.org/10.15672/hujms.909721

Cited By: 7

Abstract

References

[1] S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by mod- ified Adomian decomposition method, Appl. Math. Comput. 145, 887–893, 2003.
[2] 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.
[3] 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.
[4] M. Ali, N. Anjum and Q.T. Ain, Homotopy Perturbation Method for the Attachment Oscillator Arising in Nanotechnology, Fibers Polym. 22, 1601-1606, 2021.
[5] 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.
[6] 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.
[7] N. Anjum and J.H. He, Homotopy perturbation method for N/MEMS oscillators, Math. Methods Appl. Sci. DOI: 10.1002/mma.6583.
[8] 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.
[9] 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.
[10] 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.
[11] 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.
[12] E. Babolian and J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 132, 167–172, 2002.
[13] S. Bhalekar and V. Daftardar-Gejji, Convergence of new iterative method, Int. J. Differ. Equ. 2011, 989065, 10 pages, 2011.
[14] C. Chun, Some fourth-order iterative methods for solving nonlinear equations, Appl. Math. Comput. 195 (2), 454–459, 2018.
[15] V. Daftardar-Gejji and H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2), 753–763, 2006.
[16] M. Fahad, M. Nazeer, W. Kang and C.Y. Jung, A fixed point type iterative method and its dynamical behavior, Panam. Math. J. 27 (4), 86–99, 2017.
[17] M. Frontini and E. Sormani, Third-order methods from quadrature formulae for solv- ing systems of nonlinear equations, Appl. Math. Comput. 149, 771–782, 2004.
[18] C.H. He, An introduction to an ancient Chinese algorithm and its modification, Int. J. Num. Meth. Heat Fluid Flow, 26 (8), 2486–2491, 2016.
[19] C.H. He, C. Liu and K.A. Gepreel, Low frequency property of a fractal vibration model for a concrete beam, Fractals, 29 (05), 2021.
[20] C.H. He, Y. Shen and F.Y. Ji, Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 28 (1), 2050011, 2020.
[21] C.H. He, C. Liu, J.H. He, H. Mohammad-Sedighi, A. Shokri and K.A. Gepreel, A frac- tal model for the internal temperature response of a porous concrete, Appl. Comput. Math. 20 (2), 2021.
[22] C.H. He, C. Liu, J.H. He, A.H. Shirazi and H. Mohammad-Sedighi, Passive Atmo- spheric water harvesting utilizing an ancient Chinese ink slab and its possible ap- plications in modern architecture, Facta Univ. Ser.: Mech. Eng., 19 (2), 229–239, 2021.
[23] J.H. He, Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Eng. J. 11 (4), 1411–1414, 2020.
[24] J.H. He and Y.O. El-Dib, Homotopy perturbation method for Fangzhu oscillator, J. Math. Chem. 58 (10), 2245–2253, 2020.
[25] J.H. He and Y.O. El-Dib, Homotopy perturbation method with three expansions, J. Math. Chem. 59, 1139–1150.
[26] J.H. He and Y.O. El-Dib, Periodic property of the time-fractional Kundu–Mukherjee– Naskar equation, Results Phys. 19, 103345, 2020.
[27] J.H. He and Y.O. El-Dib, The enhanced homotopy perturbation method for axial vibration of strings, Facta Univ. Ser.: Mech. Eng. DOI: 10.22190/FUME210125033H.
[28] J.H. He and Y.O. El-Dib, The reducing rank method to solve third-order Duffing equation with the homotopy perturbation, Numer. Methods Partial Differ. Equ. 37 (2), 2021.
[29] J.H. He, N. Anjum and P.S. Skrzypacz, A Variational Principle for a Nonlinear Oscillator Arising in the Microelectromechanical System, J. Appl. Comput. Mech. 7 (1), 78–83, 2021.
[30] J.H. He, F.Y. Ji and H. Mohammad-Sedighi, Difference equation vs differential equa- tion on different scales, Int. J. Num. Meth. Heat Fluid Flow, 31 (1), 391–401, 2021.
[31] J.H. He, S.J. Kou and C.H. He, Fractal oscillation and its frequency-amplitude prop- erty, Fractals, DOI:10.1142/S0218348X2150105X.
[32] J.H. He, D. Nurakhmetov and P. Skrzypacz, Dynamic pull-in for micro- electromechanical device with a current-carrying conductor, J. Low Freq. Noise Vib. Act. Control, 40 (2), 1059–1066, 2021.
[33] J.H. He, N. Qie and C.H. He, Solitary waves travelling along an unsmooth boundary, Results Phys. 24, 104104, 2021.
[34] J.H. He, N. Qie, C.H. He and T. Saeed, On a strong minimum condition of a fractal variational principle, Appl. Math. Lett. 119, 107199, 2021.
[35] J.H. He, P.S. Skrzypacz, Y.N. Zhang and J. Pang, Approximate periodic solutions to microelectromechanical system oscillator subject to magneto static excitation, Math. Methods Appl. Sci. DOI: 10.1002/mma.7018.
[36] J.H. He, W.F. Hou, N. Qie, K.A. Gepreel, A.H. Shirazi and H. Mohammad-Sedighi, Hamiltonian-based frequency-amplitude formulation for nonlinear oscillators, Facta Univ. Ser.: Mech. Eng. DOI: 10.22190/FUME 201205002H.
[37] S. Huang, A. Rafiq, M.R. Shahzad and F. Ali, New higher order iterative methods for solving nonlinear equations, Hacettepe J. Math. Stat. 47 (1), 77–91, 2018.
[38] M.M.M. Joubari, D.D. Ganji and H.J. Jouybari, Determination of Periodic Solution for Tapered Beams with Modified Iteration Perturbation Method, J. Appl. Comput. Mech. 1 (1), 44–51, 2015.
[39] B. Kalantary, Polynomial Root-Finding and Polynomiography, World Sci. Publishing Co., Hackensack, 2009.
[40] W. Kang, M. Nazeer, A. Rafiq and C.J. Jung, A new third-order iterative method for scalar nonlinear equations, Int. J. Math. Anal. 8 (43), 2141–2150, 2014.
[41] H.T. Kung and J.F. Traub, Optimal order of one-point and multi-point iteration, Appl. Math. Comput. 21, 643–651, 1974.
[42] X.J. Li, Z. Liu and J.H. He, A fractal two-phase flow model for the fiber motion in a polymer filling process, Fractals, 28 (5), 2050093, 2020.
[43] M. Nadeem and J.H. He, He-Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics, J. Math. Chem. 59, 1234–1245, 2021.
[44] M. Nadeem, J.H. He and A. Islam, The homotopy perturbation method for fractional differential equations: part -1 Mohand transform, Int. J. Numer. Methods Heat Fluid Flow, DOI: 10.1108/HFF-11-2020-0703.
[45] M.A. Noor, Iterative methods for nonlinear equations using homotopy perturbation technique, Appl. Math. Inf. Sci. 4 (2), 227–235, 2010.
[46] M.A. Noor, and V. Gupta, Modified Householder iterative method free from second derivatives for nonlinear equations, Appl. Math. Comput. 190, 1701–1706, 2007.
[47] A.M. Ostrowski, Solution of equations in Euclidean and Banach Space, Third Ed., Academic Press, New York, 1973.
[48] F.A. Porta and V. Ptak, Nondiscrete induction and iterative process, Research Notes in Mathematics, Pitman, 103, Boston, 1994.
[49] R. Sharma and A. Bahl, An optimal fourth order iterative method for solving nonlinear equations and its dynamics, J. Complex Anal. 9, 259167–259176, 2015.
[50] S. Li, "Fourth-order iterative method without calculating the higher derivatives for nonlinear equation", J. Algorithm Comput. Technol. 13, 1–8, 2019.
[51] P. Skrzypacz, J.H. He, G. Ellis and M. Kuanyshbay, A simple approximation of pe- riodic solutions to microelectromechanical system model of oscillating parallel plate capacitor, Math. Meth. App. Sci. DOI: 10.1002/mma.6898.
[52] D. Tian, C.H. He and J.H. He, Fractal Pull-in Stability Theory for Microelectrome- chanical Systems, Front. Phys. DOI: 10.3389/fphy.2021.606011.
[53] J.F. Traub, Iterative Methods for the Solution of Equations, Printice Hall, New Jersey, 1964.
[54] M. Turkyilmazoglu, An optimal variational iteration method, Appl. Math. Lett. 24 (5), 762–765, 2011.
[55] M. Turkyilmazoglu, Is homotopy perturbation method the traditional Taylor series expansion, Hacettepe J. Math. Stat. 44 (3), 651–657, 2015.
[56] M. Turkyilmazoglu, Parametrized Adomian Decomposition Method with Optimum Convergence, ACM Trans. Model. Comput. Simul. 27 (4), 2017.
[57] M. Turkyilmazoglu, A simple algorithm for high order Newton iteration formulae and some new variants, Hacettepe J. Math. Stat. 49 (1), 425–438, 2020.
[58] S. Weerakon and T.J.I. Fernando, A variant of Newton s method with accelerated third-order convergence, Appl. Math. Lett. 13, 87–93, 2000.

New optimal fourth-order iterative method based on linear combination technique

Year 2021, , 1692 - 1708, 14.12.2021

Akbar Nadeem , Faisal Ali , Ji-huan He

https://doi.org/10.15672/hujms.909721

Cited By: 7

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.

Keywords

Iterative method, nonlinear equation, order of convergence, linear combination

References

[1] S. Abbasbandy, Improving Newton-Raphson method for nonlinear equations by mod- ified Adomian decomposition method, Appl. Math. Comput. 145, 887–893, 2003.
[2] 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.
[3] 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.
[4] M. Ali, N. Anjum and Q.T. Ain, Homotopy Perturbation Method for the Attachment Oscillator Arising in Nanotechnology, Fibers Polym. 22, 1601-1606, 2021.
[5] 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.
[6] 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.
[7] N. Anjum and J.H. He, Homotopy perturbation method for N/MEMS oscillators, Math. Methods Appl. Sci. DOI: 10.1002/mma.6583.
[8] 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.
[9] 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.
[10] 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.
[11] 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.
[12] E. Babolian and J. Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 132, 167–172, 2002.
[13] S. Bhalekar and V. Daftardar-Gejji, Convergence of new iterative method, Int. J. Differ. Equ. 2011, 989065, 10 pages, 2011.
[14] C. Chun, Some fourth-order iterative methods for solving nonlinear equations, Appl. Math. Comput. 195 (2), 454–459, 2018.
[15] V. Daftardar-Gejji and H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2), 753–763, 2006.
[16] M. Fahad, M. Nazeer, W. Kang and C.Y. Jung, A fixed point type iterative method and its dynamical behavior, Panam. Math. J. 27 (4), 86–99, 2017.
[17] M. Frontini and E. Sormani, Third-order methods from quadrature formulae for solv- ing systems of nonlinear equations, Appl. Math. Comput. 149, 771–782, 2004.
[18] C.H. He, An introduction to an ancient Chinese algorithm and its modification, Int. J. Num. Meth. Heat Fluid Flow, 26 (8), 2486–2491, 2016.
[19] C.H. He, C. Liu and K.A. Gepreel, Low frequency property of a fractal vibration model for a concrete beam, Fractals, 29 (05), 2021.
[20] C.H. He, Y. Shen and F.Y. Ji, Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, 28 (1), 2050011, 2020.
[21] C.H. He, C. Liu, J.H. He, H. Mohammad-Sedighi, A. Shokri and K.A. Gepreel, A frac- tal model for the internal temperature response of a porous concrete, Appl. Comput. Math. 20 (2), 2021.
[22] C.H. He, C. Liu, J.H. He, A.H. Shirazi and H. Mohammad-Sedighi, Passive Atmo- spheric water harvesting utilizing an ancient Chinese ink slab and its possible ap- plications in modern architecture, Facta Univ. Ser.: Mech. Eng., 19 (2), 229–239, 2021.
[23] J.H. He, Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Eng. J. 11 (4), 1411–1414, 2020.
[24] J.H. He and Y.O. El-Dib, Homotopy perturbation method for Fangzhu oscillator, J. Math. Chem. 58 (10), 2245–2253, 2020.
[25] J.H. He and Y.O. El-Dib, Homotopy perturbation method with three expansions, J. Math. Chem. 59, 1139–1150.
[26] J.H. He and Y.O. El-Dib, Periodic property of the time-fractional Kundu–Mukherjee– Naskar equation, Results Phys. 19, 103345, 2020.
[27] J.H. He and Y.O. El-Dib, The enhanced homotopy perturbation method for axial vibration of strings, Facta Univ. Ser.: Mech. Eng. DOI: 10.22190/FUME210125033H.
[28] J.H. He and Y.O. El-Dib, The reducing rank method to solve third-order Duffing equation with the homotopy perturbation, Numer. Methods Partial Differ. Equ. 37 (2), 2021.
[29] J.H. He, N. Anjum and P.S. Skrzypacz, A Variational Principle for a Nonlinear Oscillator Arising in the Microelectromechanical System, J. Appl. Comput. Mech. 7 (1), 78–83, 2021.
[30] J.H. He, F.Y. Ji and H. Mohammad-Sedighi, Difference equation vs differential equa- tion on different scales, Int. J. Num. Meth. Heat Fluid Flow, 31 (1), 391–401, 2021.
[31] J.H. He, S.J. Kou and C.H. He, Fractal oscillation and its frequency-amplitude prop- erty, Fractals, DOI:10.1142/S0218348X2150105X.
[32] J.H. He, D. Nurakhmetov and P. Skrzypacz, Dynamic pull-in for micro- electromechanical device with a current-carrying conductor, J. Low Freq. Noise Vib. Act. Control, 40 (2), 1059–1066, 2021.
[33] J.H. He, N. Qie and C.H. He, Solitary waves travelling along an unsmooth boundary, Results Phys. 24, 104104, 2021.
[34] J.H. He, N. Qie, C.H. He and T. Saeed, On a strong minimum condition of a fractal variational principle, Appl. Math. Lett. 119, 107199, 2021.
[35] J.H. He, P.S. Skrzypacz, Y.N. Zhang and J. Pang, Approximate periodic solutions to microelectromechanical system oscillator subject to magneto static excitation, Math. Methods Appl. Sci. DOI: 10.1002/mma.7018.
[36] J.H. He, W.F. Hou, N. Qie, K.A. Gepreel, A.H. Shirazi and H. Mohammad-Sedighi, Hamiltonian-based frequency-amplitude formulation for nonlinear oscillators, Facta Univ. Ser.: Mech. Eng. DOI: 10.22190/FUME 201205002H.
[37] S. Huang, A. Rafiq, M.R. Shahzad and F. Ali, New higher order iterative methods for solving nonlinear equations, Hacettepe J. Math. Stat. 47 (1), 77–91, 2018.
[38] M.M.M. Joubari, D.D. Ganji and H.J. Jouybari, Determination of Periodic Solution for Tapered Beams with Modified Iteration Perturbation Method, J. Appl. Comput. Mech. 1 (1), 44–51, 2015.
[39] B. Kalantary, Polynomial Root-Finding and Polynomiography, World Sci. Publishing Co., Hackensack, 2009.
[40] W. Kang, M. Nazeer, A. Rafiq and C.J. Jung, A new third-order iterative method for scalar nonlinear equations, Int. J. Math. Anal. 8 (43), 2141–2150, 2014.
[41] H.T. Kung and J.F. Traub, Optimal order of one-point and multi-point iteration, Appl. Math. Comput. 21, 643–651, 1974.
[42] X.J. Li, Z. Liu and J.H. He, A fractal two-phase flow model for the fiber motion in a polymer filling process, Fractals, 28 (5), 2050093, 2020.
[43] M. Nadeem and J.H. He, He-Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics, J. Math. Chem. 59, 1234–1245, 2021.
[44] M. Nadeem, J.H. He and A. Islam, The homotopy perturbation method for fractional differential equations: part -1 Mohand transform, Int. J. Numer. Methods Heat Fluid Flow, DOI: 10.1108/HFF-11-2020-0703.
[45] M.A. Noor, Iterative methods for nonlinear equations using homotopy perturbation technique, Appl. Math. Inf. Sci. 4 (2), 227–235, 2010.
[46] M.A. Noor, and V. Gupta, Modified Householder iterative method free from second derivatives for nonlinear equations, Appl. Math. Comput. 190, 1701–1706, 2007.
[47] A.M. Ostrowski, Solution of equations in Euclidean and Banach Space, Third Ed., Academic Press, New York, 1973.
[48] F.A. Porta and V. Ptak, Nondiscrete induction and iterative process, Research Notes in Mathematics, Pitman, 103, Boston, 1994.
[49] R. Sharma and A. Bahl, An optimal fourth order iterative method for solving nonlinear equations and its dynamics, J. Complex Anal. 9, 259167–259176, 2015.
[50] S. Li, "Fourth-order iterative method without calculating the higher derivatives for nonlinear equation", J. Algorithm Comput. Technol. 13, 1–8, 2019.
[51] P. Skrzypacz, J.H. He, G. Ellis and M. Kuanyshbay, A simple approximation of pe- riodic solutions to microelectromechanical system model of oscillating parallel plate capacitor, Math. Meth. App. Sci. DOI: 10.1002/mma.6898.
[52] D. Tian, C.H. He and J.H. He, Fractal Pull-in Stability Theory for Microelectrome- chanical Systems, Front. Phys. DOI: 10.3389/fphy.2021.606011.
[53] J.F. Traub, Iterative Methods for the Solution of Equations, Printice Hall, New Jersey, 1964.
[54] M. Turkyilmazoglu, An optimal variational iteration method, Appl. Math. Lett. 24 (5), 762–765, 2011.
[55] M. Turkyilmazoglu, Is homotopy perturbation method the traditional Taylor series expansion, Hacettepe J. Math. Stat. 44 (3), 651–657, 2015.
[56] M. Turkyilmazoglu, Parametrized Adomian Decomposition Method with Optimum Convergence, ACM Trans. Model. Comput. Simul. 27 (4), 2017.
[57] M. Turkyilmazoglu, A simple algorithm for high order Newton iteration formulae and some new variants, Hacettepe J. Math. Stat. 49 (1), 425–438, 2020.
[58] S. Weerakon and T.J.I. Fernando, A variant of Newton s method with accelerated third-order convergence, Appl. Math. Lett. 13, 87–93, 2000.

There are 58 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	Akbar Nadeem 0000-0003-2361-4606 Faisal Ali 0000-0002-4893-0616 Ji-huan He 0000-0002-1636-0559
Publication Date	December 14, 2021
Published in Issue	Year 2021

Cite

APA	Nadeem, A., Ali, F., & He, J.-h. (2021). New optimal fourth-order iterative method based on linear combination technique. Hacettepe Journal of Mathematics and Statistics, 50(6), 1692-1708. https://doi.org/10.15672/hujms.909721
AMA	Nadeem A, Ali F, He Jh. New optimal fourth-order iterative method based on linear combination technique. Hacettepe Journal of Mathematics and Statistics. December 2021;50(6):1692-1708. doi:10.15672/hujms.909721
Chicago	Nadeem, Akbar, Faisal Ali, and Ji-huan He. “New Optimal Fourth-Order Iterative Method Based on Linear Combination Technique”. Hacettepe Journal of Mathematics and Statistics 50, no. 6 (December 2021): 1692-1708. https://doi.org/10.15672/hujms.909721.
EndNote	Nadeem A, Ali F, He J-h (December 1, 2021) New optimal fourth-order iterative method based on linear combination technique. Hacettepe Journal of Mathematics and Statistics 50 6 1692–1708.
IEEE	A. Nadeem, F. Ali, and J.-h. He, “New optimal fourth-order iterative method based on linear combination technique”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 6, pp. 1692–1708, 2021, doi: 10.15672/hujms.909721.
ISNAD	Nadeem, Akbar et al. “New Optimal Fourth-Order Iterative Method Based on Linear Combination Technique”. Hacettepe Journal of Mathematics and Statistics 50/6 (December 2021), 1692-1708. https://doi.org/10.15672/hujms.909721.
JAMA	Nadeem A, Ali F, He J-h. New optimal fourth-order iterative method based on linear combination technique. Hacettepe Journal of Mathematics and Statistics. 2021;50:1692–1708.
MLA	Nadeem, Akbar et al. “New Optimal Fourth-Order Iterative Method Based on Linear Combination Technique”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 6, 2021, pp. 1692-08, doi:10.15672/hujms.909721.
Vancouver	Nadeem A, Ali F, He J-h. New optimal fourth-order iterative method based on linear combination technique. Hacettepe Journal of Mathematics and Statistics. 2021;50(6):1692-708.