New Family of MultiStep Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations
Abstract
:1. Introduction
2. The New Iterative Methods Construction
Algorithm 1: A suggested onestep iterative method. 
For a suitable initial approximation ${x}_{0}$, calculate the next solution ${x}_{i+1}$ using the below iterative method
$$\begin{array}{c}\hfill {x}_{i+1}={x}_{i}\frac{F\left(\right)open="("\; close=")">{x}_{i}}{}{F}^{\prime}\left(\right)open="("\; close=")">{x}_{i}& \frac{{F}^{2}\left(\right)open="("\; close=")">{x}_{i}}{{F}^{\u2033}}\\ 2{\left(\right)}^{{F}^{\prime}}3\end{array}$$

Algorithm 2: A suggested twostep iterative method (HM2). 
For a suitable initial approximation ${x}_{0}$, calculate the next solution ${x}_{i+1}$ using the following iterative method:
$$\begin{array}{c}{y}_{i}={x}_{i}\frac{F\left(\right)open="("\; close=")">{x}_{i}}{}{\lambda}_{i}F\left(\right)open="("\; close=")">{x}_{i}+{F}^{\prime}\left(\right)open="("\; close=")">{x}_{i}\\ ,\phantom{\rule{1.em}{0ex}}i=0,1,2,\dots ,\end{array}$$

Algorithm 3: A suggested threestep iterative method (HM3). 
For a suitable initial approximation ${x}_{0}$, calculate the next solution ${x}_{i+1}$ using the following iterative method:
$$\begin{array}{c}{y}_{i}={x}_{i}\frac{F\left(\right)open="("\; close=")">{x}_{i}}{}{\lambda}_{i}F\left(\right)open="("\; close=")">{x}_{i}+{F}^{\prime}\left(\right)open="("\; close=")">{x}_{i}\\ ,\phantom{\rule{1.em}{0ex}}i=0,1,2,\dots ,\end{array}$$

3. Convergence Investigation
4. Numerical Applications
5. Discussion and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
 Noor, K.I.; Noor, M.A. Predictorcorrector Hally method for nonlinear equations. Appl. Math. Comput. 2007, 188, 1587–1591. [Google Scholar]
 Noor, M.A.; Noor, K.I.; MohyudDin, S.T.; Shabbir, A. An iterative method with cubic convergence for nonlinear equations. Appl. Math. Comput. 2006, 183, 1249–1255. [Google Scholar] [CrossRef]
 Noor, M.A. Some iterative methods for solving nonlinear equations using homotopy perturbation method. Int. J. Comput. Math. 2010, 1, 141149. [Google Scholar] [CrossRef]
 Noor, M.A.; Khan, W.A. New iterative methods for solving nonlinear equation by using homotopy perturbation method. Appl. Math. Comput. 2012, 219, 3565–3574. [Google Scholar] [CrossRef]
 AbdulHassan, N.Y.; Ali, A.H.; Park, C. A new fifthorder iterative method free from second derivative for solving nonlinear equations. J. Appl. Math. Comput. 2022, 68, 2877–2886. [Google Scholar] [CrossRef]
 Liao, S.J. The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, China, 1992. [Google Scholar]
 He, J.H. Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 1999, 178, 257–262. [Google Scholar] [CrossRef]
 Liao, S.J. Comparison between the homotopy analysis method and homotopy perturbation method. Appl. Math. Comput. 2005, 169, 1186–1194. [Google Scholar] [CrossRef]
 AzZo’bi, E.A.; AlKhaled, K.; Darweesh, A. Numericanalytic solutions for nonlinear oscillators via the modified multistage decomposition method. Mathematics 2019, 7, 550. [Google Scholar] [CrossRef] [Green Version]
 Alim, M.A.; Kawser, M.A. Illustration of the homotopy perturbation method to the modified nonlinear single degree of freedom system. Chaos Solitons Fractals 2023, 171, 113481. [Google Scholar] [CrossRef]
 Sehati, M.M.; Karbassi, S.M.; Heydari, M.; Loghmani, G.B. several new iterative methods for solving nonlinear algebraic equations incorporating homotopy perturbation method (HPM). Int. J. Phys. Sci. 2012, 12, 5017–5025. [Google Scholar] [CrossRef]
 Waheed, A.; Din, S.T.M.; Zeb, M.; Usman, M. Some Higher Order Algorithms for Solving Fixed Point Problems. Commun. Math. Appl. 2018, 1, 41–52. [Google Scholar]
 Freno, B.A.; Carlberg, T.K. Machinelearning error models for approximate solutions to parameterized systems of nonlinear equations. Comput. Methods Appl. Mech. Eng. 2019, 348, 250–296. [Google Scholar] [CrossRef] [Green Version]
 Gong, W.; Liao, Z.; Mi, X.; Wang, L.; Guo, Y. Nonlinear equations solving with intelligent optimization algorithms: A survey. Complex Syst. Model. Simul. 2021, 1, 15–32. [Google Scholar] [CrossRef]
 Abbasbandy, S. Improving NewtonRaphson method for nonlinear equations by modified Adomian decomposition method. Appl. Math. Comput. 2023, 145, 887–893. [Google Scholar] [CrossRef]
 Saeed, H.J.; AbdulHassan, N.Y. An Efficient ThreeStep Iterative Methods Based on Bernstein Quadrature Formula for Solving Nonlinear Equations. Basrah J. Sci. 2021, 3, 355–383. [Google Scholar] [CrossRef]
 Wu, X.Y. A New Continuation NewtonLike Method and its Deformation. Appl. Math. Comput. 2000, 112, 75–78. [Google Scholar] [CrossRef]
 Aziz, I.; SirajulIslam; Khan, W. Quadrature Rules for Numerical Integration Based on Haar wavelets and Hybrid Functions. Comput. Math. Appl. 2011, 61, 2770–2781. [Google Scholar] [CrossRef] [Green Version]
 Ali, A.H.; Páles, Z. Taylortype Expansions in Terms of Exponential Polynomials. Math. Inequalities Appl. 2022, 25, 1123–1141. [Google Scholar] [CrossRef]
 Abbasbandy, S. Modified homotopy perturbation method for nonlinear equations and comparison with Adomian decomposition method. Appl. Math. Comput. 2006, 172, 431–438. [Google Scholar] [CrossRef]
 Ljajko, E.; Tosic, M.; Kevkic, T.; Stojanovic, V. Application of the Homotopy Perturbations Method in Approximation Probability Distributions of Nonlinear Time Series. Univ. Politeh. Buchar. Sci.Bull.Ser.Appl. Math. Phys. 2021, 83, 177–186. [Google Scholar]
 Khan, K.; Syed, H.Z. Semi Analytic Solution of HodgkinHuxley Model by Homotopy Perturbation Method. Punjab Univ. J. Math. 2021, 53, 825–842. [Google Scholar] [CrossRef]
 Chun, C. A new iterative method for solving nonlinear equations. Appl. Math. Comput. 2006, 178, 415–422. [Google Scholar] [CrossRef]
$\mathit{F}\left(\mathit{x}\right)$  $\mathit{\alpha}$ 

${F}_{1}\left(x\right)=x{e}^{x}2=0$  2.12002823898764122948468710 
${F}_{2}\left(x\right)=x{e}^{{x}^{2}}{(sinx)}^{2}+3cosx+5=0$  −1.2076478271309189270094168 
${F}_{3}\left(x\right)={x}^{2}{(1x)}^{5}=0$  0.3459548158482420179582044 
${F}_{4}\left(x\right)=ln\left(\right)open="("\; close=")">{x}^{2}+x+2$  16.6951567675073761976927688 
${F}_{5}\left(x\right)=(1+cosx)\left(\right)open="("\; close=")">{e}^{x}2$  0.6931471805599453094172322 
$\mathit{F}\left(\mathit{x}\right)$  ${\mathit{x}}_{0}$  Methods  NI  $\left(\right)open=""\; close="">{\mathit{x}}_{\mathit{i}+1}{\mathit{x}}_{\mathit{i}}$  $\left(\right)open=""\; close="">\mathit{F}\left(\right)open="("\; close=")">{\mathit{x}}_{\mathit{i}+1}$  COC 

${F}_{1}$  2.0  NR2  3  6.8094693 $\times {10}^{16}$  6.3164342 $\times {10}^{48}$  3.0104059 
CM  3  8.93125110 $\times {10}^{31}$  5.4817818 $\times {10}^{124}$  4.01084410  
AL2  3  2.2199937 $\times {10}^{78}$  2.0350825 $\times {10}^{548}$  7.0116691  
HM2  3  9.0762879 $\times {10}^{26}$  6.0604694 $\times {10}^{103}$  4.0094105  
HM3  3  3.9771853 $\times {10}^{98}$  3.1045139 $\times {10}^{784}$  8.0082867  
${F}_{2}$  −1.0  NR2  5  1.8095423 $\times {10}^{39}$  2.4092523 $\times {10}^{115}$  3.0000063 
CM  5  1.5057542 $\times {10}^{44}$  1.7691495 $\times {10}^{173}$  3.9998224  
AL2  3  1.65614610 $\times {10}^{22}$  6.6848624 $\times {10}^{150}$  6.7759406  
HM2  4  4.0189653 $\times {10}^{42}$  1.0630597 $\times {10}^{164}$  3.9996505  
HM3  3  1.9180050 $\times {10}^{49}$  1.8554791 $\times {10}^{389}$  7.9462676  
${F}_{3}$  0.2  NR2  4  8.2499542 $\times {10}^{20}$  2.4019841 $\times {10}^{57}$  2.9998339 
CM  4  5.4788184 $\times {10}^{36}$  1.0139405 $\times {10}^{140}$  4.0007370  
AL2  4  2.6904210 $\times {10}^{100}$  3.71201210 $\times {10}^{695}$  6.9990345  
HM2  4  3.7572186 $\times {10}^{44}$  5.2755324 $\times {10}^{174}$  3.9993329  
HM3  3  1.1771498 $\times {10}^{55}$  2.3415694 $\times {10}^{440}$  8.0457563  
${F}_{4}$  16.0  NR2  3  2.0752478 $\times {10}^{17}$  1.1309175 $\times {10}^{54}$  3.0094031 
CM  3  7.0787489 $\times {10}^{36}$  5.8881265 $\times {10}^{148}$  4.0092711  
AL2  diverge  diverge  diverge  diverge  
HM2  3  3.8856993 $\times {10}^{23}$  1.4531784 $\times {10}^{94}$  4.0468837  
HM3  3  1.9058709 $\times {10}^{84}$  4.3697445 $\times {10}^{679}$  8.0447728  
${F}_{5}$  0.5  NR2  4  1.8521494 $\times {10}^{29}$  5.2002262 $\times {10}^{87}$  2.9999240 
CM  3  4.8721383 $\times {10}^{19}$  2.6686961 $\times {10}^{75}$  4.1825251  
AL2  3  2.9291152 $\times {10}^{44}$  9.7204429 $\times {10}^{306}$  6.9492016  
HM2  3  5.1650339 $\times {10}^{16}$  3.7212218 $\times {10}^{62}$  4.0243631  
HM3  3  9.3624134 $\times {10}^{65}$  1.5920864 $\times {10}^{514}$  8.0130984 
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. 
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Saeed, H.J.; Ali, A.H.; Menzer, R.; Poțclean, A.D.; Arora, H. New Family of MultiStep Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations. Mathematics 2023, 11, 2603. https://doi.org/10.3390/math11122603
Saeed HJ, Ali AH, Menzer R, Poțclean AD, Arora H. New Family of MultiStep Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations. Mathematics. 2023; 11(12):2603. https://doi.org/10.3390/math11122603
Chicago/Turabian StyleSaeed, Huda J., Ali Hasan Ali, Rayene Menzer, Ana Danca Poțclean, and Himani Arora. 2023. "New Family of MultiStep Iterative Methods Based on Homotopy Perturbation Technique for Solving Nonlinear Equations" Mathematics 11, no. 12: 2603. https://doi.org/10.3390/math11122603