Authors:
Sehrish Umar,Muhammad Mujtaba Shaikh,Abdul Wasim Shaikh,DOI NO:
https://doi.org/10.26782/jmcms.2020.10.00006Keywords:
Cost-efficient,Quadrature,Nonlinear equations,Order of convergence,Efficiency index,Abstract
Nonlinear equations and their efficient numerical solution is a fundamental issue in the field of research in mathematics because nature is full of nonlinear models demanding careful solution and consideration. In this work, a new two-step iterative method for solving nonlinear equations has been developed by using quadrature formula so that the cost of evaluations is considerably reduced. The proposed strategy successfully removes the use of an additional derivative in an existing method in literature so that there is no compromise at all on the cubic convergence rate. The developed scheme is cubically convergent and uses a functional and three derivative evaluations only as compared to some other methods in the literature using much higher evaluations. The theorems concerning the derivation of the proposed method and its third order of convergence have been discussed with proofs. Performance evaluation of the new proposed scheme has been discussed with some methods from literature including well-known traditional methods. An exhaustive numerical verification has been done under the same numerical conditions on ten examples from literature. The efficiency index is found to be higher for the new proposed scheme than some schemes with order more than three, and comparable with some methods. The comparison using observed absolute errors, number of iterations, functional and derivative evaluations, and observed convergence reveals that the proposed method finds the solutions quickly and with lesser computational cost as compared to most of the other methods used in the comparison. The results show the encouraging performance of the proposed method.Refference:
I. Alamin Khan Md., Abu Hashan Md. Mashud, M. A. Halim, “NUMEROUS EXACT SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY TAN–COT METHOD”, J. Mech. Cont. & Math. Sci., Vol.-11, No.-2, January (2017) Pages 37-48
II. Abro H. A., Shaikh M. M., (2019), A new time efficient and convergent nonlinear solver , Applied Mathematics and Computation 355, 516-536.
III. Akram, S. and Q. U. Ann.,(2015). Newton Raphson Method, International Journal of Scientific & Engineering Research, Volume 6.
IV. Allame M., and N. Azad, 2012.On Modified Newton Method for Solving a Nonlinear Algebraic Equations by Mid-Point, World Applied Sciences Journal 17 (12): 1546-1548, ISSN 1818-4952 IDOSI Publications.
V. Biswa N. D. (2012), Lecture Notes on Numerical Solution of root Finding Problems.
VI. C. Chun and Y. Ham, (2007) “A one-parameter fourth-order family of iterative methods for nonlinear equations,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 610–614
VII. C. Chun and Y. Ham, (2008). “Some fourth-order modifications of Newton’s method,” Applied Mathematics and Computation, vol. 197, no. 2, pp.654–658
VIII. Chapra, S. C., & Canale, R. P. (1998). Numerical methods for engineers (Vol. 2). New York: Mcgraw-hill.
IX. Chitra S., P. Thapliyal, K.Tomar, (2014), “Role of Bisection Method”, International Journal of Computer Applications Technology and Research, vol, 3, 533-535.
X. Dunn, S., Constantinides, A., & Moghe, P. V. (2005). Numerical methods in biomedical engineering. Elsevier.
XI. Farooq Ahmed Shah, Muhammad Aslam Noor and Moneeza Batool, (2014) Derivative-Free Iterative Methods for Solving Nonlinear Equations, Appl. Math. Inf . Sci. 8, No. 5, 2189-2193.
XII. Golbabai, A., Javidi, M., “A Third-Order Newton Type Method for Nonlinear Equations Based on Modified Homotopy Perturbation Method”, Appl. Math. And Comput., 191, 199–205, 2007.
XIII. Iwetan, C. N., I. A. Fuwape, M. S. Olajide, and R. A. Adenodi, (2012), Comparative Study of the Bisection and Newton Methods in solving for Zero andExtremes of a Single-Variable Function. J. of NAMP Vol.21 173-176.
XIV. Khoso, Amjad Hussain, Muhammad Mujtaba Shaikh, and Ashfaque Ahmed Hashmani. “A New and Efficient Nonlinear Solver for Load Flow Problems.” Engineering, Technology & Applied Science Research 10, no. 3 (2020): 5851-5856.
XV. Liang Fang, Li Sun and Goping He, (2008), On An efficient Newton-type method with fifth-order convergence for solving nonlinear equations, Comp. Appl. Math., Vol. 27, N. 3,
XVI. M. A. Hafiz & Mohamed S. M. Bahgat, An Efficient Two-step Iterative Method for Solving System of Nonlinear EquationsJournal of Mathematics Research; Vol. 4, No. 4; 2012.
XVII. M. Aslam Noor, K. Inayat Noor, and M. Waseem, (2010).“Fourth-order iterative methods for solving nonlinear equations,” International Journal of Applied Mathematics and Engineering Sciences, vol. 4, pp. 43–52
XVIII. Muhammad Aslam Noor, Khalida Inayat Noor and Kshif Aftab(2012), Some New Iterative Methods for Solving Nonlinear Equations, World Applied Sciences Journal 20 (6): 870-874, 2012
XIX. Muhammad Aslam Noor, Khalida Inayat Noor, Eisa Al-Said and Muhammad Waseem. Volume 2010 .Some New Iterative Methods for Nonlinear Equations, Hindawi Publishing Corporation Mathematical Problems in Engineering
XX. Noor, M. A., F. Ahmad, Numerical compression of iterative method for solving nonlinear equation Applied Mathematics and Computation, 167-172, (2006).
XXI. Rafiq, A., S. M. Kang and Y. C. Kwun., 2013. A New Second-Order Iteration Method for Solving Nonlinear Equations, Hindawi Publishing Corporation Abstract and Applied Analysis Volume2013, Article ID 487062.
XXII. Sanyal D. C., “On The Solvability Of a Class Of Nonlinear Functional Equations”, J. Mech. Cont.& Math. Sci., Vol.-10, No.-1, October (2015) Pages 1435-1450
XXIII. Shaikh, M. M. , Massan, S-u-R. and Wagan, A. I. (2019). A sixteen decimal places’ accurate Darcy friction factor database using non-linear Colebrook’s equation with a million nodes: a way forward to the soft computing techniques. Data in brief, 27 (Decemebr 2019), 104733.
XXIV. Shaikh, M. M., Massan, S-u-R. and Wagan, A. I. (2015). A new explicit approximation to Colebrook’s friction factor in rough pipes under highly turbulent cases. International Journal of Heat and Mass Transfer, 88, 538-543.
XXV. Shin Min Kang et al.(2015). An Improvement in Newton –Raphson Method for Nonlinear –equations using Modified Adomian Decomposition Method, International Journal of Mathematical Analysis Vol. 9, 2015, no. 39, 1919 – 1928
XXVI. Tanakan, S., (2013), A New Algorithm of Modified Bisection Method for Nonlinear Equations. Applied Mathematical Sciences”, Vol. 7, no. 123, 6107 – 6114 HIKARI Ltd
XXVII. Yasmin, N., M.U.D. Junjua, (2012). Some Derivative Free Iterative Methods for Solving Nonlinear Equations, ISSN-L: 2223-9553, ISSN: 2223-9944 Vol. 2, No.1. 75-82