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Title:	New newton-like methods for nonlinear equations with their basins of attraction
Authors:	Solaiman Obadah M. Ashraf (P86228)
Supervisor:	Ishak Hashim, Prof. Dr.
Keywords:	Nonlinear equations Universiti Kebangsaan Malaysia -- Dissertations Dissertations, Academic -- Malaysia
Issue Date:	15-Jun-2020
Description:	Many problems in applied sciences yield nonlinear equations or systems of nonlinear equations that need to be solved. Finding the exact solution for such equations is usually not an easy mission.Because of that,finding an approximate solution becomes very important. The desired approximate solution has to be of small error with low computational and time costs.In this thesis,firstly,new King-type iterative schemes are presented; three iterative schemes of orders four and eight are proposed such that all of them are optimal and derivative-free. Different examples are used in comparisons with other methods of the same orders to check the efficiency of the proposed techniques.Next,a new Halley-type method of order six is proposed.Next,to increase the efficiency index of the proposed method,suitable approximation is used to introduce a second-derivative-free iterative scheme. Subsequently,by using Hermite's interpolating polynomial, the proposed method is improved to an optimal method of order eight.Several numerical examples are selected to compare the proposed schemes with other schemes of the same order.Next,the basins of attraction of several iterative techniques of order three are studied to see whether or not the efficiency index is important to the behaviour of the iterative scheme.After that,we try to clarify the answer to the question:are optimal methods always better than non-optimal methods?To do so,a new non-optimal iterative scheme of order sixteen is proposed.Different types of comparisons with otheroptimalandnon-optimal schemes of equivalent order are used,in addition to the dynamical comparison by illustrating the basins of attraction of the selected iterative techniques.Finally,anewfifth-order iterative method for solving systems of non linear equations is proposed. A generalisation of the proposed technique to be of arbitrarily odd orderintheformof 2= 1 is presented.The computational cost of the proposed scheme is found and compared with other schemes ofthe same order.Also,the basins of attraction of the proposed iterative scheme are illustrated and compared with other equivalent schemes. Several numerical examples are tested to see the efficiency and applicability of the proposed technique. It is empirically shown that the accuracy of the solutions obtained by the proposed methods is more accurate and needless computational time than the other methods used in the comparisons.Moreover,it is proved that optimal methods are not necessarily more efficient than non-optimal method seven though they needless computational cost. Further more, comparisons show that the common efficiency index and the computational efficiency index are not efficient tools to be used in comparing the efficiency of the iterative techniques because they ignore many important factors that may affect the efficiency of iterative methods.The proposed methods improve the accuracy of the obtained approximate solutions compared to other selected methods in the literature by at least 5%. Furthermore,the proposed methods reduce the needed CPU-time by at least 10%. These improvements are significant in the field of computational mathematics, chemistry, physics and any applied sciences that produce nonlinearproblems.,Ph.D
Pages:	144
Publisher:	UKM, Bangi
Appears in Collections:	Faculty of Science and Technology / Fakulti Sains dan Teknologi

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