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Some Novel Multistep Iterative Methods for Solving Nonlinear Equations and Their Dynamics

Thesis Info

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Author

Junjua, Moin-Ud-Din

Program

PhD

Institute

Bahauddin Zakariya University

City

Multan

Province

KPK

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/12353/1/Moin-ud-Din%20Junjua%202019%20maths%20bzu%20prr.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727205053

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Researchers have been contributing a lot to develop root …nding methods for solving nonlinear equations and system of nonlinear equations from many decades. The research started growing since the publication of the books by Traub in 1964 and by Ortega and Rheinboldt in 1970. Finding root of these equations have remained a very important problem in mechanical, electrical and aeronautical engineering. Some complicated techniques exist for solving cubic or quartic equations but higher nonlinear equations are rarely of a form that allows the roots to be determined exactly. So, numerical techniques must be used to solve complex nonlinear equations. Many numerical techniques have been developed earlier in literature to …nd the zero of a nonlinear equation to a speci…ed accuracy. These methods start with an initial approximation of the exact root and iteratively improve this approximation until the required accuracy is obtained. There are several contributors to this problem; Newton, Laguerre, Grae¤e, Baristow, Mueller, Traub and many others. The methods developed by all these researchers are single step. Among these techniques, Newton’s method [15, 104] is most popular method for …nding roots of the nonlinear equations. Newton’s method is quadratically convergent but it may not converge to real root if the initial guess does not lie in the vicinity of root or f 0 is zero in the neighborhood of the real root. Multipoint iterative methods allow us not to discard information that had already been computed. These methods require evaluations of the nonlinear function and derivatives of nonlinear function at several values of the independent variable [104]. The root …nding methods that use only information from the current iteration are called methods without-memory and the root …nding methods that use information from the current and previous iteration are termed as methods with-memory. Ostrowski [77] de…ned the e¢ ciency index of an iterative method as q 1 nf ; where q is the convergence order of the method and nf is the number of function evaluations required per iteration. Kung and Traub [56] conjectured that a without-memory multipoint method requiring n + 1 function evaluations per iteration have optimal order at most 2n and it can attain the e¢ ciency index at most 2 n n+1 : The methods satisfying above hypothesis of Kung and Traub are known as optimal. The main aim of this thesis is to investigate and develop some new optimal and computationally e¢ cient iterative schemes to …nd simple and multiple roots of nonlinear equations as well as for …nding roots of systems of nonlinear equations using various techniques. We have developed some novel multistep with and without-memory iterative methods for solving nonlinear equations by using the weight function approach, with-memorization, rational and inverse interpolation techniques. The basins of attractions and stability analysis of the methods have also been investigated for deep study. A large number of real world applications are reduced to solve systems of nonlinear equations numerically. Solution of such systems has been one of the most challenging problems in numerical analysis. Newton’s method is a basic method for this problem which is also extended for solving systems of nonlinear equations. Several iterative methods for solving systems of nonlinear equations are brought forward. One of the main advantages of these schemes was to achieve high order of convergence with few Jacobian and function evaluations. We have established in this thesis, a new family of optimal fourth order Jarratt type methods for solving nonlinear equations and have extended it to solve systems of nonlinear equations. Convergence analysis for both cases shows that the order of convergence of the new methods is at least four. Cost of computations, numerical tests and basins of attraction are presented which show that the new methods are better alternates to existing methods of similar kind. In addition stability analysis shows the stable behavior of new methods. We have also given applications of the proposed methods to well known Burger’s equation and global positioning system (GPS). In this thesis, we have developed two new classes of optimal eighth order without-memory methods for …nding simple roots of nonlinear equations using weight function approach and four parameters. These methods are extendable to with-memory scheme as well. We have also developed general classes of optimal derivative-free npoint iterative methods based on inverse and rational interpolations that satisfy Kung–Tarub’s Hypothesis [56]. The proposed schemes require n + 1 function evaluations to acquire the convergence order 2n and e¢ - ciency index 2 n n+1. Some dynamical aspects and basins of attraction are studied for the presented methods. Moreover, we have studied the stability analysis of the proposed methods by using the polynomial p(z) = z21. With-memory multi-step iterative methods that use information from the current and previous iterations, increase the convergence order and computational e¢ ciency of the multi-step iterative methods without-memory without any additional function evaluations. The increase in the order of convergence is based on one or more accelerator parameters which appear in the error equations of the without-memory methods. For this reason, several multi-step withand without-memory iterative methods have been developed in recent years. For a background study regarding the acceleration of convergence order via withmemorization, one may see e.g. [78,79]. In this work, we have presented two new e¢ cient with-memory iterative methods for simple roots of nonlinear equations based on newly developed optimal eighth order derivative-free without-memory methods involving four parameters. iiipoint iterative methods based on inverse and rational interpolations that satisfy Kung–Tarub’s Hypothesis [56]. The proposed schemes require n + 1 function evaluations to acquire the convergence order 2n and e¢ - ciency index 2 n n+1. Some dynamical aspects and basins of attraction are studied for the presented methods. Moreover, we have studied the stability analysis of the proposed methods by using the polynomial p(z) = z21. With-memory multi-step iterative methods that use information from the current and previous iterations, increase the convergence order and computational e¢ ciency of the multi-step iterative methods without-memory without any additional function evaluations. The increase in the order of convergence is based on one or more accelerator parameters which appear in the error equations of the without-memory methods. For this reason, several multi-step withand without-memory iterative methods have been developed in recent years. For a background study regarding the acceleration of convergence order via withmemorization, one may see e.g. [78,79]. In this work, we have presented two new e¢ cient with-memory iterative methods for simple roots of nonlinear equations based on newly developed optimal eighth order derivative-free without-memory methods involving four parameters. For this, we approximate the involved parameters with the help of Newton’s interpolating polynomials passing through best saved iterative points to construct highly e¢ cient with-memory methods. This is a novel idea since there are a few with-memory iterative methods in the literature involving four accelerators. The R-order of convergence [73] of the new with-memory methods raises from 8 to 15:5156 without additional function evaluations and e¢ ciency index is signi…cantly improved from 81=4 1:68179 to 15:515601=4 1:9847. We have also presented a general class of with-memory methods as an extension of newly developed derivative-free family of npoint without-memory optimal methods employing a self-accelerating parameter. At each iterative step, we use a suitable variation of the free parameter. The convergence order of the existing family is improved from 2n to 2n + 2n1 without additional function evaluations. An extensive comparison of our with-memory methods is done with the existing withand without-memory methods in terms of e¢ ciency index, residual error and computational order of convergence using some nonlinear equations. In this thesis, we have also established some new families of methods to …nd multiple roots of univariate nonlinear equations. Two families are of sixth order convergent methods and the other two are of optimal eighth order convergent methods. These families are based on modi…ed Newton’s method and weight function approach. An extensive convergence analysis is presented for each of the presented schemes with the help of symbolic computations on programming package Mathematica 8. In addition, we have also demonstrated the applicability of the presented schemes on some real-life problems and illustrated that the proposed methods are more e¢ cient among the available multiple root …nding techniques. The numerical tests of all the problems considered in this thesis have been carried out by using the programming package Maple 16 based on highprecision calculations on few initial estimations. Comparison of the performance of proposed and existing methods has also been carried out by drawing dynamical phase portraits of the stability behavior of the methods on the complex plane, that allows us to know how wide is the set of initial guesses that leads us to the required roots. Both of the comparisons give us complementary information that ivpoint without-memory optimal methods employing a self-accelerating parameter. At each iterative step, we use a suitable variation of the free parameter. The convergence order of the existing family is improved from 2n to 2n + 2n1 without additional function evaluations. An extensive comparison of our with-memory methods is done with the existing withand without-memory methods in terms of e¢ ciency index, residual error and computational order of convergence using some nonlinear equations. In this thesis, we have also established some new families of methods to …nd multiple roots of univariate nonlinear equations. Two families are of sixth order convergent methods and the other two are of optimal eighth order convergent methods. These families are based on modi…ed Newton’s method and weight function approach. An extensive convergence analysis is presented for each of the presented schemes with the help of symbolic computations on programming package Mathematica 8. In addition, we have also demonstrated the applicability of the presented schemes on some real-life problems and illustrated that the proposed methods are more e¢ cient among the available multiple root …nding techniques. The numerical tests of all the problems considered in this thesis have been carried out by using the programming package Maple 16 based on highprecision calculations on few initial estimations. Comparison of the performance of proposed and existing methods has also been carried out by drawing dynamical phase portraits of the stability behavior of the methods on the complex plane, that allows us to know how wide is the set of initial guesses that leads us to the required roots. Both of the comparisons give us complementary information that iv1 without additional function evaluations. An extensive comparison of our with-memory methods is done with the existing withand without-memory methods in terms of e¢ ciency index, residual error and computational order of convergence using some nonlinear equations. In this thesis, we have also established some new families of methods to …nd multiple roots of univariate nonlinear equations. Two families are of sixth order convergent methods and the other two are of optimal eighth order convergent methods. These families are based on modi…ed Newton’s method and weight function approach. An extensive convergence analysis is presented for each of the presented schemes with the help of symbolic computations on programming package Mathematica 8. In addition, we have also demonstrated the applicability of the presented schemes on some real-life problems and illustrated that the proposed methods are more e¢ cient among the available multiple root …nding techniques. The numerical tests of all the problems considered in this thesis have been carried out by using the programming package Maple 16 based on highprecision calculations on few initial estimations. Comparison of the performance of proposed and existing methods has also been carried out by drawing dynamical phase portraits of the stability behavior of the methods on the complex plane, that allows us to know how wide is the set of initial guesses that leads us to the required roots. Both of the comparisons give us complementary information that helps us to fully understand the numerical performance of the iterative schemes and to establish the conclusions." xml:lang="en_US
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سی حرفی : ۱۰

سی حرفی ۔۸
(تن بیتاں وچ مکمل)
الف
آماہی، ’ب‘ بہت تھکی، ت تاہنگ تیری پئی مار دی اے
ث
ثابتی نہیں، ’ج‘ جگر باہجوں، ’ح‘ حالت گئی گھر بار دی اے
خ
خوشی گئی، ’د‘ دکھ بہتے، ’ذ‘ ذکر تے فکر سب یار دی اے
ر
رب وارث، ’ز‘ زاریاں دا، ’س‘ سک حنیف دیدار دی اے

ش
شوق لگا، ’ص‘ صادقاں دا، ’ض‘ ضعف نہیں کجھ نتار دا اے
ط
طوق پیا، ’ظ‘ ظالماں دا، ’ع‘ عاشقاں ہانگرا دار دا اے
غ
غم لگا، ’ف‘ فکر ڈاہڈا، ’ق‘ قسم مینوں شوق یار دا اے
ک
کون کٹے، ’ل‘ لکھ دتا،’م‘ مویاں نوں یار کیوں مار دا اے

ن
نیہہ ڈونگھی، چڑھی گھٹ کالی، اساں لنگھنا پہلڑے پور یارو
و
واہ کوئی نہیں، ہور راہ کوئی نہیں، ’ہ‘ ہڑ دا سماں ضرور یارو
لا
لا مکان دا پتہ دسے، ’ی‘ یاد نہ مان غرور یارو
ے
یار حنیف بھلائی دنیا، کیڈ پائے نیں عشق فتور یارو

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التفكير الاستراتيجي وعلاقته بالاداء الوظيفي في وزارة التربية والتعليم الفلسطينية

هدفت هذه الدراسة إلى التعرّف على علاقة التفكير الإستراتيجي بالأداء الوظيفي في وزارة التربية والتعليم الفلسطينية ومديريّاتها، إضافة إلى التعرّف إلى واقع ممارسة أبعاد التفكير الإستراتيجي في الوزارة، ومعرّفة مستوى الأداء الوظيفي في الوزارة، كما تمثلت مشكلة الدراسة في ندرة الدراسات والبحوث التطبيقية التي تتناول التفكير الإستراتيجي ومستوى الاهتمام به وعلاقته بالمتغيرات الأخرى، خاصة في القطاع العام في فلسطين، ولتحقيق أهداف الدراسة استخدم المنهج الوصفي التحليلي، واستخدمت الإستبانة أداة لجمع البيانات اللازمة من مجتمع الدراسة وعدده 550 موظفًا، وقد اعتمدت العينة العشوائية الطبقية النسبية ، وقد توصلت الدراسة إلى مجموعة من النتائج أهمّها أنّ درجة تطبيق موظفي الإدارة العليا في وزارة التربية والتعليم للتفكير الإستراتيجي متوسطة، حيث بلغت نسبة تطبيق موظفي الإدارة العليا للتفكير الإستراتيجي (61.40%)، وأنّ مستوى الأداء الوظيفي لموظفي الإدارة العليا جاء بدرجة مرتفعة وبنسبة تقييم بلغت (70.8%) ، كما وتبين وجود علاقة طردية ذات دلالة احصائية بين تطبيق التفكير الإستراتيجي ومستوى الأداء الوظيفي في وزارة التربية والتعليم الفلسطينية. وفي ضوء النتائج التي توصلت إليها الدراسة، قُدمت مجموعة من التوصيات أبرزها على وزارة التربية والتعليم الفلسطينية تبني ثقافة التفكير الاستراتيجي ونشرها بين العاملين في كافة المستويات ومنحنهم صلاحيات تمكنهم من ممارسة التفكير الاستراتيجي. كما وعليها تعزيز العاملين ذوي الأداء الوظيفي المرتفع وتحفيزهم باستمرار ورصد احتياجاتهم وتدريبهم والذي ينعكس بشكل ايجابي في تحقيق أهداف وزارة التربية والتعليم الفلسطينية. الكلمات المفتاحية: التفكير الاستراتيجي ، الاداء الوظيفي ، التعليم في فلسطين

Stability Analysis of Self-Gravitating Collapsing Systems

This thesis deals with the dynamical instability as well as inhomogeneities in self- gravitating collapsing objects. For this purpose, the matter distribution is considered imperfect due to anisotropic pressure, shear and bulk viscosity, dissipation in di®usion and streaming out limits and electromagnetic e®ects. For instability regimes, the interior region is taken as spherical, cylindrical and axially symmetric spacetimes which are matched with suitable exterior to explore Darmois conditions. The ¯eld equations and conservation laws are formulated and then perturbed up to ¯rst order in perturbation parameter to construct the collapse equation. The instability regimes are investigated under both N and pN approximations. A crucial role of adiabatic index has been analyzed in the presence of expansion scalar. For spherical con¯guration, the matter distribution is considered to be charged anisotropic dissipative with shear viscosity. The charged cylindrical geometry is dis- cussed with three types of °uid con¯gurations. In the ¯rst case, we take anisotropic pressure, bulk viscosity and dissipation only in di®usion approximation. The second case studies the instability epochs with anisotropic pressure under zero expansion con- dition while in the third case we take isotropic °uid with dissipation in streaming out limit. The dynamical instability for axially symmetric geometry includes two kinds of matter con¯gurations. Initially, this analysis is done only with anisotropic matter, but later we discuss the role of heat °ux and shear viscosity as well. We conclude that the radial pro¯le of material variables like energy density, principal stresses, dissipation, viscosity and electric charge control the stability of self-gravitating objects. The inhomogeneity factors have been identi¯ed for charged plane symmetric space- time with some particular cases of non-dissipative and dissipative °uids. In the non- dissipative case, we analyze inhomogeneity factor for dust, isotropic and anisotropic matter distributions while dissipative matter distribution includes only geodesic dust °uid. We ¯nd that electric charge increases inhomogeneity in the energy density, which is due to shear scalar, anisotropy and dissipation. Also, we explore some dynamical variables, structure scalars as well as an explicit expression for the super- Poynting vector associated with tilted and non-tilted Szekeres spacetime.