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Computer Geometries for Finding All Real Zeros of Polynomial Equations Simultaneously

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Computers, Materials & Continua 2021年第11期69卷 2635-2651页

作者： Naila Rafiq Saima Akram mudassir shams Nazir Ahmad Mir Department of Mathematics NUMLIslamabad44000Pakistan Centre for Advanced Studies in Pure&Applied Mathematics Bahauddin Zakariya UniversityMultan Department of Mathematics and Statistics Riphah International University I-14Islamabad44000Pakistan

In this research article,we construct a family of derivative free simultaneous numerical schemes to approximate all real zero of non-linear polynomial *** make a comparative analysis of the newly constructed numerical schemes with a well-known existing simultaneous method for determining all the distinct real zeros of polynomial equations using computer algebra system Mat *** bound of convergence of simultaneous schemes is calculated using *** convergence property of the numerical schemes is presented by taking random starting initial approximation and their convergence history are graphically *** real life engineering applications along with some higher degree polynomials are considered as numerical test problems to show performance and efficiency of the derivative free family of numerical methods with comparison of an existing method of same order in *** computational order of convergence,CPU time,graph of computational order of convergence and residual error graphs elaborate efficiency,robustness and authentication of the suggested family of numerical methods in its domain.

关键词： Polynomials simultaneous iterative methods random initial guesses lower bound local computational order CAS-mathematica and mat lab

Stable Computer Method for Solving Initial Value Problems with Engineering Applications

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Computer Systems Science & Engineering 2023年第6期45卷 2617-2633页

作者： mudassir shams Nasreen Kausar Ebru Ozbilge Alper Bulut Department of Mathematics and Statistics Riphah International UniversityI-14Islamabad44000Pakistan Department of Mathematics Yildiz Technical UniversityFaculty of Arts and ScienceEsenler34210IstanbulTurkey Department of Mathematics&Statistics American University of the Middle EastEgaila54200Kuwait

Engineering and applied mathematics disciplines that involve differential equations in general,and initial value problems in particular,include classical mechanics,thermodynamics,electromagnetism,and the general theory of relativity.A reliable,stable,efficient,and consistent numerical scheme is frequently required for modelling and simulation of a wide range of real-world problems using differential *** this study,the tangent slope is assumed to be the contra-harmonic mean,in which the arithmetic mean is used as a correction instead of Euler’s method to improve the efficiency of the improved Euler’s technique for solving ordinary differential equations with initial *** stability,consistency,and efficiency of the system were evaluated,and the conclusions were supported by the presentation of numerical test applications in *** to the stability analysis,the proposed method has a wider stability region than other well-known methods that are currently used in the literature for solving initial-value *** validate the rate convergence of the numerical technique,a few initial value problems of both scalar and vector valued types were *** proposed method,modified Euler explicit method,and other methods known in the literature have all been used to calculate the absolute maximum error,absolute error at the last grid point of the integration interval under consideration,and computational time in seconds to test the *** Lorentz system was used as an example to illustrate the validity of the solution provided by the newly developed *** method is determined to be more reliable than the commonly existing methods with the same order of convergence,as mentioned in the literature for numerical calculations and visualization of the results produced by all the methods discussed,Mat Lab-R2011b has been used.

关键词： Local truncation error consistency computational time stability lorentz system

Computer Oriented Numerical Scheme for Solving Engineering Problems

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Computer Systems Science & Engineering 2022年第8期42卷 689-701页

作者： mudassir shams Naila Rafiq Nasreen Kausar Nazir Ahmad Mir Ahmad Alalyani Department of Mathematica and Statistics Riphah International UniversityI-14Islamabad 44000Pakistan Department of Mathematics NUMLIslamabad 44000Pakistan Department of Mathematics Yildiz Technical UniversityFaculty of Arts and ScienceEsenler34210IstanbulTurkey Department of Mathematics Faculty of Science and Arts in ALmandaqAl-Baha UniversityAl-BahaSaudi Arabia

In this study,we construct a family of single root finding method of optimal order four and then generalize this family for estimating of all roots of non-linear equation *** analysis proves that the local order of convergence is four in case of single root finding iterative method and six for simultaneous determination of all roots of non-linear *** non-linear equations are taken from physics,chemistry and engineering to present the performance and efficiency of the newly constructed *** real world applications are taken from fluid mechanics,i.e.,fluid permeability in biogels and biomedical engineering which includes blood Rheology-Model which as an intermediate result give some nonlinear *** non-linear equations are then solved using newly developed simultaneous iterative *** developed simultaneous iterative schemes reach to exact values on initial guessed values within given tolerance,using very less number of function evaluations in each *** convergence order of single root finding method is computed using *** computational order of convergence,CPU-time,absolute residuals errors are calculated to elaborate the efficiency,robustness and authentication of the iterative simultaneous method in its domain.

关键词： Biomedical engineering convergence order iterative method CPU-time simultaneous method

Computer Methodologies for the Comparison of Some Efficient Derivative FreeSimultaneous Iterative Methods for Finding Roots of Non-Linear Equations

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Computers, Materials & Continua 2021年第1期66卷 275-290页

作者： Yuming Chu Naila Rafiq mudassir shams Saima Akram Nazir Ahmad Mir Humaira Kalsoom Department of Mathematics Huzhou UniversityHuzhou313000China Department of Mathematics National University of Modern LanguagesIslamabadPakistan Department of Mathematics and Statistics Riphah International University I-14Islamabad44000Pakistan Center for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University MultanPakistan School of Mathematical Sciences Zhejiang UniversityHanghou310027China

In this article,we construct the most powerful family of simultaneous iterative method with global convergence behavior among all the existing methods in literature for finding all roots of non-linear *** analysis proved that the order of convergence of the family of derivative free simultaneous iterative method is *** main aim is to check out the most regularly used simultaneous iterative methods for finding all roots of non-linear equations by studying their dynamical planes,numerical experiments and CPU *** planes of iterative methods are drawn by using MATLAB for the comparison of global convergence properties of simultaneous iterative *** behavior of the higher order simultaneous iterative methods are also illustrated by residual graph obtained from some numerical test *** test examples,dynamical behavior and computational efficiency are provided to present the performance and dominant efficiency of the newly constructed derivative free family of simultaneous iterative method over existing higher order simultaneous methods in literature.

关键词： Non-linear equation iterative method simultaneous method basins of attractions computational efficiency

On Computer Implementation for Comparison of Inverse Numerical Schemes for Non-Linear Equations

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Computer Systems Science & Engineering 2021年第3期36卷 493-507页

作者： mudassir shams Naila Rafiq Nazir Ahmad Mir Babar Ahmad Saqib Abbasi Mutee-Ur-Rehman Kayani Department of Mathematics and Statistics Riphah International UniversityI-14Islamabad44000Pakistan Department of Mathematics NUMLIslamabadPakistan Department of Mathematics Comsats UniversityIslamabad44000Pakistan

In this research article,we interrogate two new modifications in inverse Weierstrass iterative method for estimating all roots of non-linear equation *** modifications enables us to accelerate the convergence order of inverse Weierstrass method from 2 to *** analysis proves that the orders of convergence of the two newly constructed inverse methods are *** computer algebra system Mathematica,we find the lower bound of the convergence order and verify it *** planes of the inverse simultaneous methods and classical iterative methods are generated using MATLAB(R2011b),to present the global convergence properties of inverse simultaneous iterative methods as compared to classical *** non-linear models are taken from Physics,Chemistry and engineering to demonstrate the performance and efficiency of the newly constructed *** CPU time,and residual graphs of the methods are provided to present the dominance behavior of our newly constructed methods as compared to existing inverse and classical simultaneous iterative methods in the literature.

关键词： Non-linear equation inverse iterative method simultaneous method basins of attraction lower bound of convergence