Amir Naseem - Academia.edu (original) (raw)
Papers by Amir Naseem
Complexity
The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be ... more The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second-and sixth-order convergence are carefully merged to obtain a hybrid multi-step numerical method with twelfth-order convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absol...
Open journal of mathematical sciences, Apr 16, 2019
The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mo... more The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mostly reduced in solving single variable nonlinear equations. Hence, the problem of approximating a solution of the nonlinear equations is important. The numerical methods for finding roots of such equations are called iterative methods. There are two type of iterative methods in literature: involving higher derivatives and free from higher derivatives. The methods which do not require higher derivatives have less order of convergence and the methods having high convergence order require higher derivatives. The aim of present report is to develop an iterative method having high order of convergence but not involving higher derivatives. We propose three new methods to solve nonlinear equations and solve text examples to check validity and efficiency of our iterative methods.
Open Journal of Mathematical Sciences, 2019
The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mo... more The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mostly reduced in solving single variable nonlinear equations. Hence, the problem of approximating a solution of the nonlinear equations is important. The numerical methods for finding roots of such equations are called iterative methods. There are two type of iterative methods in literature: involving higher derivatives and free from higher derivatives. The methods which do not require higher derivatives have less order of convergence and the methods having high convergence order require higher derivatives. The aim of present report is to develop an iterative method having high order of convergence but not involving higher derivatives. We propose three new methods to solve nonlinear equations and solve text examples to check validity and efficiency of our iterative methods.
Open Journal of Mathematical Analysis, 2018
In this report we present new sixth order iterative methods for solving non-linear equations. The... more In this report we present new sixth order iterative methods for solving non-linear equations. The derivation of these methods is purely based on variational iteration technique. To check the validity and efficiency we compare of methods with Newton's method, Ostrowski's method, Traub's method and modified Halleys's method by solving some test examples. Numerical results shows that our developed methods are more effective. Finally, we compare polynomigraphs of our developed methods with Newton's method, Ostrowski's method, Traub's method and modified Halleys's method.
IEEE Access
The primary objective of this paper is to develop a new method for root-finding by combining forw... more The primary objective of this paper is to develop a new method for root-finding by combining forward and finite-difference techniques in order to provide an efficient, derivative-free algorithm with a lower processing cost per iteration. This will be accomplished by combining forward and finite-difference techniques. We also detail the convergence criterion that was devised for the root-finding approach, and we show that the method that was recommended is quintic-order convergent. We addressed a few engineering issues in order to illustrate the validity and application of the developed root-finding algorithm. The quantitative results justified the constructed root-finding algorithm's robust performance in comparison to other quintic-order methods that can be found in the literature. For the graphical analysis, we make use of the newly discovered method to plot some novel polynomiographs that are attractive to the eye, and then we evaluate these new plots in relation to previously established quintic-order root-finding strategies. The graphic analysis demonstrates that the newly created method for root-finding has better convergence with the larger area than the other comparable methods do.
Complexity
In the present work, we introduce two novel root-finding algorithms for nonlinear scalar equation... more In the present work, we introduce two novel root-finding algorithms for nonlinear scalar equations. Among these algorithms, the second one is optimal according to Kung-Traub’s conjecture. It is established that the newly proposed algorithms bear the fourth- and sixth-order of convergence. To show the effectiveness of the suggested methods, we provide several real-life problems associated with engineering sciences. These problems have been solved through the suggested methods, and their numerical results proved the superiority of these methods over the other ones. Finally, we study the dynamics of the proposed methods using polynomiographs created with the help of a computer program using six cubic-degree polynomials and then give a detailed graphical comparison with similar existing methods which shows the supremacy of the presented iteration schemes with respect to convergence speed and other dynamical aspects.
Complexity
The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be ... more The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second-and sixth-order convergence are carefully merged to obtain a hybrid multi-step numerical method with twelfth-order convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absol...
Journal of Chemistry
Dendrimers are spherical three-dimensional molecules with a repetitively branching core. They are... more Dendrimers are spherical three-dimensional molecules with a repetitively branching core. They are normally symmetric around the core. Bismuth (III) iodide has the formula B i I 3 and is an inorganic chemical. The reaction between bismuth and iodine produces this gray-black solid, which was of great interest in qualitative inorganic analysis. Mathematical chemistry is an area of mathematics that employs mathematical methods to tackle chemical-related problems. One of these tools is a graphical representation of chemical molecules, known as the molecular graph of a chemical substance. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. In this work, we will calculate a new topological index, namely, Sombor index, multiplicative Sombor index, and its reduced version for bismuth (III) iodide and dendrimers. We also plot our com...
Journal of Function Spaces
In this paper, we propose two novel iteration schemes for computing zeros of nonlinear equations ... more In this paper, we propose two novel iteration schemes for computing zeros of nonlinear equations in one dimension. We develop these iteration schemes with the help of Taylor’s series expansion, generalized Newton-Raphson’s method, and interpolation technique. The convergence analysis of the proposed iteration schemes is discussed. It is established that the newly developed iteration schemes have sixth order of convergence. Several numerical examples have been solved to illustrate the applicability and validity of the suggested schemes. These problems also include some real-life applications associated with the chemical and civil engineering such as adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia, the van der Wall’s equation, and the open channel flow problem whose numerical results prove the better efficiency of these methods as compared to other well-known existing iterative methods of the same kind.
Journal of Chemistry
COVID-19 is causing havoc to human health and the world economy right now. It is a single standar... more COVID-19 is causing havoc to human health and the world economy right now. It is a single standard positive-sense RNA virus which is transferred by inhalation of a viral droplet. Its genome forms four structural proteins such as nucleocapsid protein, membrane protein, spike protein, and envelop protein. The capsid of coronavirus is a protein shell within which a positive strand of RNA is present which enables the virus to control the machinery of human cells. It has several variants, e.g., SARS, MERS, and now a new variant identified in 2019, which is a novel coronavirus that causes novel coronavirus disease (COVID-19). COVID-19 is a novel coronavirus disease that originally arose in Wuhan, China, and quickly spread around the world. Clinically, we identified the virus presence by a PCR-based test. Preventive measures and vaccination are the only treatment against coronavirus. Some of these include Remdesivir (GS-5734), Chloroquine, Hydroxychloroquine, and Theaflavin. A topological ...
Mathematical Problems in Engineering
Chemical graph theory is a field of mathematical chemistry that links mathematics, chemistry, and... more Chemical graph theory is a field of mathematical chemistry that links mathematics, chemistry, and graph theory to solve chemistry-related issues quantitatively. Mathematical chemistry is an area of mathematics that employs mathematical methods to tackle chemical-related problems. A graphical representation of chemical molecules, known as the molecular graph of the chemical substance, is one of these tools. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. In this work, we calculate a new topological index namely, the Sombor index, the Super Sombor index, and its reduced version for chemical networks. We also plot our computed results to examine how they were affected by the parameters involved. This document lists the distinct degrees and degree sums of enhanced mesh network, triangular mesh network, star of silicate netw...
Complexity, 2021
Nowadays, the use of computers is becoming very important in various fields of mathematics and en... more Nowadays, the use of computers is becoming very important in various fields of mathematics and engineering sciences. Many complex statistics can be sorted out easily with the help of different computer programs in seconds, especially in computational and applied Mathematics. With the help of different computer tools and languages, a variety of iterative algorithms can be operated in computers for solving different nonlinear problems. The most important factor of an iterative algorithm is its efficiency that relies upon the convergence rate and computational cost per iteration. Taking these facts into account, this article aims to design a new iterative algorithm that is derivative-free and performs better. We construct this algorithm by applying the forward- and finite-difference schemes on Golbabai–Javidi’s method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algor...
IEEE Access, 2022
Root-finding of non-linear equations is one of the most appearing problems in engineering science... more Root-finding of non-linear equations is one of the most appearing problems in engineering sciences. Most of the complicated engineering problems can be modeled easily by means of non-linear functions. The role of iterative algorithms via computers for solving such functions is much important and cannot be denied in the modern age. In an iterative algorithm, the convergence order and the computational cost per iteration are the main characteristics that depict its efficiency and performance i.e., a method with higher-order and lower computational cost will be more efficient and vice versa. Keeping these facts into consideration, the main goal of this paper is to introduce a new derivative-free iterative method that performs better. We develop this algorithm by utilizing the forward-and finite-difference schemes on well-known Househölder's method, resulting in an efficient and derivative-free algorithm with a low per iteration computing cost. We also look at the developed algorithm's convergence criterion and show that it is quartic-order convergent. We investigate nine test-examples and solve them to demonstrate its correctness, validity, and efficiency numerically. Some real-world engineering problems in civil and chemical engineering are also included in these examples. The numerical results of the test-examples reveal that the newly constructed method outperforms the existing similar algorithms found in the literature. We consider various different-degrees complex polynomials for the graphical analysis and used a computer tool to create the polynomiographs of the proposed quartic-order algorithm and compare it to other comparable existing approaches. The graphical findings show that the developed method has a faster convergence speed than the other comparable algorithms.
Complexity, 2021
In this article, we design a novel fourth-order and derivative free root-finding algorithm. We co... more In this article, we design a novel fourth-order and derivative free root-finding algorithm. We construct this algorithm by applying the finite difference scheme on the well-known Ostrowski’s method. The convergence analysis shows that the newly designed algorithm possesses fourth-order convergence. To demonstrate the applicability of the designed algorithm, we consider five real-life engineering problems in the form of nonlinear scalar functions and then solve them via computer tools. The numerical results show that the new algorithm outperforms the other fourth-order comparable algorithms in the literature in terms of performance, applicability, and efficiency. Finally, we present the dynamics of the designed algorithm via computer tools by examining certain complex polynomials that depict the convergence and other graphical features of the designed algorithm.
IEEE Access, 2021
Solving non-linear equations in different scientific disciplines is one of the most important and... more Solving non-linear equations in different scientific disciplines is one of the most important and frequently appearing problems. A variety of real-world problems in different scientific fields can be modeled via non-linear equations. Iterative algorithms play a vital role in finding the solution of such nonlinear problems. This article aims to design a new iterative algorithm that is derivative-free and performing better. We construct this algorithm by applying the forward-and finite-difference schemes on the wellknown Traubs's method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algorithm and prove its fifth-order convergence. To demonstrate the accuracy, validity and applicability of the designed algorithm, we consider eleven different types of numerical test examples and solve them. The considered problems also involve some real-life applications of civil and chemical engineering. The obtained numerical results of the test examples show that the newly designed algorithm is working better against the other similarorder algorithms in the literature. For the graphical analysis, we consider some different-degree complex polynomials and draw polynomiographs of the designed fifth-order algorithm and compare them with the other fifth-order methods with the help of a computer program Mathematica 12.0. The graphical results show the convergence speed and other graphical characteristics of the designed algorithm and prove its supremacy over the other comparable ones.
4 Division of Science and Technology, University of Education, Lahore Pakistan. mmunir@ue.edu.pk ... more 4 Division of Science and Technology, University of Education, Lahore Pakistan. mmunir@ue.edu.pk ABSTRACT: The aim of this paper is to present polynomiography using fouth order iterative method for solving nonlinear equations. Polynomiography is the art and science of visualization in approximation of zeros of complex polynomials. The images thus obtained are called polynomiographs. In this paper, we obtained polynomiographs of different complex polynomials . The obtained polynomiographs reflect interesting patterns of complex polynomials. We believe that the results of this paper enrich the functionality of the existing polynomiography software.
Polynomiography is the art and science of visualization in approximation of zeros of complex poly... more Polynomiography is the art and science of visualization in approximation of zeros of complex polynomials. It allows one to obtain many colorful images of polynomials. These images can subsequently be re-colored in many ways to create artwork. Polynomiography has tremendous applications in the visual, education, and science. The aim of this paper is to present polynomiography using an iterative method for finding the roots of a given complex polynomial Presented examples show that we obtain very interesting patterns for complex polynomial equations.
IEEE Access, 2021
The task of root-finding of the non-linear equations is perhaps, one of the most complicated prob... more The task of root-finding of the non-linear equations is perhaps, one of the most complicated problems in applied mathematics especially in a diverse range of engineering applications. The characteristics of the root-finding methods such as convergence rate, performance, efficiency, etc., are directly relied upon the initial guess of the solution to execute the process in most of the systems of non-linear equations. Keeping these facts into mind, based on Taylor's series expansion, we present some new modifications of Halley, Househölder and Golbabai and Javidi's methods and then making them second derivative free by applying Taylor's series. The convergence analysis of the suggested methods is discussed. It is established that the proposed methods possess convergence of orders five and six. Several numerical problems have been tested to demonstrate the validity and applicability of the proposed methods. These test examples also include some real-life problems associated with chemical and civil engineering such as open channel flow problem, the adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia and the van der Wall's equation whose numerical results prove the better performance of the suggested methods as compared to other well-known existing methods of the same kind in the literature. Finally, the dynamics of the presented algorithms in the form of polynomiographs have been shown with the aid of computer program by considering some complex polynomials and compared them with the other well-known iterative algorithms that revealed the convergence speed and other dynamical aspects of the presented methods.
Intelligent Automation & Soft Computing, 2021
Complexity
The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be ... more The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second-and sixth-order convergence are carefully merged to obtain a hybrid multi-step numerical method with twelfth-order convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absol...
Open journal of mathematical sciences, Apr 16, 2019
The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mo... more The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mostly reduced in solving single variable nonlinear equations. Hence, the problem of approximating a solution of the nonlinear equations is important. The numerical methods for finding roots of such equations are called iterative methods. There are two type of iterative methods in literature: involving higher derivatives and free from higher derivatives. The methods which do not require higher derivatives have less order of convergence and the methods having high convergence order require higher derivatives. The aim of present report is to develop an iterative method having high order of convergence but not involving higher derivatives. We propose three new methods to solve nonlinear equations and solve text examples to check validity and efficiency of our iterative methods.
Open Journal of Mathematical Sciences, 2019
The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mo... more The boundary value problems in Kinetic theory of gases, elasticity and other applied areas are mostly reduced in solving single variable nonlinear equations. Hence, the problem of approximating a solution of the nonlinear equations is important. The numerical methods for finding roots of such equations are called iterative methods. There are two type of iterative methods in literature: involving higher derivatives and free from higher derivatives. The methods which do not require higher derivatives have less order of convergence and the methods having high convergence order require higher derivatives. The aim of present report is to develop an iterative method having high order of convergence but not involving higher derivatives. We propose three new methods to solve nonlinear equations and solve text examples to check validity and efficiency of our iterative methods.
Open Journal of Mathematical Analysis, 2018
In this report we present new sixth order iterative methods for solving non-linear equations. The... more In this report we present new sixth order iterative methods for solving non-linear equations. The derivation of these methods is purely based on variational iteration technique. To check the validity and efficiency we compare of methods with Newton's method, Ostrowski's method, Traub's method and modified Halleys's method by solving some test examples. Numerical results shows that our developed methods are more effective. Finally, we compare polynomigraphs of our developed methods with Newton's method, Ostrowski's method, Traub's method and modified Halleys's method.
IEEE Access
The primary objective of this paper is to develop a new method for root-finding by combining forw... more The primary objective of this paper is to develop a new method for root-finding by combining forward and finite-difference techniques in order to provide an efficient, derivative-free algorithm with a lower processing cost per iteration. This will be accomplished by combining forward and finite-difference techniques. We also detail the convergence criterion that was devised for the root-finding approach, and we show that the method that was recommended is quintic-order convergent. We addressed a few engineering issues in order to illustrate the validity and application of the developed root-finding algorithm. The quantitative results justified the constructed root-finding algorithm's robust performance in comparison to other quintic-order methods that can be found in the literature. For the graphical analysis, we make use of the newly discovered method to plot some novel polynomiographs that are attractive to the eye, and then we evaluate these new plots in relation to previously established quintic-order root-finding strategies. The graphic analysis demonstrates that the newly created method for root-finding has better convergence with the larger area than the other comparable methods do.
Complexity
In the present work, we introduce two novel root-finding algorithms for nonlinear scalar equation... more In the present work, we introduce two novel root-finding algorithms for nonlinear scalar equations. Among these algorithms, the second one is optimal according to Kung-Traub’s conjecture. It is established that the newly proposed algorithms bear the fourth- and sixth-order of convergence. To show the effectiveness of the suggested methods, we provide several real-life problems associated with engineering sciences. These problems have been solved through the suggested methods, and their numerical results proved the superiority of these methods over the other ones. Finally, we study the dynamics of the proposed methods using polynomiographs created with the help of a computer program using six cubic-degree polynomials and then give a detailed graphical comparison with similar existing methods which shows the supremacy of the presented iteration schemes with respect to convergence speed and other dynamical aspects.
Complexity
The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be ... more The appearance of nonlinear equations in science, engineering, economics, and medicine cannot be denied. Solving such equations requires numerical methods having higher-order convergence with cost-effectiveness, for the equations do not have exact solutions. In the pursuit of efficient numerical methods, an attempt is made to devise a modified strategy for approximating the solution of nonlinear models in either scalar or vector versions. Two numerical methods of second-and sixth-order convergence are carefully merged to obtain a hybrid multi-step numerical method with twelfth-order convergence while using seven function evaluations per iteration, resulting in the efficiency index of about 1.4262. The convergence is also ascertained theoretically, and the asymptotic error constant is computed. Furthermore, various numerical methods of varying orders are used to compare the numerical results obtained with the proposed hybrid method in approximate solution, number of iterations, absol...
Journal of Chemistry
Dendrimers are spherical three-dimensional molecules with a repetitively branching core. They are... more Dendrimers are spherical three-dimensional molecules with a repetitively branching core. They are normally symmetric around the core. Bismuth (III) iodide has the formula B i I 3 and is an inorganic chemical. The reaction between bismuth and iodine produces this gray-black solid, which was of great interest in qualitative inorganic analysis. Mathematical chemistry is an area of mathematics that employs mathematical methods to tackle chemical-related problems. One of these tools is a graphical representation of chemical molecules, known as the molecular graph of a chemical substance. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. In this work, we will calculate a new topological index, namely, Sombor index, multiplicative Sombor index, and its reduced version for bismuth (III) iodide and dendrimers. We also plot our com...
Journal of Function Spaces
In this paper, we propose two novel iteration schemes for computing zeros of nonlinear equations ... more In this paper, we propose two novel iteration schemes for computing zeros of nonlinear equations in one dimension. We develop these iteration schemes with the help of Taylor’s series expansion, generalized Newton-Raphson’s method, and interpolation technique. The convergence analysis of the proposed iteration schemes is discussed. It is established that the newly developed iteration schemes have sixth order of convergence. Several numerical examples have been solved to illustrate the applicability and validity of the suggested schemes. These problems also include some real-life applications associated with the chemical and civil engineering such as adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia, the van der Wall’s equation, and the open channel flow problem whose numerical results prove the better efficiency of these methods as compared to other well-known existing iterative methods of the same kind.
Journal of Chemistry
COVID-19 is causing havoc to human health and the world economy right now. It is a single standar... more COVID-19 is causing havoc to human health and the world economy right now. It is a single standard positive-sense RNA virus which is transferred by inhalation of a viral droplet. Its genome forms four structural proteins such as nucleocapsid protein, membrane protein, spike protein, and envelop protein. The capsid of coronavirus is a protein shell within which a positive strand of RNA is present which enables the virus to control the machinery of human cells. It has several variants, e.g., SARS, MERS, and now a new variant identified in 2019, which is a novel coronavirus that causes novel coronavirus disease (COVID-19). COVID-19 is a novel coronavirus disease that originally arose in Wuhan, China, and quickly spread around the world. Clinically, we identified the virus presence by a PCR-based test. Preventive measures and vaccination are the only treatment against coronavirus. Some of these include Remdesivir (GS-5734), Chloroquine, Hydroxychloroquine, and Theaflavin. A topological ...
Mathematical Problems in Engineering
Chemical graph theory is a field of mathematical chemistry that links mathematics, chemistry, and... more Chemical graph theory is a field of mathematical chemistry that links mathematics, chemistry, and graph theory to solve chemistry-related issues quantitatively. Mathematical chemistry is an area of mathematics that employs mathematical methods to tackle chemical-related problems. A graphical representation of chemical molecules, known as the molecular graph of the chemical substance, is one of these tools. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. In this work, we calculate a new topological index namely, the Sombor index, the Super Sombor index, and its reduced version for chemical networks. We also plot our computed results to examine how they were affected by the parameters involved. This document lists the distinct degrees and degree sums of enhanced mesh network, triangular mesh network, star of silicate netw...
Complexity, 2021
Nowadays, the use of computers is becoming very important in various fields of mathematics and en... more Nowadays, the use of computers is becoming very important in various fields of mathematics and engineering sciences. Many complex statistics can be sorted out easily with the help of different computer programs in seconds, especially in computational and applied Mathematics. With the help of different computer tools and languages, a variety of iterative algorithms can be operated in computers for solving different nonlinear problems. The most important factor of an iterative algorithm is its efficiency that relies upon the convergence rate and computational cost per iteration. Taking these facts into account, this article aims to design a new iterative algorithm that is derivative-free and performs better. We construct this algorithm by applying the forward- and finite-difference schemes on Golbabai–Javidi’s method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algor...
IEEE Access, 2022
Root-finding of non-linear equations is one of the most appearing problems in engineering science... more Root-finding of non-linear equations is one of the most appearing problems in engineering sciences. Most of the complicated engineering problems can be modeled easily by means of non-linear functions. The role of iterative algorithms via computers for solving such functions is much important and cannot be denied in the modern age. In an iterative algorithm, the convergence order and the computational cost per iteration are the main characteristics that depict its efficiency and performance i.e., a method with higher-order and lower computational cost will be more efficient and vice versa. Keeping these facts into consideration, the main goal of this paper is to introduce a new derivative-free iterative method that performs better. We develop this algorithm by utilizing the forward-and finite-difference schemes on well-known Househölder's method, resulting in an efficient and derivative-free algorithm with a low per iteration computing cost. We also look at the developed algorithm's convergence criterion and show that it is quartic-order convergent. We investigate nine test-examples and solve them to demonstrate its correctness, validity, and efficiency numerically. Some real-world engineering problems in civil and chemical engineering are also included in these examples. The numerical results of the test-examples reveal that the newly constructed method outperforms the existing similar algorithms found in the literature. We consider various different-degrees complex polynomials for the graphical analysis and used a computer tool to create the polynomiographs of the proposed quartic-order algorithm and compare it to other comparable existing approaches. The graphical findings show that the developed method has a faster convergence speed than the other comparable algorithms.
Complexity, 2021
In this article, we design a novel fourth-order and derivative free root-finding algorithm. We co... more In this article, we design a novel fourth-order and derivative free root-finding algorithm. We construct this algorithm by applying the finite difference scheme on the well-known Ostrowski’s method. The convergence analysis shows that the newly designed algorithm possesses fourth-order convergence. To demonstrate the applicability of the designed algorithm, we consider five real-life engineering problems in the form of nonlinear scalar functions and then solve them via computer tools. The numerical results show that the new algorithm outperforms the other fourth-order comparable algorithms in the literature in terms of performance, applicability, and efficiency. Finally, we present the dynamics of the designed algorithm via computer tools by examining certain complex polynomials that depict the convergence and other graphical features of the designed algorithm.
IEEE Access, 2021
Solving non-linear equations in different scientific disciplines is one of the most important and... more Solving non-linear equations in different scientific disciplines is one of the most important and frequently appearing problems. A variety of real-world problems in different scientific fields can be modeled via non-linear equations. Iterative algorithms play a vital role in finding the solution of such nonlinear problems. This article aims to design a new iterative algorithm that is derivative-free and performing better. We construct this algorithm by applying the forward-and finite-difference schemes on the wellknown Traubs's method which yields us an efficient and derivative-free algorithm whose computational cost is low as per iteration. We also study the convergence criterion of the designed algorithm and prove its fifth-order convergence. To demonstrate the accuracy, validity and applicability of the designed algorithm, we consider eleven different types of numerical test examples and solve them. The considered problems also involve some real-life applications of civil and chemical engineering. The obtained numerical results of the test examples show that the newly designed algorithm is working better against the other similarorder algorithms in the literature. For the graphical analysis, we consider some different-degree complex polynomials and draw polynomiographs of the designed fifth-order algorithm and compare them with the other fifth-order methods with the help of a computer program Mathematica 12.0. The graphical results show the convergence speed and other graphical characteristics of the designed algorithm and prove its supremacy over the other comparable ones.
4 Division of Science and Technology, University of Education, Lahore Pakistan. mmunir@ue.edu.pk ... more 4 Division of Science and Technology, University of Education, Lahore Pakistan. mmunir@ue.edu.pk ABSTRACT: The aim of this paper is to present polynomiography using fouth order iterative method for solving nonlinear equations. Polynomiography is the art and science of visualization in approximation of zeros of complex polynomials. The images thus obtained are called polynomiographs. In this paper, we obtained polynomiographs of different complex polynomials . The obtained polynomiographs reflect interesting patterns of complex polynomials. We believe that the results of this paper enrich the functionality of the existing polynomiography software.
Polynomiography is the art and science of visualization in approximation of zeros of complex poly... more Polynomiography is the art and science of visualization in approximation of zeros of complex polynomials. It allows one to obtain many colorful images of polynomials. These images can subsequently be re-colored in many ways to create artwork. Polynomiography has tremendous applications in the visual, education, and science. The aim of this paper is to present polynomiography using an iterative method for finding the roots of a given complex polynomial Presented examples show that we obtain very interesting patterns for complex polynomial equations.
IEEE Access, 2021
The task of root-finding of the non-linear equations is perhaps, one of the most complicated prob... more The task of root-finding of the non-linear equations is perhaps, one of the most complicated problems in applied mathematics especially in a diverse range of engineering applications. The characteristics of the root-finding methods such as convergence rate, performance, efficiency, etc., are directly relied upon the initial guess of the solution to execute the process in most of the systems of non-linear equations. Keeping these facts into mind, based on Taylor's series expansion, we present some new modifications of Halley, Househölder and Golbabai and Javidi's methods and then making them second derivative free by applying Taylor's series. The convergence analysis of the suggested methods is discussed. It is established that the proposed methods possess convergence of orders five and six. Several numerical problems have been tested to demonstrate the validity and applicability of the proposed methods. These test examples also include some real-life problems associated with chemical and civil engineering such as open channel flow problem, the adiabatic flame temperature equation, conversion of nitrogen-hydrogen feed to ammonia and the van der Wall's equation whose numerical results prove the better performance of the suggested methods as compared to other well-known existing methods of the same kind in the literature. Finally, the dynamics of the presented algorithms in the form of polynomiographs have been shown with the aid of computer program by considering some complex polynomials and compared them with the other well-known iterative algorithms that revealed the convergence speed and other dynamical aspects of the presented methods.
Intelligent Automation & Soft Computing, 2021