International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document discusses stochastic partial differential equations (SPDEs). It outlines several approaches that have been used to solve SPDEs, including methods based on diffusion processes, stochastic characteristic systems, direct methods from mathematical physics, and substitution of integral equations. It also discusses using backward stochastic differential equations to study SPDEs and introduces notation for the analysis of an Ito SDE with inverse time. The document is technical in nature and outlines the mathematical frameworks and equations involved in solving SPDEs through various probabilistic methods.
Derivation and Application of Multistep Methods to a Class of First-order Ord...AI Publications
Of concern in this work is the derivation and implementation of the multistep methods through Taylor’s expansion and numerical integration. For the Taylor’s expansion method, the series is truncated after some terms to give the needed approximations which allows for the necessary substitutions for the derivatives to be evaluated on the differential equations. For the numerical integration technique, an interpolating polynomial that is determined by some data points replaces the differential equation function and it is integrated over a specified interval. The methods show that they are only convergent if and only if they are consistent and stable. In our numerical examples, the methods are applied on non-stiff initial value problems of first-order ordinary differential equations, where it is established that the multistep methods show superiority over the single-step methods in terms of robustness, efficiency, stability and accuracy, the only setback being that the multi-step methods require more computational effort than the single-step methods.
This document presents an internship project report on multistep methods for solving initial value problems of ordinary differential equations. It introduces the basic problem of finding the function y(t) that satisfies a given differential equation and initial condition. It discusses existence and uniqueness theorems, Picard's method of successive approximations, and approaches for approximating the required integrations, including the derivative, Taylor series, and Euler's methods. The report appears to evaluate various one-step and multistep numerical methods for solving initial value problems, including Runge-Kutta, Adams-Bashforth, and Adams-Moulton methods.
The document analyzes the analytic solution of Burger's equations using the variational iteration method. It begins by introducing the variational iteration method and how it can be used to solve differential equations. It then applies the method to obtain exact solutions for the (1+1), (1+2), and (1+3) dimensional Burger equations. Lengthy iterative solutions are presented for each case. The variational iteration method is shown to provide exact solutions to these Burger equations without requiring linearization.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
Numerical Solution Of Delay Differential Equations Using The Adomian Decompos...theijes
Adomian Decomposition Method has been applied to obtain approximate solution to a wide class of ordinary and partial differential equation problems arising from Physics, Chemistry, Biology and Engineering. In this paper, a numerical solution of delay differential Equations (DDE) based on the Adomian Decomposition Method (ADM) is presented. The solutions obtained were without discretization nor linearization. Example problems were solved for demonstration. Keywords: Adomian Decomposition, Delay Differential Equations (DDE), Functional Equations , Method of Characteristic.
In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem where as all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP.
On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore, the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful.
This document discusses regularization methods for solving inverse problems. It begins with an introduction to inverse problems, explaining that they are typically ill-posed and lack stability. Regularization is introduced as a way to approximate an ill-posed inverse problem with a family of nearby well-posed problems. Two main categories of regularization methods are described: classical regularization methods, which include singular value decomposition, Tikhonov regularization, and truncated iterative methods; and local regularization methods, designed for Volterra integral equations of the first kind. The document provides mathematical definitions and explanations of these various regularization techniques.
This document discusses stochastic partial differential equations (SPDEs). It outlines several approaches that have been used to solve SPDEs, including methods based on diffusion processes, stochastic characteristic systems, direct methods from mathematical physics, and substitution of integral equations. It also discusses using backward stochastic differential equations to study SPDEs and introduces notation for the analysis of an Ito SDE with inverse time. The document is technical in nature and outlines the mathematical frameworks and equations involved in solving SPDEs through various probabilistic methods.
Derivation and Application of Multistep Methods to a Class of First-order Ord...AI Publications
Of concern in this work is the derivation and implementation of the multistep methods through Taylor’s expansion and numerical integration. For the Taylor’s expansion method, the series is truncated after some terms to give the needed approximations which allows for the necessary substitutions for the derivatives to be evaluated on the differential equations. For the numerical integration technique, an interpolating polynomial that is determined by some data points replaces the differential equation function and it is integrated over a specified interval. The methods show that they are only convergent if and only if they are consistent and stable. In our numerical examples, the methods are applied on non-stiff initial value problems of first-order ordinary differential equations, where it is established that the multistep methods show superiority over the single-step methods in terms of robustness, efficiency, stability and accuracy, the only setback being that the multi-step methods require more computational effort than the single-step methods.
This document presents an internship project report on multistep methods for solving initial value problems of ordinary differential equations. It introduces the basic problem of finding the function y(t) that satisfies a given differential equation and initial condition. It discusses existence and uniqueness theorems, Picard's method of successive approximations, and approaches for approximating the required integrations, including the derivative, Taylor series, and Euler's methods. The report appears to evaluate various one-step and multistep numerical methods for solving initial value problems, including Runge-Kutta, Adams-Bashforth, and Adams-Moulton methods.
The document analyzes the analytic solution of Burger's equations using the variational iteration method. It begins by introducing the variational iteration method and how it can be used to solve differential equations. It then applies the method to obtain exact solutions for the (1+1), (1+2), and (1+3) dimensional Burger equations. Lengthy iterative solutions are presented for each case. The variational iteration method is shown to provide exact solutions to these Burger equations without requiring linearization.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
Numerical Solution Of Delay Differential Equations Using The Adomian Decompos...theijes
Adomian Decomposition Method has been applied to obtain approximate solution to a wide class of ordinary and partial differential equation problems arising from Physics, Chemistry, Biology and Engineering. In this paper, a numerical solution of delay differential Equations (DDE) based on the Adomian Decomposition Method (ADM) is presented. The solutions obtained were without discretization nor linearization. Example problems were solved for demonstration. Keywords: Adomian Decomposition, Delay Differential Equations (DDE), Functional Equations , Method of Characteristic.
In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem where as all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP.
On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore, the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful.
This document discusses regularization methods for solving inverse problems. It begins with an introduction to inverse problems, explaining that they are typically ill-posed and lack stability. Regularization is introduced as a way to approximate an ill-posed inverse problem with a family of nearby well-posed problems. Two main categories of regularization methods are described: classical regularization methods, which include singular value decomposition, Tikhonov regularization, and truncated iterative methods; and local regularization methods, designed for Volterra integral equations of the first kind. The document provides mathematical definitions and explanations of these various regularization techniques.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
This document discusses numerical methods for solving nonlinear equations. It begins by stating that nonlinear equations cannot generally be solved analytically and must instead be approached using iterative methods. It then covers the general principles of iterative methods, including convergence conditions, order of convergence, and stop criteria. The document focuses on several common iterative methods, including first-order methods like the chord, classic chord, and regula falsi methods, as well as the second-order Newton-Raphson method. It also briefly discusses solving systems of nonlinear equations.
This document discusses conformal field theories and three point functions. It begins by introducing conformal field theories and noting that three point functions of stress tensors are constrained by conformal invariance. For parity even CFTs in d=4 dimensions, the three point functions of the stress tensor T can be written in terms of coefficients that correspond to effective numbers of scalars, fermions, and vectors. Positivity of energy flux from collider experiments imposes constraints on these coefficients. The document then discusses extending such analyses to parity violating CFTs in d=3 dimensions, where the three point functions of the stress tensor T and conserved current j contain both parity even and odd terms. It proposes studying modifications to existing constraints from applying coll
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IVRai University
This document discusses finite difference and interpolation methods. It covers topics like finite differences, difference tables, Newton's forward and backward interpolation formulas, Stirling's interpolation formula, Newton's divided difference formula for unequal intervals, and Lagrange's divided difference formula for unequal intervals. Examples are provided to demonstrate calculating finite differences, constructing difference tables, and using interpolation formulas to estimate values between given data points.
The document summarizes techniques for solving linear systems of equations. It discusses direct solution methods like Gaussian elimination that transform the system into an upper triangular system and then use back substitution to solve. Gaussian elimination involves using elementary row operations to eliminate values below the diagonal of the coefficient matrix. The document also discusses concepts like consistency, uniqueness of solutions, and ill-conditioned systems. It provides examples of applying elementary row operations during the Gaussian elimination process.
International Journal of Engineering Research and Applications (IJERA) aims to cover the latest outstanding developments in the field of all Engineering Technologies & science.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
This document provides an overview of geometrical optimal control theory for dynamical systems. It discusses several problems in optimal control theory where geometrical ideas can provide insights, including singular optimal control, implicit optimal control, integrability of optimal control problems, and feedback linearizability. For singular optimal control problems, the document analyzes the behavior at both regular and singular points, and describes how singular problems can be treated as singularly perturbed systems.
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERS...ijfcstjournal
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated.
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document provides information about solving systems of linear equations through various methods such as graphing, substitution, and elimination. It defines what a linear system is and explains the concepts of consistent and inconsistent systems. Graphing is discussed as a way to find the point where two lines intersect. The substitution and elimination methods are described step-by-step with examples shown of using each method to solve sample systems of equations. Additional topics covered include slope, matrix notation, and an example of using a matrix to perform a Hill cipher encryption on a short plaintext message.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIIRai University
This document discusses methods for finding the roots or zeros of equations, including the bisection method, Newton-Raphson method, and regula-falsi method. It provides definitions and steps for each method. The bisection method works by repeatedly bisecting the interval that contains the root. Newton-Raphson uses successive approximations to iteratively find better estimates for the root. Regula-falsi is based on finding the x-intercept of the chord between two points on the function graph. Examples are provided to demonstrate applying each method to find the roots of equations.
Numerical method for pricing american options under regime Alexander Decker
This document presents a numerical method for pricing American options under regime-switching jump-diffusion models. It begins with an abstract that describes using a cubic spline collocation method to solve a set of coupled partial integro-differential equations (PIDEs) with the free boundary feature. The document then provides background on regime-switching Lévy processes and derives the PIDEs that describe the American option price under different regimes. It presents the time and spatial discretization methods, using Crank-Nicolson for time stepping and cubic spline collocation for the spatial variable. The method is shown to exhibit second order convergence in space and time.
The document discusses polynomial normal matrices and polynomial unitary matrices. Some key points:
- Polynomial normal matrices are polynomial matrices whose coefficient matrices are normal matrices. Properties of polynomial normal matrices are discussed, including that the product of two polynomial normal matrices is also polynomial normal if they commute.
- A polynomial matrix is unitary if each coefficient matrix is unitary. It is shown that a polynomial matrix is polynomial normal if and only if every polynomial matrix unitarily equivalent to it is also polynomial normal.
- Results are presented, including that the product of the coefficient matrices of two polynomial matrices is zero if the coefficient matrices are normal and the products of the coefficients are zero.
This lecture discusses information inequalities and their proofs, including:
1) Conditioning does not increase entropy - conditioning adds information, reducing uncertainty.
2) Independence bound - mutual entropy is highest when variables are independent.
3) Data processing inequality - processing data cannot increase information about the original variables.
Fano's inequality relates error probability in estimating a random variable to the conditional entropy. It bounds the uncertainty remaining about a variable after observing its estimate.
The document discusses various methods for finding the roots of equations, including:
1. The bisection method, which repeatedly bisects an interval containing a root until the interval is sufficiently small.
2. The fixed point method, which generates successive approximations that converge to a root by iterating a function.
3. The Newton-Raphson method, which uses the tangent line at an initial guess to generate a new, closer approximation for the root through an iterative formula.
On the Numerical Fixed Point Iterative Methods of Solution for the Boundary V...BRNSS Publication Hub
In this research work, we have studied the finite difference method and used it to solve elliptic partial differential equation (PDE). The effect of the mesh size on typical elliptic PDE has been investigated. The effect of tolerance on the numerical methods used, speed of convergence, and number of iterations was also examined. Three different elliptic PDE’s; the Laplace’s equation, Poisons equation with the linear inhomogeneous term, and Poisons equations with non-linear inhomogeneous term were used in the study. Computer program was written and implemented in MATLAB to carry out lengthy calculations. It was found that the application of the finite difference methods to an elliptic PDE transforms the PDE to a system of algebraic equations whose coefficient matrix has a block tri-diagonal form. The analysis carried out shows that the accuracy of solutions increases as the mesh is decreased and that the solutions are affected by round off errors. The accuracy of solutions increases as the number of the iterations increases, also the more efficient iterative method to use is the SOR method due to its high degree of accuracy and speed of convergence
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
An Improved Regression Type Estimator of Finite Population Mean using Coeffic...inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document provides a brief history of mathematics from ancient to modern periods. It covers the development of numeration systems and arithmetic, geometry, algebra, trigonometry, calculus, and analytic geometry. Key developments include ancient civilizations developing practical math for trade and construction, Greeks establishing logic-based math and Euclidean geometry, Hindus and Arabs advancing the decimal numeral system and algebra, and Europeans in the early modern period making advances in trigonometry, logarithms, analytic geometry, and calculus.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
This document discusses numerical methods for solving nonlinear equations. It begins by stating that nonlinear equations cannot generally be solved analytically and must instead be approached using iterative methods. It then covers the general principles of iterative methods, including convergence conditions, order of convergence, and stop criteria. The document focuses on several common iterative methods, including first-order methods like the chord, classic chord, and regula falsi methods, as well as the second-order Newton-Raphson method. It also briefly discusses solving systems of nonlinear equations.
This document discusses conformal field theories and three point functions. It begins by introducing conformal field theories and noting that three point functions of stress tensors are constrained by conformal invariance. For parity even CFTs in d=4 dimensions, the three point functions of the stress tensor T can be written in terms of coefficients that correspond to effective numbers of scalars, fermions, and vectors. Positivity of energy flux from collider experiments imposes constraints on these coefficients. The document then discusses extending such analyses to parity violating CFTs in d=3 dimensions, where the three point functions of the stress tensor T and conserved current j contain both parity even and odd terms. It proposes studying modifications to existing constraints from applying coll
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IVRai University
This document discusses finite difference and interpolation methods. It covers topics like finite differences, difference tables, Newton's forward and backward interpolation formulas, Stirling's interpolation formula, Newton's divided difference formula for unequal intervals, and Lagrange's divided difference formula for unequal intervals. Examples are provided to demonstrate calculating finite differences, constructing difference tables, and using interpolation formulas to estimate values between given data points.
The document summarizes techniques for solving linear systems of equations. It discusses direct solution methods like Gaussian elimination that transform the system into an upper triangular system and then use back substitution to solve. Gaussian elimination involves using elementary row operations to eliminate values below the diagonal of the coefficient matrix. The document also discusses concepts like consistency, uniqueness of solutions, and ill-conditioned systems. It provides examples of applying elementary row operations during the Gaussian elimination process.
International Journal of Engineering Research and Applications (IJERA) aims to cover the latest outstanding developments in the field of all Engineering Technologies & science.
International Journal of Engineering Research and Applications (IJERA) is a team of researchers not publication services or private publications running the journals for monetary benefits, we are association of scientists and academia who focus only on supporting authors who want to publish their work. The articles published in our journal can be accessed online, all the articles will be archived for real time access.
Our journal system primarily aims to bring out the research talent and the works done by sciaentists, academia, engineers, practitioners, scholars, post graduate students of engineering and science. This journal aims to cover the scientific research in a broader sense and not publishing a niche area of research facilitating researchers from various verticals to publish their papers. It is also aimed to provide a platform for the researchers to publish in a shorter of time, enabling them to continue further All articles published are freely available to scientific researchers in the Government agencies,educators and the general public. We are taking serious efforts to promote our journal across the globe in various ways, we are sure that our journal will act as a scientific platform for all researchers to publish their works online.
This document provides an overview of geometrical optimal control theory for dynamical systems. It discusses several problems in optimal control theory where geometrical ideas can provide insights, including singular optimal control, implicit optimal control, integrability of optimal control problems, and feedback linearizability. For singular optimal control problems, the document analyzes the behavior at both regular and singular points, and describes how singular problems can be treated as singularly perturbed systems.
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERS...ijfcstjournal
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated.
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document provides information about solving systems of linear equations through various methods such as graphing, substitution, and elimination. It defines what a linear system is and explains the concepts of consistent and inconsistent systems. Graphing is discussed as a way to find the point where two lines intersect. The substitution and elimination methods are described step-by-step with examples shown of using each method to solve sample systems of equations. Additional topics covered include slope, matrix notation, and an example of using a matrix to perform a Hill cipher encryption on a short plaintext message.
Engineering Mathematics-IV_B.Tech_Semester-IV_Unit-IIIRai University
This document discusses methods for finding the roots or zeros of equations, including the bisection method, Newton-Raphson method, and regula-falsi method. It provides definitions and steps for each method. The bisection method works by repeatedly bisecting the interval that contains the root. Newton-Raphson uses successive approximations to iteratively find better estimates for the root. Regula-falsi is based on finding the x-intercept of the chord between two points on the function graph. Examples are provided to demonstrate applying each method to find the roots of equations.
Numerical method for pricing american options under regime Alexander Decker
This document presents a numerical method for pricing American options under regime-switching jump-diffusion models. It begins with an abstract that describes using a cubic spline collocation method to solve a set of coupled partial integro-differential equations (PIDEs) with the free boundary feature. The document then provides background on regime-switching Lévy processes and derives the PIDEs that describe the American option price under different regimes. It presents the time and spatial discretization methods, using Crank-Nicolson for time stepping and cubic spline collocation for the spatial variable. The method is shown to exhibit second order convergence in space and time.
The document discusses polynomial normal matrices and polynomial unitary matrices. Some key points:
- Polynomial normal matrices are polynomial matrices whose coefficient matrices are normal matrices. Properties of polynomial normal matrices are discussed, including that the product of two polynomial normal matrices is also polynomial normal if they commute.
- A polynomial matrix is unitary if each coefficient matrix is unitary. It is shown that a polynomial matrix is polynomial normal if and only if every polynomial matrix unitarily equivalent to it is also polynomial normal.
- Results are presented, including that the product of the coefficient matrices of two polynomial matrices is zero if the coefficient matrices are normal and the products of the coefficients are zero.
This lecture discusses information inequalities and their proofs, including:
1) Conditioning does not increase entropy - conditioning adds information, reducing uncertainty.
2) Independence bound - mutual entropy is highest when variables are independent.
3) Data processing inequality - processing data cannot increase information about the original variables.
Fano's inequality relates error probability in estimating a random variable to the conditional entropy. It bounds the uncertainty remaining about a variable after observing its estimate.
The document discusses various methods for finding the roots of equations, including:
1. The bisection method, which repeatedly bisects an interval containing a root until the interval is sufficiently small.
2. The fixed point method, which generates successive approximations that converge to a root by iterating a function.
3. The Newton-Raphson method, which uses the tangent line at an initial guess to generate a new, closer approximation for the root through an iterative formula.
On the Numerical Fixed Point Iterative Methods of Solution for the Boundary V...BRNSS Publication Hub
In this research work, we have studied the finite difference method and used it to solve elliptic partial differential equation (PDE). The effect of the mesh size on typical elliptic PDE has been investigated. The effect of tolerance on the numerical methods used, speed of convergence, and number of iterations was also examined. Three different elliptic PDE’s; the Laplace’s equation, Poisons equation with the linear inhomogeneous term, and Poisons equations with non-linear inhomogeneous term were used in the study. Computer program was written and implemented in MATLAB to carry out lengthy calculations. It was found that the application of the finite difference methods to an elliptic PDE transforms the PDE to a system of algebraic equations whose coefficient matrix has a block tri-diagonal form. The analysis carried out shows that the accuracy of solutions increases as the mesh is decreased and that the solutions are affected by round off errors. The accuracy of solutions increases as the number of the iterations increases, also the more efficient iterative method to use is the SOR method due to its high degree of accuracy and speed of convergence
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
An Improved Regression Type Estimator of Finite Population Mean using Coeffic...inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document provides a brief history of mathematics from ancient to modern periods. It covers the development of numeration systems and arithmetic, geometry, algebra, trigonometry, calculus, and analytic geometry. Key developments include ancient civilizations developing practical math for trade and construction, Greeks establishing logic-based math and Euclidean geometry, Hindus and Arabs advancing the decimal numeral system and algebra, and Europeans in the early modern period making advances in trigonometry, logarithms, analytic geometry, and calculus.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Inventory Model with Different Deterioration Rates with Stock and Price Depen...inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
On Estimation of Population Variance Using Auxiliary Informationinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
SUCCESSIVE LINEARIZATION SOLUTION OF A BOUNDARY LAYER CONVECTIVE HEAT TRANSFE...ijcsa
The purpose of this paper is to discuss the flow of forced convection over a flat plate. The governing partial
differential equations are transformed into ordinary differential equations using suitable transformations.
The resulting equations were solved using a recent semi-numerical scheme known as the successive
linearization method (SLM). A comparison between the obtained results with homotopy perturbation method and numerical method (NM) has been included to test the accuracy and convergence of the method.
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International Journal of Mathematics and Statistics Invention (IJMSI)
1. International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 2 Issue 1 ǁ January. 2014ǁ PP-30-40
On New Iterative Methods for Numerical Solution of HigherOrder Parametric Differetial Equations
1,
1,2,
O.A TAIWO and 2,A. O.ADEWUMI
Department of mathematics, university of Ilorin, Ilorin, Nigeria
ABSTRACT : In this paper, we attempt to answer the question “Is it possible to reduce the order of the
Homotopy Analysis Method (HAM ) approximation to obtain the required approximation analytical solution to
a given accuracy “? YES. Based on the Homotopy Analysis Method, we developed two iterative methods,
namely; Integrated Chebyshev Homotopy Analysis Methods (HC-HAM) and Integrated Chebyshev-Tau
Homotopy Analysis Method (HC-THAM) for solving higher-order parametric boundary-value problems.
Homotopy Aalysis Method is blended with Integrated Chebyshev Polynomials and Tau Methods and this is
done by using Integrated Chebyshev Polynomials to represent the initial approximation and the derivative
corresponding to m=1 and also by introducing a perturbation terms in the deformation equation. The
performance of the proposed methods is validated through examples from literature. Apart from ease of
implementation, better accuracy is obtained. Comparison with existing methods such as Standard Homotopy
Analysis Method, Adomian Decomposition Method, Extended Adomian Decomposition Method, Optimal
Homotopy Asymptotic Method and Homotopy Perturbation Method are made to show the superiority and simple
applicability of the proposed iterative methods.
KEYWORDS: Homotopy analysis method, integrated Chebyshev polynomial, Tau method, Parametric
Differential equations, Perturbation term
I.
INTRODUCTION
Higher-order parametric differential equations [14] appear frequently in physical problems and there
are numbers of real time phenomena which are modeled by such equations [13]. Since exact solutions to these
differential equations are very rare, so researchers always look for the best approximation solution [1]. The
recent literature for the solutions of differential equations includes: the Adomian Decomposition Method
(ADM) [3], the Differential Transform Method (DTM) [5], the Variation Iteration Method (VIM) [9],the
Homotopy Perturbation Method (HPM) [7,8], the Extended Adomian Decomposition Method (EADM) [4],
Homotopy Analysis Method [2,13]. etc. The classical Perturbation Methods are restricted to small or large
parameters and hence their use is confined to a limited class of problems. The HPM as well as HAM, which are
the elegant combination of Homotopy from topology and perturbation techniques, overcomes the restrictions of
small or large parameters in the problems [1]. Liao [11,12] developed Homotopy Analysis Method as this
method has been applied on a wide class of initial and boundary value problems [2]. Also, Marinca and
Herisanu [15,16 ] introduced the Optimal Homotopy Asymptotic Method (OHAM), which uses the more
generalized auxiliary function (HCP). They reported different forms of auxiliary that can be expressed in a
compact form as H(P)=f(r)g(P, C i ) is the power series in P, and the unknown constants C i , which control the
convergence of the approximating series solution, are optimally determined [1]. G. Ebadietal [4] used Extended
Adomian Decomposition Method for the solutions of fourth-order parametric boundary value problems, J. A li.
et al [1] applied Optimal Homotopy Asymptotic Method for solving parameterized sixth-order boundary-value
problem and S.T. Mohyus Din [13] solved higher-order parametric differential equations by Homotopy Analysis
Method. In this paper, we solved higher-order parametric differential equations by IC-HAM and IC-THAM.
The results are then compared with those of exact solution and the solution obtained by HAM, HPM, OHAM,
ADM and EADM. The structure of this paper is organized as follows; brief discussion on Chebyshev
Polynomials is presented in sections 2. Section 3 is devoted to the construction of the proposed methods. In
sections 4, the new methods are applied to some numerical examples and finally, section 5 is devoted to
conclude the paper.
II.
CONSTRUCTION OF CHEBYSHEV POLYNOMIALS
The Chebyshev Polynomial of degree n over {-1,1} is defined by the relation
Tn x cos n cos 1 x
(2.1)
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30 | P a g e
2. On New Iterative Methods for Numerical Solution…
And,
Tn 1 x 2 xTn x Tn1 x ,
n 1
(2.2)
Equation (2.2) is the recurrence relation of the Chebyshev Polynomials in the interval [-1, 1]. Few terms are:
T0 x 1 , T1 x x, T2 x 2 x 2 1, T3 x 4 x 3 3x
etc
These could be converted into any interval of consideration. For example, in [a , b],we have
2 x a b
Tn x cosnCos 1
b a
(2.3)
And the recurrence relation is given as
2x a b
Tn x 2
Tn x Tn 1 x , n 1
ba
(2.4)
Numerical Solution Techniques
It is necessary in the first instance to give a brief review of the Homotopy Analysis Method since our
techniques build on this method and serve to improve the accuracy of the HAM. The new methods refined the
HAM by using a more accurate initial approximation solution and other derivatives corresponding to m=1 by
integrating truncation Chebyshev polynomial and solving the higher-order deformation equations using tau
methods, known for better higher accuracy.
Basic idea of HAM
Consider the following differential equation
N[U(t)]=0
(3.1)
Where N is a nonlinear operator, t denotes independent variable, U(t) is an unknown function respectively. By
means of generalizing the traditional Homotopy Method, Liao [11,12] constructs the so-called zero-order
deformation equation
1 q L t , q
Where
U 0 t qc0 H t N t , q
(3.2)
q 0,1 is the embedding parameter, C0 0 is a non-zero auxiliary parameter, H(t) 0 is an auxiliary
function, L is an auxiliary linear operator, U o t is an initial guess of U(t),
t, q is an unknown function
respectively. Obviously, when q=0 and q=1, it holds for
t ,0 U o t , t ,1
U t
Thus, as q increase from 0 to 1, the solution
solution U t ,
(3.3)
t, q
varies from initial guess U 0 t to the
exp anding U 0 t in Taylor series with respect to q, we have
t , q U 0 t U m t q m
(3.4)
m 1
Where
1 m t , q
U m t
m! q m
(3.5)
q 0
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31 | P a g e
3. On New Iterative Methods for Numerical Solution…
If the auxiliary linear operator, the initial guess, the auxiliary function and convergence-control parameter, are
properly chosen, the series (3.4), converges at q=1, then we have
U t U 0 t U m t
(3.6)
m 1
According to the definition (3.5), the governing equation can be deduced from the zero-order deformation
equation (3.2)
Define the vector
U m 0 t , U 1 t , .......... , U n t
U
(3.7)
Differentiating equation (3.2) m times with respect to the embedding parameter q and then setting q=0 and
finally dividing them by m!, we have the so-called mth-order deformation equation
LU m t X mU m1 t Co H t Rm U m1 t
(3.8)
Where,
1
m1 N t , q
Rm U m1
m 1!
q m1
(3.9)
And,
m 1
0 ,
Xm
1,
m 1
(3.10)
If we multiply with
equation
L1 each side of the equation (3.8), we obtained the following mth-order deformation
U m t X mU m1 t C0 H t L1 Rm U m1
(3.11)
It should be emphasized that U m t for m 1 is governed by the linear equation (3.8) with the boundary
conditions, which is easily solved by symbolic composition software such as Maple or Mathematics.
3.2 Construct of the NHAMs Algorithms
We consider the general higher-order boundary-value problem of the form:
y n x f
x, y, y ,......., y ,
1
n 1
a xb
(3.12)
Subject to the two-point boundary conditions
y a 0 ,
y 1 a 1 ,........, y r a r
(3.13)
y b 0 ,
y 1 b 1 ,........, y r a r
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32 | P a g e
4. On New Iterative Methods for Numerical Solution…
Where 0 r n 2 is an integer, f is a polynomial in
x, y x , y 1 x ,...... y n 1 x , and a, b, 0 , 1 ,......., r , 0 , 1 ,.... n r 2 are real cons tan ts.
The zeroth-order deformation equations are given as
1 q Lr x, q
y 0 x qc0 H N Y , q
(3.14)
And
1 q LY x, q
y 0 x qc0 H
N Y , q H
1
DH N , q 0,1
(3.15)
Where,
DH N
d
T1TN T2TN 1 T3TN 2 ....... TN TN 1 ,
d
( x ) n1
H ( )
,
(n 1) !
N Y , q
L1
x
( . )d
0
n y
y 2 y
n1 y
f , y, , 2 ,......... n1
,
n
(3.16)
It should be emphasized that y o x of the solution
y x and other derivatives corresponding to m 1 are
determined as follows.
Following [10], we have
N
d n y0 x
ai Ti x
dx n
i 0
(3.17)
Integrating equation (3.17) successively, we obtain
N
N 1
d n 1 y 0 x
ai Ti x dx c1 Qi i n 1
dx n 1
i 0
i 1
N 1
d n2 y0 x
n 1
Qi i dx c1 x c2
n2
dx
i 1
N 1
Q
i 0
i
(3.18 )
in 2
(3.19)
N
dy0 x
Qi
dx
i 2dx c1
x n 2
x n 3
c2
........ cn2 x cn1
n 2!
n 3!
www.ijmsi.org
N n 1
Q
i 0
i
i0
(3.20)
33 | P a g e
5. On New Iterative Methods for Numerical Solution…
N
y 0 x Qi
i 0
i 1dx c1
x n 1
x n2
c2
........ c n 1 x c n
n 1!
n 2!
N n
Q
i 0
i
i0
(3.21)
Unlike in the case of the HAM, the auxiliary function and convergence-control parameter are not necessary as
there is no need for the solution of the higher-order deformation to confirm to some rules of solution expression.
METHOD
Following the HAM procedure, we formulate the higher-order deformation equation by differentiating
the zero-order deformation equation m-times with respect to q and then dividing by m! to get
L y m x X m y m1 x H Rm y m1
(3.22)
1
Operating the operator L , the inverse of
d
to both sides of (3.22), then the mth –order deformation have
d
the following form:
y m x X m y m1 x L1 ( H Rm y m1
(3.23)
Where
n y
Rm y m1 n
y
n1 y
f , y, ,........, n1
Thus, the recursive formula for the Integrated Chebysher Homotopy Analysis Method (IC-HAM) is formulated
as:
y0 x
N n
Q
i 0
y1 x
i
i 0
L1 H Rm y m 1
.
.
(3.24)
y m x y m 1 x L1 H Rm y m1 , m 2,3..........
Method :
Following the same procedure as discussed in method 1 (IC-HAM), the mth-order deformation of method 2 (ICTHAM) has the form:
y m x X m y m 1 x L1 H Rm y m1
(3.25)
Where,
n y
y
n1 y
Rm y m1 n f , y, ,........, n1
1 X m H 1 DH N
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6. On New Iterative Methods for Numerical Solution…
Also, the recursive formula for the IC-THAM is given as:
y0 x
N n
Q
i 0
i
y1 x L1 H
i 0
Rm
y m1
.
(3.26)
y m x y m1 x L1 H
Rm
y m1 , m 2,3..........
Application of methods on some Examples
In this section, we apply the techniques described in section 3.To some illustrative example of fourth, sixth and
eighth-order parameter boundary-value problems.
Example:
Consider the following linear problem [4]
y iv x 1 c y ii x cyx
1
cx 2 1 ,0 x 1
2
Subject to
y 0 1 ,
y 1 0 1
y 1 1.5 smh 1 ,
y 1 1 1.5 cosh1
The exact solution for this problem is
yx 1
1 2
x smh x
2
According to (3.24) the zeroth-order deformation is given by
d 4 y , q
d 2 y , q
1
1 q LY x, q y0 x qH
1 c
cy ; q c 2 1
4
d 2
2
d
Now, our initial approximation has the form of equation (3.21), we chose the auxiliary linear operation
L y ; q
x
dy ; q
and H
d
6
3
Hence, the mth-order deformation can be given by
L y m x xm y m1 x H Rm y m1
Where
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7. On New Iterative Methods for Numerical Solution…
d 4 y 0 ; y
R1 y 0
d 4
1 c d
2
y 0 ; q
1
cy 0 ; q c 2 1
2
d
2
And
d 4 y m1 ; y
d 2 y m1 ; q
Rm y m1
1 c
cy m1 ; q 1 xm
d 4
d 2
1 2
c 1; m 2
2
Now, the solution of the m-th order deformation equation for
m 1 becomes y m x X m y m1 x
x
0
1
3
x Rm y m1 d
6
Consequently, the first few terms of the IC-HAM series solution for N=4 and C=1 are as follows;
1
1
1
1 5
1 6
1
1
y 0 x c4 c3 x c2 x 2 c1 x 3 a0 x 4 x 4
x a1 x 4
x a 2
2
6
24
60
45
2
24
4 7
4
4
32 7
8 8
1 4 3 5 2 6
1
x
x x
x a3 x 4 x 5 x 6
x
x a 4
20
15
105
15
a
105
105
24
24
1 5
1 5
1 4
1 4
1 6
1
1 6 1 4
y1 x
x7
x c1
x x c 2
x c3
x c4
x
x
60
12
120
24
24
720
5040
720
1 6
1
1 6
1 7
1
1
1 4
1 4 1 5
x
x 8 a 0
x
x
x
x
x8
x 9 a1
x
360
40320
60
360
1260
40320
181440
24
24
7 6
1 7
31
1
1
1 4 1 5
x
x
x8
x9
x10 a 2
x x
15
360
315
40320
45360
226800
24
191 8
61
1
1
1 4 3 5 47 6 13 7
x
x
x
x
x9
x10
x11 a3
x
20
360
420
40320
60480
37800
207900
24
811 8
19 9
13 10
2
1
1 4 4 5 53 6 92 7
x
x
x
x
x
x11
x12 a 4
x x
15
120
315
13440
2268
8100
51975
155925
24
and so on.
The first order approximation solution by IC-HAM is
y x y 0 x y1 x
And the residual of the solution is
1
R y iv x 1 c y ii x cy x cx 2 1
2
Using the boundary conditions, we obtained c1 , c2 , ..... c4 and minimizing the residual error by using Least
Square Method, we obtained the following values of a0 , a1 , a3 and a 4 for c 1
a0 0.5541761136 ; a1 0.5816075870 ; a2 0.0332521244 2 ; a3 0.0059547394 61 ;
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8. On New Iterative Methods for Numerical Solution…
a4 0.0001715598 043 .
Thus, the approximation solution becomes:
y x 1 0.4999999964 x 2 0.1666666688 x 3 0.0000031231 3x 4 0.0083148926 99 x 5
0.0000423351 8090 x 6 0.0001511510 114 x 7 0.0000253855 3147 x 8 1.100271312 x10 19 x12
2.204069758 x10 8 x11 2.862613806 x10 7 x10 0.0000020962 70555 x 9
For c = 10, the following values a 0 , a1 , a3 and a 4 are obtained
a0 0.5541759788 ; a1 0.5816065302 ; a2 0.03325208987 ; a3 0.005953393889 ;
a4 0.0001715410873
In this case, the approximate solution is
y x 1 x 0.4999999868 x 2 0.1666666666 x 3 0.0000033190 90608 x 4 0.0083144687 0 x 5
0.00004217325 x 6 0.0001530912179x 7 0.00002194783736x 8 1.100151273 X 10 8 x12
2.203494560 X 10 7 x11 0.000001554979248x10 8.63843375 X 10 7 x 9
Method 2:
Following the same procedures as discussed in method I (IC-HAM) and using equation (3.2.14), we obtained
C1 ,.....C 4 in terms of a0 , a1 ,... a 4, 1 ,..., 4 and minimizing of the residual error, we obtained the following
values of a0 ,... a 4 , 1 ... 4 for C 1 :
a0 0.1480912750 ; a1 1.677563749; a2 0.004350300326 ; a3 0.008606762145 ;
a4 0.00008410994856; 1 0.0003445697303; 2 0.02176231068 ; 3 0.008251027514 ;
4 1 0552333902 .
Also, by substituting these values into first order approximation
y 0 y1 , the approximation solution
becomes:
y x 1 0.4999999979x 2 0.166666680x 3 0.0000015534x 4 0.00832419551x 5
0.000020901367 x 6 0.0001751188113x 7 1.000000000x 0.00001264705882x 8
5.394256762X 10 x12 4.463512138X 10 8 x11 1.118820631X 10 7 x10 4.36488315X 10 8 x 9
Similarly, for C=10, the following values of a0 ; a1 ........ a 4 ;
1 ;......... . 4
are obtained;
a0 0.03731442487 ; a1 0.4615749665 ; a2 0.08481031518 ; a3 0.003695192408 ;
a4 0.002389928628 ; 1 0.00007793043043047; 2 0.0008494323998 ;
3 0.0004018919498; 4 0.05881308214 .
Thus, we have the following approximate solution:
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9. On New Iterative Methods for Numerical Solution…
y x 1 0.4999999997x 2 0.1666666647x 3 0.0000042137x 4 0.00829840437 x 5
0.00011579718 x 6 7.64551X 10 7 x 7 1.000000000x 0.0001836807083x 8
1.53274246X 10 7 x12 0.000001097384388x11 0.00001543090318x10 0.00008470636057x 9
Table 1:
Absolute errors of the first-order approximate solution when C=1 and the error for third order approximation
solution of HAM [13]
x
Analytical solution
[4]
EHPM
[4]
EADM
[4]
EEADM
[13]
EHAM
E IC -HAM
E IC -
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.000000000
1.105166750
1.221336002
1.349520293
1.490752326
1.646095306
1.816653582
2.003583702
2.208105982
2.431516726
2.675201194
0.0000
7.4e-05
2.5e-04
4.6e-04
6.5e-04
7.6e-04
7.5e-04
6.1e-04
3.8e-04
1.3e-04
0.0000
0.0000
7.4e-05
2.5e-04
4.6e-04
6.5e-04
7.6e-04
7.5e-04
6.1e-04
3.8e-04
1.3e-04
0.0000
0.0000
1.1488e-06
3.2027e-07
1.1328e-05
3.4636e-05
6.6411e-05
9.6330e-05
1.1038e-04
9.6471e-05
5.2931e-05
0.0000
0.0000
4.8e-06
8.6e-05
3.2e-04
6.5e-04
9.6e-04
1.1e-03
1.00e-03
7.0e-04
2.5e-04
0.0000
0.0000
0.0000
2.000e-09
3.00e-09
2.00e-09
1.00e-09
3.00e-09
4.00e-09
1.00e-09
0.0000
0.0000
THAM
0.0000
0.0000
0.0000
2.00e-09
2.00e-09
0.0000
3.00e-09
1.00e-09
1.00e-09
1.00e-09
0.0000
Table 2:
Absolute errors of the first-order approximate solution when C=10 and errors for third order approximation of
HAM [13]
X
Analytical solution
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.000000000
1.105166750
1.221336002
1.349520293
1.490752326
1.646095306
1.816653582
2.003583702
2.208105982
2.431516726
2.675201194
[4]
EHPM
0.0000
1.7e-04
5.7e-04
1.0e-03
1.4e-03
1.6e-03
1.6e-03
1.2e-03
7.6e-03
2.5e-03
0.0000
[4]
EADM
0.0000
1.7e-04
5.7e-04
1.0e-03
1.4e-03
1.6e-03
1.6e-03
1.2e-03
7.6e-03
2.5e-03
0.0000
[4]
EEADM
0.0000
4.50410e-06
3.02581e-05
8.72832e-05
1.67419e-04
2.44493e-04
2.83793e-04
2.58064e-04
1.66169e-04
4.94701e-05
0.0000
[13]
EHAM
0.0000
2.9e-06
9.2e-05
3.1e-04
6.2e-04
9.2e-04
1.1e-03
1.0e-03
7.2e-04
2.7e-04
0.0000
EIC-HAM
0.0000
0.0000
1.00e-09
2.00e-09
2.00e-09
1.00e-09
0.0000
1.00e-09
1.00e-09
1.00e-09
0.0000
EICTHAM
0.0000
0.0000
0.0000
0.0000
0.0000
1.00e-09
0.0000
2.00e-09
1.00e-09
1.00e-09
1.30e-09
Example 2: Consider the following problem [1]
y vi x 1 c y iv x Cy ii x Cx ; 0 x 1
with the boundary conditions
y 0 1 ; y i 0 1 ; y ii 0 0
y1
7
1
sinh1 ; y i 1 cosh1 ; y ii 1 1 sinh1
6
2
The exact solution is given as:
yx 1
1 3
x sinhx
6
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10. On New Iterative Methods for Numerical Solution…
This problem is solved by the method applied in Example 1 and for each test point, the absolute error between
the analytical solution and the results obtained by the HAM [13], OHAM [1], and the IC-HAM and IC-THAM
when N=10 are compared in Table3 for C=1000. With only one iteration, a better approximation is obtained.
Table 3: Absolute errors of the first –order approximate solution when C=1000 and errors of the third-order
OHAM [1] and fifth-order HAM [1] are tabulated below for comparison.
x
Analytical solution
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.100333417
1.202669336
1.309020293
1.421418993
1.541928638
1.672653587
1.815750369
1.973439315
2.148016726
Error of
IC-HAM
1.0e-09
1.0e-09
0.0000
1.0e-09
2.0e-09
1.0e-09
2.0e-09
2.0e-09
1.0e-09
Error of
IC-THAM
1.0e-09
2.0e-09
1.0e-09
4.0e-09
4.0e-09
3.0e-09
2.0e-09
2.0e-09
2.0e-09
[1]
EHAM
9.1e-06
1.6e-04
4.4e-04
6.8e-04
7.3e-04
5.8e-04
3.2e-04
9.8e-05
4.7e-06
[1]
EOHAM
1.1e-05
3.3e-06
1.4e-05
5.2e-06
4.2e-05
5.7e-05
4.9e-05
4.5e-05
2.4e-05
Example 3:
Consider the following eighth-order parametric differential equation type [13]
d 8 yx
d 4 y x
c 4
1 c
Cy x
x 1
8
4
24
dx
dx
Subject to the boundary conditions
y 0 1;
y 1
dy0
d 2 y 0
1;
0
dx
dx 2
d 3 y 0
1
24
25
dy1 1
d 2 y 1 1
d 3 y 1
sinh 1;
cosh1;
sinh 1;
1 cosh1
24
dx
6
2
dx 2
dx 3
The exact solution for is given as:
yx 1
1 4
x sinhx
24
The numerical results obtained by IC-HAM and IC-THAM are compared with the analytical solution and the
results obtained by HAM [13] are presented in Table 4. These results are evaluated at m=1and it is seen from the
numerical results in the table that IC-HAM and IC-THAM are more accurate than the second order application
of the HAM solution in [13]
Table 4:
Absolute errors of the first-order approximate solution obtained by IC-HAM, IC-THAM (N=10) and the secondorder approximation of HAM [13] for example 3 when C=1000
X
0
0.1
0.2
0.3
0.4
0.5
0.6
Analytical solution
1.000000000
1.100170917
1.201402670
1.304857793
1.411818993
1.523699472
1.642053582
EHAM [13]
0.0000
8.7e-09
9.8e-08
3.3e-07
6.7e-07
9.4e-07
9.9e-07
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EIC-HAM
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
EIC-THAM
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
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11. On New Iterative Methods for Numerical Solution…
0.7
0.8
0.9
1.0
0
1.768587869
1.905172649
2.053854226
2.216867860
Denotes less than 10
7.8e-07
3.9e-07
7.8e-07
0.0000
1.0e-9
0.0000
1.0e-09
0.0000
1.0e-09
1.0e-09
2.0e-09
0.0000
10
III.
CONCLUSION
The main concern of this work is to develop efficient algorithms for the numerical solution of higherorder parametric differential equation. The goal and the question raised in the beginning of the paper are
achieved by blending integrated Chebyshev Polynomials and tau Methods with Homotopy Analysis Method to
solve this class of problems in question. The proposed algorithm produced rapidly convergent series and the
results obtained by the new methods agreed well with the analytical solutions with less computational work.
These confirm the belief that the efficiency of the proposed methods give much wider applicability for general
classes of parametric differential equations.
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