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.
This document discusses applications of first order ordinary differential equations (ODEs) as mathematical models. It provides examples of using first order ODEs to model population growth and decay, predator-prey interactions, and mixing problems. The modeling of logistic population growth with a first order ODE is shown to be more powerful than exponential modeling. Basic principles for modeling like mass action and conservation of mass are also outlined.
Large sample property of the bayes factor in a spline semiparametric regressi...Alexander Decker
This document summarizes a research paper about investigating the large sample property of the Bayes factor for testing the polynomial component of a spline semiparametric regression model against a fully spline alternative model. It considers a semiparametric regression model where the mean function has two parts - a parametric linear component and a nonparametric penalized spline component. By representing the model as a mixed model, it obtains the closed form of the Bayes factor and proves that the Bayes factor is consistent under certain conditions on the prior and design matrix. It establishes that the Bayes factor converges to infinity under the pure polynomial model and converges to zero almost surely under the spline semiparametric alternative model.
This document discusses fuzzy soft sets and soft set theory. It begins with an introduction to soft set theory as a mathematical tool for dealing with uncertainties. It then provides definitions and examples related to soft set theory, including the definition of a soft set, operations on soft sets like union and intersection, and concepts like null soft sets and absolute soft sets. The document aims to lay the foundations of soft set theory.
This document introduces fuzzy sets and provides definitions of key concepts. It begins with an overview of fuzzy set theory and its development. Several fundamental definitions are then given, including membership function, universe of discourse, fuzzy set, support, crossover point, height, α-cut, and level set. Examples are provided to illustrate each definition. Common operations on fuzzy sets like union, intersection, and complement are also defined. The purpose is to lay the groundwork for understanding fuzzy set theory and its basic elements.
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.
11.solution of a singular class of boundary value problems by variation itera...Alexander Decker
1) The document proposes an effective methodology called the variation iteration method to find solutions to singular second order linear and nonlinear boundary value problems.
2) The variation iteration method constructs a sequence of correction functionals to iteratively solve the boundary value problem. It is shown that the limit of the convergent iterative sequence obtained from this method is the exact solution.
3) The convergence of the iterative sequence generated by the variation iteration method is analyzed. It is established that the sequence converges to the exact solution under certain continuity conditions on the problem functions.
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.
This document discusses applications of first order ordinary differential equations (ODEs) as mathematical models. It provides examples of using first order ODEs to model population growth and decay, predator-prey interactions, and mixing problems. The modeling of logistic population growth with a first order ODE is shown to be more powerful than exponential modeling. Basic principles for modeling like mass action and conservation of mass are also outlined.
Large sample property of the bayes factor in a spline semiparametric regressi...Alexander Decker
This document summarizes a research paper about investigating the large sample property of the Bayes factor for testing the polynomial component of a spline semiparametric regression model against a fully spline alternative model. It considers a semiparametric regression model where the mean function has two parts - a parametric linear component and a nonparametric penalized spline component. By representing the model as a mixed model, it obtains the closed form of the Bayes factor and proves that the Bayes factor is consistent under certain conditions on the prior and design matrix. It establishes that the Bayes factor converges to infinity under the pure polynomial model and converges to zero almost surely under the spline semiparametric alternative model.
This document discusses fuzzy soft sets and soft set theory. It begins with an introduction to soft set theory as a mathematical tool for dealing with uncertainties. It then provides definitions and examples related to soft set theory, including the definition of a soft set, operations on soft sets like union and intersection, and concepts like null soft sets and absolute soft sets. The document aims to lay the foundations of soft set theory.
This document introduces fuzzy sets and provides definitions of key concepts. It begins with an overview of fuzzy set theory and its development. Several fundamental definitions are then given, including membership function, universe of discourse, fuzzy set, support, crossover point, height, α-cut, and level set. Examples are provided to illustrate each definition. Common operations on fuzzy sets like union, intersection, and complement are also defined. The purpose is to lay the groundwork for understanding fuzzy set theory and its basic elements.
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.
11.solution of a singular class of boundary value problems by variation itera...Alexander Decker
1) The document proposes an effective methodology called the variation iteration method to find solutions to singular second order linear and nonlinear boundary value problems.
2) The variation iteration method constructs a sequence of correction functionals to iteratively solve the boundary value problem. It is shown that the limit of the convergent iterative sequence obtained from this method is the exact solution.
3) The convergence of the iterative sequence generated by the variation iteration method is analyzed. It is established that the sequence converges to the exact solution under certain continuity conditions on the problem functions.
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
The document proposes a correlation measure for intuitionistic fuzzy multi sets (IFMS). IFMS are an extension of intuitionistic fuzzy sets that allow elements to have multiple membership and non-membership values. The document defines IFMS and reviews existing correlation measures for fuzzy and intuitionistic fuzzy sets. It then proposes a new correlation similarity measure for IFMS that is an extension of existing intuitionistic fuzzy set correlation measures. This new measure takes into account the multiple membership and non-membership values of elements in IFMS. Examples are provided to demonstrate properties of the new correlation measure.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Solution of a singular class of boundary value problems by variation iteratio...Alexander Decker
1. The document proposes an effective methodology called the variation iteration method to find solutions to a general class of singular second-order linear and nonlinear boundary value problems.
2. The variation iteration method generates a sequence of correction functionals that converges to the exact solution of the boundary value problem.
3. The author applies the variation iteration method to solve a specific class of boundary value problems and derives the sequence of correction functionals. Convergence of the iterative sequence is also analyzed.
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.
Expectation Maximization Algorithm with Combinatorial AssumptionLoc Nguyen
Expectation maximization (EM) algorithm is a popular and powerful mathematical method for parameter estimation in case that there exist both observed data and hidden data. The EM process depends on an implicit relationship between observed data and hidden data which is specified by a mapping function in traditional EM and a joint probability density function (PDF) in practical EM. However, the mapping function is vague and impractical whereas the joint PDF is not easy to be defined because of heterogeneity between observed data and hidden data. The research aims to improve competency of EM by making it more feasible and easier to be specified, which removes the vagueness. Therefore, the research proposes an assumption that observed data is the combination of hidden data which is realized as an analytic function where data points are numerical. In other words, observed points are supposedly calculated from hidden points via regression model. Mathematical computations and proofs indicate feasibility and clearness of the proposed method which can be considered as an extension of EM.
Exact solutions for weakly singular Volterra integral equations by using an e...iosrjce
In this paper, an iterative method proposed by Daftardar-Gejji and Jafari namely (DJM) will be
presented to solve the weakly singular Volterra integral equation (WSVIE) of the second kind. This method is
able to solve large class of linear and nonlinear equations effectively, more accurately and easily. In this
iterative method the solution is obtained in the series form that converges to the exact solution if it exists. The
main contribution of the current paper is to obtain the exact solution rather than numerical solution as done by
some existing techniques. The results demonstrate that the method has many merits such as being derivativefree,
overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in
Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier as in Variational
Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations
as in Homotopy Perturbation Method (HPM). The results reveal that the method is accurate and easy to
implement. The software used for the calculations in this study was MATHEMATICA®
10.0.
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
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
On solving fuzzy delay differential equationusing bezier curves IJECEIAES
In this article, we plan to use Bezier curves method to solve linear fuzzy delay differential equations. A Bezier curves method is presented and modified to solve fuzzy delay problems taking the advantages of the fuzzy set theory properties. The approximate solution with different degrees is compared to the exact solution to confirm that the linear fuzzy delay differential equations process is accurate and efficient. Numerical example is explained and analyzed involved first order linear fuzzy delay differential equations to demonstrate these proper features of this proposed problem.
This research discussed nonoscillatory properties for resolve of nonlinear neutral
differences equation of second order with positive and negative coefficients. The
various new conditions which is ensure that all nonoscillatory solutions tend to zero
or infinity like are given two examples are illustrate the ordinary results.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
This document provides an overview of topics covered in a differential equations course, including:
1. Review of integration by parts and partial fractions.
2. Discussion of integral curves and the existence and uniqueness theorem for differential equations.
3. Classification and methods for solving first and higher order linear differential equations, including separable, exact, integrating factors, Bernoulli, homogeneous with constant coefficients, and undetermined coefficients.
4. Brief introduction to additional solution methods like Euler's method, power series, and Laplace transforms.
5. Mention of solving systems of linear differential equations.
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
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as extensions of classical set theory and differential equations to account for uncertainty. Several examples of fuzzy initial value problems are analyzed, comparing their behaviors under different types of differentiability. The solutions exhibit very different properties, even though the original crisp equations were equivalent, showing that different fuzzy representations can model the same real-world problem very differently.
This document presents an overview of the differential transform method (DTM) for solving differential equations. DTM uses Taylor series to obtain approximate or exact solutions. The document defines 1D, 2D, and 3D DTM and lists common operations. Examples are provided of applying DTM to solve systems of linear/non-linear differential equations. The document concludes with references for further applications of DTM in engineering and mathematics.
In this paper, natural inner product structure for the space of fuzzy n−tuples is introduced. Also we have
introduced the ortho vector, stochastic fuzzy vectors, ortho- stochastic fuzzy vectors, ortho-stochastic fuzzy
matrices and the concept of orthogonal complement of fuzzy vector subspace of a fuzzy vector space.
The document discusses the topic of matroids. It begins with an introduction and outline. It then provides definitions of independence systems and matroids. Examples of matroids are given such as minimal spanning trees, matchings, and matrix matroids. The greedy algorithm is discussed as it relates to matroids. Bases of matroids are defined and their properties explained. Partition matroids and their use in modeling problems like Hamiltonian paths are also covered.
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.
A Fast Numerical Method For Solving Calculus Of Variation ProblemsSara Alvarez
This document presents a numerical method called differential transform method (DTM) for solving calculus of variation problems. DTM finds the solution of variational problems in the form of a polynomial series without discretization. The method is applied to obtain the solution of the Euler-Lagrange equation arising from variational problems by considering it as an initial value problem. Some examples are presented to demonstrate the efficiency and accuracy of DTM for solving calculus of variation problems.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
The document proposes a correlation measure for intuitionistic fuzzy multi sets (IFMS). IFMS are an extension of intuitionistic fuzzy sets that allow elements to have multiple membership and non-membership values. The document defines IFMS and reviews existing correlation measures for fuzzy and intuitionistic fuzzy sets. It then proposes a new correlation similarity measure for IFMS that is an extension of existing intuitionistic fuzzy set correlation measures. This new measure takes into account the multiple membership and non-membership values of elements in IFMS. Examples are provided to demonstrate properties of the new correlation measure.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Solution of a singular class of boundary value problems by variation iteratio...Alexander Decker
1. The document proposes an effective methodology called the variation iteration method to find solutions to a general class of singular second-order linear and nonlinear boundary value problems.
2. The variation iteration method generates a sequence of correction functionals that converges to the exact solution of the boundary value problem.
3. The author applies the variation iteration method to solve a specific class of boundary value problems and derives the sequence of correction functionals. Convergence of the iterative sequence is also analyzed.
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.
Expectation Maximization Algorithm with Combinatorial AssumptionLoc Nguyen
Expectation maximization (EM) algorithm is a popular and powerful mathematical method for parameter estimation in case that there exist both observed data and hidden data. The EM process depends on an implicit relationship between observed data and hidden data which is specified by a mapping function in traditional EM and a joint probability density function (PDF) in practical EM. However, the mapping function is vague and impractical whereas the joint PDF is not easy to be defined because of heterogeneity between observed data and hidden data. The research aims to improve competency of EM by making it more feasible and easier to be specified, which removes the vagueness. Therefore, the research proposes an assumption that observed data is the combination of hidden data which is realized as an analytic function where data points are numerical. In other words, observed points are supposedly calculated from hidden points via regression model. Mathematical computations and proofs indicate feasibility and clearness of the proposed method which can be considered as an extension of EM.
Exact solutions for weakly singular Volterra integral equations by using an e...iosrjce
In this paper, an iterative method proposed by Daftardar-Gejji and Jafari namely (DJM) will be
presented to solve the weakly singular Volterra integral equation (WSVIE) of the second kind. This method is
able to solve large class of linear and nonlinear equations effectively, more accurately and easily. In this
iterative method the solution is obtained in the series form that converges to the exact solution if it exists. The
main contribution of the current paper is to obtain the exact solution rather than numerical solution as done by
some existing techniques. The results demonstrate that the method has many merits such as being derivativefree,
overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in
Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier as in Variational
Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations
as in Homotopy Perturbation Method (HPM). The results reveal that the method is accurate and easy to
implement. The software used for the calculations in this study was MATHEMATICA®
10.0.
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
Comparative Study of the Effect of Different Collocation Points on Legendre-C...IOSR Journals
We seek to explore the effects of three basic types of Collocation points namely points at zeros of Legendre polynomials, equally-spaced points with boundary points inclusive and equally-spaced points with boundary point non-inclusive. Established in literature is the fact that type of collocation point influences to a large extent the results produced via collocation method (using orthogonal polynomials as basis function). We
analyse the effect of these points on the accuracy of collocation method of solving second order BVP. For equally-spaced points we further consider the effect of including the boundary points as collocation points. Numerical results are presented to depict the effect of these points and the nature of problem that is best handled by each.
On solving fuzzy delay differential equationusing bezier curves IJECEIAES
In this article, we plan to use Bezier curves method to solve linear fuzzy delay differential equations. A Bezier curves method is presented and modified to solve fuzzy delay problems taking the advantages of the fuzzy set theory properties. The approximate solution with different degrees is compared to the exact solution to confirm that the linear fuzzy delay differential equations process is accurate and efficient. Numerical example is explained and analyzed involved first order linear fuzzy delay differential equations to demonstrate these proper features of this proposed problem.
This research discussed nonoscillatory properties for resolve of nonlinear neutral
differences equation of second order with positive and negative coefficients. The
various new conditions which is ensure that all nonoscillatory solutions tend to zero
or infinity like are given two examples are illustrate the ordinary results.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
This document provides an overview of topics covered in a differential equations course, including:
1. Review of integration by parts and partial fractions.
2. Discussion of integral curves and the existence and uniqueness theorem for differential equations.
3. Classification and methods for solving first and higher order linear differential equations, including separable, exact, integrating factors, Bernoulli, homogeneous with constant coefficients, and undetermined coefficients.
4. Brief introduction to additional solution methods like Euler's method, power series, and Laplace transforms.
5. Mention of solving systems of linear differential equations.
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
Perspectives and application of fuzzy initial value problemsRAJKRISHNA MONDAL
This document discusses fuzzy differential equations and fuzzy initial value problems. It introduces fuzzy sets and fuzzy differential equations as extensions of classical set theory and differential equations to account for uncertainty. Several examples of fuzzy initial value problems are analyzed, comparing their behaviors under different types of differentiability. The solutions exhibit very different properties, even though the original crisp equations were equivalent, showing that different fuzzy representations can model the same real-world problem very differently.
This document presents an overview of the differential transform method (DTM) for solving differential equations. DTM uses Taylor series to obtain approximate or exact solutions. The document defines 1D, 2D, and 3D DTM and lists common operations. Examples are provided of applying DTM to solve systems of linear/non-linear differential equations. The document concludes with references for further applications of DTM in engineering and mathematics.
In this paper, natural inner product structure for the space of fuzzy n−tuples is introduced. Also we have
introduced the ortho vector, stochastic fuzzy vectors, ortho- stochastic fuzzy vectors, ortho-stochastic fuzzy
matrices and the concept of orthogonal complement of fuzzy vector subspace of a fuzzy vector space.
The document discusses the topic of matroids. It begins with an introduction and outline. It then provides definitions of independence systems and matroids. Examples of matroids are given such as minimal spanning trees, matchings, and matrix matroids. The greedy algorithm is discussed as it relates to matroids. Bases of matroids are defined and their properties explained. Partition matroids and their use in modeling problems like Hamiltonian paths are also covered.
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.
A Fast Numerical Method For Solving Calculus Of Variation ProblemsSara Alvarez
This document presents a numerical method called differential transform method (DTM) for solving calculus of variation problems. DTM finds the solution of variational problems in the form of a polynomial series without discretization. The method is applied to obtain the solution of the Euler-Lagrange equation arising from variational problems by considering it as an initial value problem. Some examples are presented to demonstrate the efficiency and accuracy of DTM for solving calculus of variation problems.
Catalan Tau Collocation for Numerical Solution of 2-Dimentional Nonlinear Par...IJERA Editor
Tau method which is an economized polynomial technique for solving ordinary and partial differential
equations with smooth solutions is modified in this paper for easy computation, accuracy and speed. The
modification is based on the systematic use of „Catalan polynomial‟ in collocation tau method and the
linearizing the nonlinear part by the use of Adomian‟s polynomial to approximate the solution of 2-dimentional
Nonlinear Partial differential equation. The method involves the direct use of Catalan Polynomial in the solution
of linearizedPartial differential Equation without first rewriting them in terms of other known functions as
commonly practiced. The linearization process was done through adopting the Adomian Polynomial technique.
The results obtained are quite comparable with the standard collocation tau methods for nonlinear partial
differential equations.
He laplace method for special nonlinear partial differential equationsAlexander Decker
This document presents the He-Laplace method for solving nonlinear partial differential equations. The method combines Laplace transforms, homotopy perturbation method, and He's polynomials. It is shown that the He-Laplace method can easily handle nonlinear terms through the use of He's polynomials and provides better results than traditional methods. An example demonstrates the application of the He-Laplace method to solve a nonlinear parabolic-hyperbolic partial differential equation.
Accuracy Study On Numerical Solutions Of Initial Value Problems (IVP) In Ordi...Sheila Sinclair
This document summarizes a study on the accuracy of numerical solutions to initial value problems in ordinary differential equations using the Euler method. The authors apply the Euler method without discretization or assumptions to solve initial value problems. They consider examples of different types of ordinary differential equations and compare the approximate solutions to exact solutions. The results show that the approximate solutions converge monotonically to the exact solutions as the step size decreases, improving accuracy. The authors analyze errors for different step sizes and find that the Euler method is efficient but requires a small step size to achieve accuracy.
This paper evaluates different alignment methods for Chinese to Japanese patent translation, including sampling-based alignment and hierarchical sub-sentential alignment. Experimental results show this combined method significantly reduces training time compared to traditional GIZA++ alignment, with translation quality remaining steady. Specifically, using this method, training time was reduced to just 57 minutes while maintaining comparable BLEU scores, representing a five-fold decrease compared to GIZA++. The paper concludes this approach can effectively accelerate statistical machine translation system development for patent translation tasks.
11.[36 49]solution of a subclass of lane emden differential equation by varia...Alexander Decker
This document describes applying He's variational iteration method to solve a subclass of Lane-Emden differential equations. The method constructs a sequence of correction functionals that generate successive approximations to the solution. It introduces a polynomial initial approximation satisfying the boundary conditions. The Lagrange multiplier is determined using variational theory. Estimates are established to prove the iterative sequence converges uniformly to the exact solution. The method is illustrated on some example problems and shown to produce exact polynomial solutions, demonstrating the effectiveness of the variational iteration method.
This document presents three methods for numerically solving linear Volterra-Fredholm integro-differential equations (LVFIDEs) of the first order: original Lagrange polynomial method, barycentric Lagrange polynomial method, and modified Lagrange polynomial method. It derives the equations for approximating the solution using each method. The document also provides algorithms to implement each method. It includes some test examples and their numerical solutions to validate the accuracy of the techniques.
A semi analytic method for solving two-dimensional fractional dispersion equa...Alexander Decker
This document presents a semi-analytic method called the modified decomposition method for solving two-dimensional fractional dispersion equations. The method is applied to solve a two-dimensional fractional dispersion equation subject to initial and boundary conditions. The numerical results obtained from the modified decomposition method are shown to closely match the exact solution, demonstrating the accuracy of this approach. The method provides an efficient means of obtaining analytical solutions to fractional differential equations.
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
A novel numerical approach for odd higher order boundary value problemsAlexander Decker
This document presents a novel numerical approach for solving odd higher order boundary value problems (BVPs), specifically fifth, seventh, and ninth order linear and nonlinear BVPs. The method uses the Galerkin weighted residual method with Legendre polynomials as basis functions to satisfy the boundary conditions. Matrix formulations are derived for the fifth, seventh, and ninth order cases. Several examples are presented and the results are compared to existing methods to demonstrate the reliability and efficiency of the proposed method.
A novel numerical approach for odd higher order boundary value problemsAlexander Decker
This document presents a novel numerical approach for solving odd higher order boundary value problems (BVPs), specifically fifth, seventh, and ninth order linear and nonlinear BVPs. The method uses the Galerkin weighted residual method with Legendre polynomials as basis functions to satisfy the boundary conditions. Matrix formulations are derived for the fifth, seventh, and ninth order cases. Several examples are presented and the results are compared to existing methods to demonstrate the reliability and efficiency of the proposed method.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
This document summarizes a research paper that proposes using Milne's predictor-corrector method to solve the dependency problem in fuzzy computations when numerically solving fuzzy differential equations (FDEs). The paper first provides background on fuzzy sets, fuzzy numbers, fuzzy processes, and fuzzy initial value problems (FIVPs). It then describes Milne's predictor-corrector method and illustrates how it can be applied to solve some example FIVPs. The goal is to address issues that arise from dependencies in fuzzy computations by using this numerical method to find solutions to FDEs.
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method is presented. To illustrate the effectiveness of the Leapfrog method, an example of the differential
equations on the sphere has been considered and the solutions were obtained using methods taken taken from
the literature [17] and Leapfrog method. The obtained discrete solutions are compared with the exact solutions
of the differential equations on the sphere. Solution graphs for the differential equations on the sphere have
been presented in the graphical form to show the efficiency of this Leapfrog method. This Leapfrog method can
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of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
Some new exact Solutions for the nonlinear schrödinger equationinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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Numerical solutions for linear fredholm integro differential
1. Journal of Educational Policy and
Entrepreneurial Research (JEPER) www.iiste.org
Vol.1, N0.2, October 2014. Pp 175-185
175
http://www.iiste.org/Journals/index.php/JEPER/index Taiwo, Alimi and Akanmu
Numerical Solutions for Linear Fredholm Integro-Differential
Difference Equations with Variable Coefficients by Collocation Methods
Taiwo, O. A.; Alimi, A. T. and *Akanmu, M. A.
Department of Mathematics, Faculty of Physical Sciences, University of Ilorin,
Ilorin, Nigeria
*Department of Science Education, University of Ilorin, Ilorin, Nigeria
tomotayoadebayo@yahoo.com
Abstract
We employed an efficient numerical collocation approximation methods to obtain an approximate solution of linear
Fredholm integro-differential difference equation with variable coefficients. An assumed approximate solutions for
both collocation approximation methods are substituted into the problem considered. After simplifications and
collocations, resulted into system of linear algebraic equations which are then solved using MAPLE 18 modules to
obtain the unknown constants involved in the assumed solution. The known constants are then substituted back into
the assumed approximate solution. Numerical examples were solved to illustrate the reliability, accuracy and
efficiency of these methods on problems considered by comparing the numerical solutions obtained with the exact
solution and also with some other existing methods. We observed from the results obtained that the methods are
reliable, accurate, fast, simple to apply and less computational which makes the valid for the classes of problems
considered.
Keywords: Approximate solution, Collocation, Fredholm, Integro-differential difference and linear algebraic
equations
Introduction
The theory of integral equation is one of the most important branches of Mathematics. Basically, its importance is in
terms of boundary value problem in equation theories with partial derivatives. Integral equations have many
applications in Mathematics, chemistry and engineering e.t.c. In recent years, the studies of integro-differential
difference equations i.e equations containing shifts of unknown functions and its derivatives, are developed very
rapidly and intensively [see Gulsu and Sezer (2006), Cao and Wang (2004), Bhrawy et al., (2012)]. These equations
are classified into two types; Fredholm integro-differential-difference equations and Volterra integro-differential-
difference equations, the upper bound of the integral part of Volterra type is variable, while it is a fixed number for
that of Fredholm type which are often difficult to solve analytically, or to obtain closed form solution, therefore, a
numerical method is needed.
The study of integro-differential difference equations have great interest in contemporary research work in which
several numerical methods have been devoloped and applied to obtain their approximate solutions such as Taylor
and Bernoulli matrix methods [Gulsu and Sezer, 2006, Bhrawy et al., 2012], Chebyshev finite difference method
[Dehghan and Saadatmandi, 2008], Legendre Tau method [Dehghan and Saadatmandi, 2010], Bessel matrix method
[Yuzbas et al., 2011], and Variational Iteration Method (VIM) [Biazar and Gholami Porshokouhi, 2010]. Homotopy
analysis method (HAM) was first introduced by Liao (2004) to obtain series solutions of various linear and
2. Journal of Educational Policy and
Entrepreneurial Research (JEPER) www.iiste.org
Vol.1, N0.2, October 2014. Pp 175-185
176
http://www.iiste.org/Journals/index.php/JEPER/index Taiwo, Alimi and Akanmu
nonlinear problems of this type of equation.
In this study, the basic ideas of the above studies motivated the work to apply a numerical collocation approximation
methods that is reliable, fast, accurate and less computational to obtain an approximate solutions to the mth order
linear Fredholm integro-differential difference equation with variable coefficients of the form:
dttytxKxfxyxPxyP
b
a
r
r
n
r
k
k
m
k
)(),()(=)()()( )(*
0=
)(
0=
(1)
with the mixed conditions
,=)()()( )()()(
1
0=
i
k
ik
k
ik
k
ik
m
k
cycbybaya
bcami 1,,0,1,= (2)
where ),(),(),( *
txKxPxP rk and )(xf are given continuous smooth functions defined on bxa . The real
coefficients ikikik cba ,, and i are appropriate constants , is refer to as the delay or difference constant (Gulsu
and Sezer, 2006).
Basic Definitions
Integro-Differential Equations (IDEs)
An integro-differential equation is an equation which involves both integral and derivatives of an unknown function.
A standard integro-differential equation is of the form:
dttytxKxfxy
xh
xg
n
)(),()(=)(
)(
)(
)(
(3)
where )(),(),( xfxhxg and and the kernel ),( txK are as prescribed in definition (2.2) and n is the order of
the IDE.
Equation (3) is called Fredholm Integro-Differential Equation if both the lower and upper bounds of the region of
the integration are fixed numbers while it is called Volterra Integro-Differential Equation if the lower bound of the
region of integration is a fixed number and the upper bound is not.
Collocation Method
This is a method of evaluating a given differential equation at some points in order to nullify the values of a
differential equation or intgro-differential equation at those points.
Approximate Solution
This is the expression obtained after the unknown constants have been found and substituted back into the assumed
solution. It is referred to as an approximate solution since it is a reasonable approximation to the exact solution. It is
denoted by )(xN
y , and taken as an inexact representation of the exact solution, where N is the degree of the
approximant used in the calculation. Methods of approximate solution are usually adopted because complete
information needed to arrive at the exact solution may not be given. In this work, approximate solution used are
given as
)(=
0=
)( xay nn
N
i
xN
where x represents the independent variables in the problem, 0)( nan are the unknown constants to be
determined and 0)(),( nxn is the basis function which is either Chebyshev or Legendre Polynomials.
3. Journal of Educational Policy and
Entrepreneurial Research (JEPER) www.iiste.org
Vol.1, N0.2, October 2014. Pp 175-185
177
http://www.iiste.org/Journals/index.php/JEPER/index Taiwo, Alimi and Akanmu
Chebyshev Polynomials
The Chebyshev polynomials of degree n of first kind which is valid in the interval 11 x and is given by
)(=)( 1
xncoscosxTn
(4)
xxTxT =)(1,=)( 1
and the recurrence relation is given by
1),()(2=)( 11 nxTxxTxT nnn
bxa
ab
bax
nxTn
,
2
coscos=)( 1
and this satisfies the recurrence relation
bxanxTxT
ab
bax
xT nnn
0.),()(
2
2=)( 11 (5)
Equation (5) is the recurrence relation of the Chebyshev polynomials in the interval 1,1][ , thus we have
1=)(*
xT
xxT =)(*
1
12=)( 2*
2 xxT (6)
xxxT 34=)( 3*
3
.andsoon188=)( 24*
4 xxxT
Legendre's Polynomial
The Legendre's polynomial is defined and denoted by
)()(1)(2
1
1
=)( 11 xPxxPn
n
xP n
n
nn
and
0,1,=;1)(
!2
1
=)( 2
nx
dxn
xP n
nnn
with the first few polynomial as
1=)(xP
xxP =)(1
1)(3
2
1
=)( 2
2 xxP (7)
)3(5
2
1
=)( 3
3 xxxP
113)30(35
8
1
=)( 24
4 xxxxP
Discussion of Methods
Problem Considered
4. Journal of Educational Policy and
Entrepreneurial Research (JEPER) www.iiste.org
Vol.1, N0.2, October 2014. Pp 175-185
178
http://www.iiste.org/Journals/index.php/JEPER/index Taiwo, Alimi and Akanmu
We consider the
th
m order linear Fredholm integro-differential difference equation with variable coefficients of the
forms:
dttytxKxfxyxPxyPa
b
a
r
r
n
r
k
k
m
k
)(),()(=)()()()( )(*
0=
)(
0=
(8)
with the mixed conditions
,=)()()( )()()(
1
0=
i
k
ik
k
ik
k
ik
m
k
cycbybaya
bcami 1,,0,1,= (9)
Equation (8) is referred to as Linear Fredholm Integro-differential difference equation with variable coefficients,
where ),(),(),( *
txKxPxP rk and )(xf are given continuous smooth functions defined on bxa . The real
coefficients ikikik cba ,, and i are appropriate constants , is refer to as the delay term or difference constant
(Gulsu and Sezer, 2006).
In this section, standard collocation methods is applied to solve equation of the form (a) using the following bases
functions:
(i) Chebyshev Polynomials
(ii) Legendre Polynomials
Method I: Standard Collocation Method by Chebyshev Polynomial Basis
In order to solve equations (8)-(9) using the collocation approximation method, we used an approximate solution of
the form
)(=)(
0=
xTaxy ii
N
i
N (10)
where N is the degree of our approximant, 0)( iai are constants to be determined and 0)( iTi are the
Chebyshev Polynomials defined in equation (5). Thus, differentiating equation (10) with respect to x m -times ( m
is the order of the given problem), we obtain
)(=
)(=
)(=
0=
0=
0=
xTay
xTay
xTay
m
ii
n
i
m
ii
n
i
ii
n
i
(11)
and then substituting equation (10) and its derivatives in equation (11) into equation (8), we obtain
dttytxKxfxyxPxyP
N
b
a
r
Nr
n
r
k
Nk
m
k
)(),()(=)()()( )(*
0=
)(
0=
(12)
Evaluating the integral part of equation (12) and after simplifications, we collocate the resulting equation at the
point kxx = to get
)()(=)()()( )(*
0=
)(
0=
kkk
r
Nkr
n
r
k
k
Nk
m
k
xGxfxyxPxyP (13)
5. Journal of Educational Policy and
Entrepreneurial Research (JEPER) www.iiste.org
Vol.1, N0.2, October 2014. Pp 175-185
179
http://www.iiste.org/Journals/index.php/JEPER/index Taiwo, Alimi and Akanmu
where )(xG is the evaluated integral part and
11(1)=;
1
)(
=
Nk
N
kab
axk (14)
Thus, equation (13) gives rise to 1)( N system of linear algebraic equations in 1)( N unknown constants and
m extra equations are obtained using the conditions given in equation (9). Altogether, we now have 1)( mN
system of linear algebraic equations. These equations are then solved using MAPLE software to obtain (N+1)
unknown constants 0)( iai which are then substituted back into the approximate solution given by equation (10).
Method II: Standard Collocation Method by Legendre Polynomial Basis
We consider here also the problem of the form (a) using the collocation approximation method, we used an
approximate solution of the form
)(=)(
0=
xLaxy ii
N
i
N (15)
where N is the degree of our approximant, 0)( iai are constants to be determined and 0)( iLi are the
Legendre Polynomials defined in equation (7). Thus, differentiating equation (15) with respect to x m -times ( m
is the order of the given problem), we obtain
)(=
)(=
)(=
0=
0=
0=
xLay
xLay
xLay
m
ii
n
i
m
ii
n
i
ii
n
i
(16)
and then substituting equation (15) and its derivatives in equation (16) into equation (8), we obtain
dttytxKxfxyxPxyP
N
b
a
r
Nr
n
r
k
Nk
m
k
)(),()(=)()()( )(*
0=
)(
0=
(17)
Hence, evaluating the integral part of equation (20) and after simplification, we collocate the resulting equation at
the point kxx = to get
)()(=)()()( )(*
0=
)(
0=
kkk
r
Nkr
n
r
k
k
Nk
m
k
xGxfxyxPxyP (18)
where )(xG is the evaluated integral part and
11(1)=;
1
)(
=
Nk
N
kab
axk (19)
Thus, equation (18) gives rise to 1)( N system of linear algebraic equations in 1)( N unknown constants and
m extra equations are obtained using the conditions given in equation (9). Altogether, we now have 1)( mN
system of linear algebraic equations. These equations are then solved using MAPLE software to obtain (N+1)
unknown constants 0)( iai which are then substituted back into the approximate solution given by equation (10).
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Numerical Examples
Numerical Example 1
Consider the Second order linear Fredholm integro-differential difference equation with variable
coefficients
dtttyeexyxyxxyxxyxy x
1)(=1)(1)(1)()()(
0
1
(20)
with the initial conditions
1=(0)1,=(0) yy (21)
The exact solution is given as
x
exy
=)( [Gulsu and Sezer, 2006].
Numerical Example 2
Consider third order linear Fredholm integro-differential-difference equation with variable coefficients
))(cos1)(sin1)((=1)(1)()()( xxxxxyxyxyxxy
dtty 1)(12cos
1
1
(22)
with the initial conditions
0=(0)1,=(0)0,=(0) yyy (23)
The exact solution is given as xxy sin=)( [Gulsu and Sezer, 2006].
Numerical Example 3
Consider first order linear Fredholm integro-differential-difference equation with variable coefficients
dttytxxxyxyxxyxy 1)()(2=1)(1)()()(
1
1
(24)
with the mixed condition
0=(1)(0)21)( yyy (25)
The exact solution is given as 43=)( xxy [Gulsu and Sezer, 2006].
Remark: We defined absolute error as:
1,2,3,=,,)()(= NbxaxyxyError
N
Here, )(xy is the given exact solution and )(xy
N
is the approximate solution respectively.
Numerical Results and Eerror for Examples
Table 1: Results obtained for example 1: Case N=6
x EXACT APPROXIMATE SOLUTIONS
CHEBYSHEV LEGENDRE TAYLOR
.0
1.0000000000 1.0000000000 1.0000000000 0.00000000000
-0.1
1.1051709181 1.1052508587 1.1052508587 1.10530000000
-0.2
1.2214027582 1.2216333968 1.2216333968 1.22160000000
-0.3
1.3498588076 1.3501660185 1.3501660185 1.35000000000
-0.4
1.4918246976 1.4919754149 1.4919754149 1.49190000000
-0.5
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1.6487212707 1.6483097916 1.6483097916 1.64850000000
-0.6
1.8221188004 1.8205524611 1.8205524611 1.82110000000
-0.7
2.0137527075 2.0102358013 2.0102358013 2.01140000000
-0.8
2.2255409285 2.2190555779 2.2190555779 2.22100000000
-0.9
2.4596031112 2.4488856333 2.4488856334 2.45220000000
-1.0
2.7182818285 2.7017929401 2.7017929400 2.70690000000
Table 2: Results obtained for example 1: Case N=7
x EXACT APPROXIMATE SOLUTIONS
CHEBYSHEV LEGENDRE TAYLOR
.0
1.0000000000 1.0000000000 1.0000000000 0.00000000000
-0.1
1.1051709181 1.1052038643 1.1052038642 1.10530000000
-0.2
1.2214027582 1.2214978470 1.2214978469 1.22160000000
-0.3
1.3498588076 1.3499847405 1.3499847404 1.35000000000
-0.4
1.4918246976 1.4918832368 1.4918832367 1.49190000000
-0.5
1.6487212707 1.6485413049 1.6485413048 1.64850000000
-0.6
1.8221188004 1.8214503126 1.8214503126 1.82110000000
-0.7
2.0137527075 2.0122500401 2.0122600401 2.01140000000
-0.8
2.2255409285 2.2227947307 2.2227947307 2.22100000000
-0.9
2.4596031112 2.4550703277 2.4550703277 2.45220000000
-1.0
2.7182818285 2.7113130423 2.7113130423 2.70690000000
Table 3: Absolute Errors for Example 1: Case N=6 and 7
x
CHEBYSHEV LEGENDRE TAYLOR CHEBYSHEV LEGENDRE TAYLOR
N=6 N=7
.0
0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
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Here, we denoted CPE as the Error of results obtained using Chebyshev polynomials and LPE as the Error of
results obtained using Legendre polynomials.
Table 5: Results obtained for example 2: Case 6=N
x EXACT APPROXIMATE SOLUTIONS
CHEBYSHEV LEGENDRE TAYLOR
-1.0 -
0.8414709848
-
0.8412178340
-
0.8183268604
-
0.9273450000
-0.8 -
0.7173560909
-
0.7171927056
-
0.7064218929
-
0.7567230000
-0.6 -
0.5646424734
-
0.5645686751
-
0.5604519509
-
0.5797120000
-0.4 -
0.3894183423
-
0.3883086444
-
0.3883086444
-
0.3935390000
-0.2 -
0.1986693308
-
0.1985475309
-
0.1985475310
-
0.1991540000
.0
0.00000000000 0.0000000000 0.0000000000 0.0000000000
.2
0.19866933080 0.1985811922 0.1985811920 0.1991280000
.4
0.3894183423 0.3888427347 0.3888427345 0.3931170000
.6
0.5646424734 0.5631179243 0.5631179242 0.5774680000
.8
0.7173560909 0.7146801750 0.7146801750 0.7491370000
.0
0.8414709848 0.8379628764 0.8379628762 0.9072650000
Table 6: Results obtained for example 2: Case 7=N
x EXACT APPROXIMATE SOLUTIONS
CHEBYSHEV LEGENDRE TAYLOR
-1.0 -0.8414709848 -0.8459516158 -0.8459531673 -0.9018320000
-0.8 -0.7173560909 -0.7194740946 -0.7194748278 -0.7401870000
-0.6 -0.5646424734 -0.5654548269 -0.5654551080 -0.5712780000
-0.4 -0.3894183423 -0.3896336970 -0.3896337714 -0.3906190000
-0.2 -0.1986693308 -0.1986930013 -0.1986930094 -0.1987380000
.0 0.00000000000 0.0000000000 0.0000000000 0.0000000000
.2 0.19866933080 0.1986865239 0.1986865299 0.19861600000
.4 0.3894183423 0.3895309176 0.3895309564 0.3886090000
.6 0.5646424734 0.5649417357 0.5649418384 0.5608220000
.8 0.7173560909 0.7178844600 0.7178846405 0.7058770000
.0 0.8414709848 0.8421710782 0.8421713155 0.8140980000
Table 7: Errors obtained for example 2: Case N=6 and 7
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x
CHEBYSHEV LEGENDRE TAYLOR CHEBYSHEV LEGENDRE TAYLOR
N=6 N=7
-1.0
2.3144E-02 2.3144E-02 8.5870E-02 4.4806E-03 4.8218E-03 6.0360E-02
-0.8
1.0934E-02 1.0934E-02 3.9360E-02 2.1180E-03 2.1187E-03 2.2830E-02
-0.6
4.1905E-03 4.1905E-03 1.5070E-02 8.1235E-04 8.1263E-04 6.6360E-03
-0.4
1.1097E-03 1.1097E-03 4.1210E-03 2.1535E-04 2.1543E-04 1.2010E-03
-0.2
1.2180E-04 1.2180E-04 4.8500E-04 2.3671E-05 2.3679E-05 6.9000E-05
.0
0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
.2
8.8139E-05 8.8139E-05 4.5900E-04 1.7193E-05 1.7199E-05 5.3000E-05
.4
5.7561E-04 5.7560E-04 3.6990E-03 1.1258E-04 1.1261E-04 8.0900E-04
.6
1.5245E-03 1.5245E-03 1.2820E-02 2.9926E-04 2.9927E-04 3.8200E-03
.8
2.6759E-03 2.6759E-09 3.1780E-02 5.2837E-04 5.2855E-04 1.1470E-02
.0
3.5081E-03 3.5081E-03 6.5790E-02 7.0009E-04 7.0033E-04 2.7370E-02
NOTE:
On solving this numerical example (3) using the two methods, the same exact solution is obtained.
Presentation of Results in Graphical Forms
Conclusion
We have presented and illustrated the collocation approximation methods using two different bases functions
namely; Chebyshev and Legendre polynomials to solve linear Fredholm integro-differential difference equations
with variable coefficients which are very difficult to solve analytically. In many cases, it is required to obtain the
approximate solutions. One of the advantages of these methods is that the numerical solutions of the problems
considered is converted into system of linear algebraic equations which are very easy to solve for the constants
involved. Another considerable advantage of these methods is that if the exact solution is a polynomial function,
with the methods used, the analytical solution is obtained.
Moreover, satisfactory results of illustrative examples were obtained when the value of N increases for both
methods, the approximate solutions obtained are closer to the exact solution (where the exact solution are known in
closed form) which are compared with some other existing methods and makes these methods valid for solving
linear Fredholm Integro-differential difference and Fredholm Integral equations.
References
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Integro-differential-difference equations. International Mathematical Forum, 5(65) , 3335-3341.
Bhrawy, A. H., Tohidi, E. & Soleymani, F. (2012). A new Bernoulli Matrix Method for Solving High Order Linear
and Nonlinear Fredholm Integro-Differential Equations with Piecewise Interval. Applied Mathematics and
11. Journal of Educational Policy and
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Cao, J. and Wang, J. (2004). Delay-dependent robust stability of uncertain nonlinear systems with time delay, Appl.
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