Unknown Taylor Coefficients Research Articles

A numerical method for solving continuous population models for single and interacting species

In this study, a numerical approach is presented to obtain the approximate solutions of continuous population models for single and interacting species. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. By using Taylor polynomials and collocation points, this method transforms population models into a matrix equation. The matrix equation corresponds to a system of nonlinear equations with the unknown Taylor coefficients. To illustrate reliability and efficiency of the method, numerical examples are presented and results are compared with the other numerical methods. Additionally, residual correction procedure is applied to estimate the absolute errors. All numerical computations have been performed on the computer algebraic system Maple 15.

A Numerical Approach Based on Taylor Polynomials for Solving a Class of Nonlinear Differential Equations

In this study, a matrix method based on Taylor polynomials and collocation points is presented for the approximate solution of a class of nonlinear differential equations, which have many applications in mathematics, physics and engineering. By means of matrix forms of the Taylor polynomials and their derivatives, the technique we have used reduces the solution of the nonlinear equation with mixed conditions to the solution of a matrix equation which corresponds to a system of nonlinear algebraic equations with the unknown Taylor coefficients. On the other hand, to illustrate the validity and applicability of the method, some numerical examples together with residual error analysis are performed and the obtained results are compared with the existing results in literature.

The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs

We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given. Program summaryProgram Title: TDDSProgram Files doi:http://dx.doi.org/10.17632/twk8zjxgbz.1Licensing provisions: GNU LGPLProgramming language: MAPLE 11 to MAPLE 2017, available independently in MAPLE 2018Nature of problem: Systems of polynomially nonlinear partial differential equations are not given in a formally integrable form in general. In order to determine analytic solutions in terms of power series, symbolic manipulations are necessary to find a complete set of conditions for the unknown Taylor coefficients. A particular case of that problem is deciding consistency of a system of PDEs. Nonlinear PDEs require splitting into different cases in general. Deciding whether another PDE is a consequence of a given system depends on similar symbolic manipulations. Computing all consequences of a given system which involve only a subset of the unknown functions or a certain subset of their derivatives are instances of differential elimination problems, which arise, e.g., in detection of hidden constraints in singular dynamical systems and field theoretical models.Solution method: The solution method consists, in principle, of pseudo-division of differential polynomials, as in Euclid’s algorithm, with case distinctions according to vanishing or non-vanishing leading coefficients and discriminants, combined with completion to involution for partial differential equations. Since an enormous growth of expressions can be expected in general, efficient versions of these techniques need to be used, e.g., subresultants, Janet division, and need to be applied in an appropriate order. Factorization of polynomials, while not strictly necessary for the method, should be utilized to reduce the size of expressions whenever possible.

A NEW APPROACH FOR NUMERICAL SOLUTION OF LINEAR AND NON-LINEAR SYSTEMS

In this study, Taylor matrix algorithm is designed for the approximate solution of linear and non-linear differential equation systems. The algorithm is essentially based on the expansion of the functions in differential equation systems to Taylor series and substituting the matrix forms of these expansions into the given equation systems. Using the Mathematica program, the matrix equations are solved and the unknown Taylor coefficients are found approximately. The presented numerical approach is discussed on samples from various linear and non-linear differential equation systems as well as stiff systems. The computational data are then compared with those of some earlier numerical or exact results. As a result, this comparison demonstrates that the proposed method is accurate and reliable.

Taylor collocation approach for delayed Lotka–Volterra predator–prey system

In this study, a numerical approach is proposed to obtain approximate solutions of the system of nonlinear delay differential equations defining Lotka–Volterra prey–predator model. By using the Taylor polynomials and collocation points, this method transforms the population model into a matrix equation. The matrix equation corresponds to a system of nonlinear equations with the unknown Taylor coefficients. Numerical examples are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and produces accurate results. All numerical computations have been performed on the computer algebraic system Maple 15.

Taylor matrix method for solving linear two-dimensional Fredholm integral equations with Piecewise Intervals

This paper contributes an efficient numerical approach, which produces an approximate polynomial solution, for solving linear two-dimensional Fredholm integral equations with piecewise intervals. The main technique is based upon approximating the unknown function (i.e., expected solution) and known functions (including four variable kernel functions and two variable source functions) in terms of their Taylor expansions in a new matrix form. With the aid of these approximations, we convert the main problems into their associated matrix equations. The matrix equations correspond to systems of linear equations with the unknown Taylor coefficients. Moreover, under several mild conditions, error analysis of the proposed method is provided. Numerical examples are included to demonstrate the validity and applicability of the method and the comparisons are made with the existing results.

Taylor polynomial method and error estimation for a kind of mixed Volterra–Fredholm integral equations

In this paper, an efficient numerical method of Taylor polynomial method is developed to solve the Volterra–Fredholm integral equations. This method is based on first taking truncated Taylor series of the functions in equation and then substituting them into the given equation, thereby the equation reduces to a matrix equation, finally, we can solve the unknown Taylor coefficients and obtained the approximation of solution. Error estimation and illustrative examples are included to demonstrate the validity and applicability of the method.

Taylor collocation method for systems of high-order linear differential–difference equations with variable coefficients

A Taylor collocation method has been developed to solve the systems of high-order linear differential–difference equations in terms of the Taylor polynomials. Using the Taylor collocation points, this method transforms differential–difference equation systems and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence, by finding the Taylor coefficients easily, Taylor polynomial solutions are obtained. To illustrate the pertinent features examples are presented and results are compared. All numerical computations have been performed on the computer algebraic system Maple 9.

A Numerical Scheme to Solve Fuzzy Linear Volterra Integral Equations System

The current research attempts to offer a new method for solving fuzzy linear Volterra integral equations system. This method converts the given fuzzy system into a linear system in crisp case by using the Taylor expansion method. Now the solution of this system yields the unknown Taylor coefficients of the solution functions. The proposed method is illustrated by an example and also results are compared with the exact solution by using computer simulations.

Application of Taylor Matrix Method to the Solution of Longitudinal Vibration of Rods

The present study introduces a novel and simple matrix method for the solution of longitudinal vibration of rods in terms of Taylor polynomials. The proposed method converts the governing partial differential equation of the system into a matrix equation, which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Then the solution is obtained easily by solving these matrix equations. Both free and forced vibrations of the system are studied; particular and general solutions are determined. The method is demonstrated by an illustrative example using symbolic computation. Comparison of the numerical solution obtained in this study with the exact solution is quite good.

Approximate solution to linear complex differential equation by a new approximate approach

The purpose of this study is to give a Taylor polynomial approximation for the solution of high-order linear complex differential equations with variable coefficients and mixed conditions about any point. For this purpose, a matrix method is introduced. This method is based on first taking the truncated Taylor expansions of the functions in the complex differential equation and then substituting their matrix forms into the given equation. Hence the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately. Also, examples that illustrate the pertinent features of the method are presented and the results are discussed.

Polynomial solution of the most general linear Fredholm–Volterra integrodifferential-difference equations by means of Taylor collocation method

In this paper, a Taylor collocation method has been presented for numerically solving the high-order general linear Fredholm–Volterra integrodifferential-difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. Using the Taylor collocation points, the method transforms the equation and conditions into the matrix equations with the unknown Taylor coefficients and hence by finding the Taylor coefficients, the polynomial approach is obtained. Also, to illustrate the accuracy of the method, numerical examples are presented and the results are discussed.

Taylor collocation method for solution of systems of high-order linear Fredholm–Volterra integro-differential equations

A Taylor collocation method is presented for numerically solving the system of high-order linear Fredholm–Volterra integro-differential equations in terms of Taylor polynomials. Using the Taylor collocations points, the method transforms the system of linear integro-differential equations (IDEs) and the given conditions into a matrix equation in the unknown Taylor coefficients. The Taylor coefficients can be found easily, and hence the Taylor polynomial approach can be applied. This method is also valid for the systems of differential and integral equations. Numerical examples are presented to illusturate the accuracy of the method. The symbolic algebra program Maple is used to prove the results.

Polynomial approach for the most general linear Fredholm integrodifferential‐difference equations using Taylor matrix method

A Taylor matrix method is developed to find an approximate solution of the most general linear Fredholm integrodifferential‐difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This method transforms the given general linear Fredholm integrodifferential‐difference equations and the mixed conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equations, the Taylor coefficients can be easily computed. Hence, the finite Taylor series approach is obtained. Also, examples are presented and the results are discussed.

Approximate solution of complex differential equations for a rectangular domain with Taylor collocation method

In this study, Taylor collocation method is investigated for the approximate computation of high-order linear complex differential equations. Using the collocation points on any rectangular domain in the complex plane, the method transforms the given complex differential equation and the mixed conditions to matrix equation with unknown Taylor coefficients. By means of the obtained matrix equations, the Taylor coefficients can be easily computed. Hence, the finite Taylor series approach is obtained. Also, examples are presented and the results are discussed.

On the solution of the Riccati equation by the Taylor matrix method

In this paper, we suggest a matrix method to solve the Riccati differential equation in terms of Taylor polynomials. This method is based on first taking the truncated Taylor series of the function in equations and then substituting their matrix forms into the given equation. Hence, the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately. The solution is calculated in the form of a series with easily computable components. The numerical results show the effectiveness of the method for this type of equation. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate.

Taylor polynomial solutions of systems of linear differential equations with variable coefficients

A Taylor collocation method has been presented for numerically solving systems of high-order linear ordinary, differential equations with variable coefficients. Using the Taylor collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equation, a new system of equations corresponding to the system of linear algebraic equations is gained. Hence by finding the Taylor coefficients, the Taylor polynomial approach is obtained. Also, the method can be used for the linear systems in the normal form. To illustrate the pertinent features of the method, examples are presented and results are compared.

A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials

The purpose of this study is to give a Taylor polynomial approximation for the solution of mth-order linear difference equations with variable coefficients under the mixed conditions about any point. For this purpose, the Taylor matrix method is introduced. This method is based on first taking the truncated Taylor expansions of the functions in the difference equation and then substituting their matrix forms into the given equation. Hence the resultant matrix equation can be solved and the unknown Taylor coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of the study are discussed.

A Taylor polynomial approach for solving differential-difference equations

The purpose of this study is to give a Taylor polynomial approximation for the solution of mth-order linear differential-difference equations with variable coefficients under the mixed conditions about any point. For this purpose, Taylor matrix method is introduced. This method is based on first taking the truncated Taylor expansions of the functions in the differential-difference equations and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately. In addition, examples that illustrate the pertinent features of the method are presented, and the results of study are discussed. Also we have discussed the accuracy of the method. We use the symbolic algebra program, Maple, to prove our results.

A new polynomial approach for solving difference and Fredholm integro-difference equations with mixed argument

The purpose of this study is to give a Taylor polynomial approximation for the solution of high-order linear Fredholm difference equations with variable coefficients and mixed argument under the mixed conditions about any point. For this purpose, a Taylor matrix method is introduced. This method is based on first taking the truncated Taylor expansions of the functions in the difference equation and then substituting their matrix forms into the given equation. Hence the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately. Also, examples that illustrate the pertinent features of the method are presented and the results are discussed.