Taylor Collocation Research Articles

Analysis of the Logistic Growth Model With Taylor Matrix and Collocation Method

Early analysis of infectious diseases is very important in the spread of the disease. The main aim of this study is to make important predictions and inferences for Covid 19, which is the current epidemic disease, with mathematical modeling and numerical solution methods. So we deal with the logistic growth model. We obtain carrying capacity and growth rate with Turkey epidemic data. The obtained growth rate and carrying capacity is used in the Taylor collocation method. With this method, we estimate and making predictions close to reality with Maple. We also show the estimates made with the help of graphics and tables.

Taylor collocation method for solving two‐dimensional partial Volterra integro‐differential equations

In this paper, the numerical solution of a two‐dimensional partial Volterra integro‐differential equation (2D‐PVIDE) is provided by extending the Taylor collocation scheme of one‐dimensional Volterra integral equations. The method is based on the use of Taylor polynomials in two‐dimensional, while the approximate solution is given by using explicit schemes. The method is proved to be high‐order convergent with respect to the maximum norm. Some numerical examples are given to verify the theoretical results.

Approximate solution of linear integral equations by Taylor ordering method: Applied mathematical approach

Abstract Since obtaining an analytic solution to some mathematical and physical problems is often very difficult, academics in recent years have focused their efforts on treating these problems using numerical methods. In science and engineering, systems of integral differential equations and their solutions are extremely important. The Taylor collocation method is described as a matrix approach for solving numerically Linear Differential Equations (LDE) by using truncated Taylor series. Integral equations are used to solve problems such as radiative transmission and the oscillation of a string, membrane, or axle. Differential equations can be used to tackle oscillating difficulties. To discover approximate solutions for linear systems of integral differential equations with variable coefficients in terms of Taylor polynomials, the collocation approach, which is offered for differential and integral equation solutions, will be developed. A system of LDE will be translated into matrix equations, and a new matrix equation will be generated in terms of the Taylor coefficients matrix by employing Taylor collocation points. The needed system will be converted to a linear algebraic equation system. Finding the Taylor coefficients will lead to the Taylor series technique.

Numerical solution of two-dimensional linear and nonlinear Volterra integral equations using Taylor collocation method

The main purpose of this work is to provide a numerical approach for two-dimensional Volterra integral equations (2D-VIEs). An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution v∈Sp−1,p−1(−1)(ΠN,M) for approximating the solution of 2D-VIEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the accuracy of the presented method.

Taylor collocation method for high-order neutral delay Volterra integro-differential equations

In this paper, the Taylor collocation method is applied to numerically solve a kth-order neutral linear Volterra integro-differential equation with constant delay and variable coefficients.We also provide a rigorous analysis to estimate the difference between the exact and approximate solution and their derivatives up to order k-1. Numerical examples are included to prove the performance of the presented convergent algorithm.

Analysis of Covid 19 disease with SIR model and Taylor matrix method

<abstract><p>Covid 19 emerged in Wuhan, China in December 2019 has continued to spread by affecting the whole world. The pandemic has affected over 328 million people with more than 5 million deaths in over 200 countries which has severely disrupted the healthcare system and halted economies of the countries. The aim of this study is to discuss the numerical solution of the SIR model on the spread of Covid 19 by the Taylor matrix and collocation method for Turkey. Predicting COVID-19 through appropriate models can help us to understand the potential spread in the population so that appropriate action can be taken to prevent further transmission and prepare health systems for medical management of the disease. We deal with Susceptible–Infected–Recovered (SIR) model. One of the proposed model's improvements is to reflect the societal feedback on the disease and confinement features. We obtain the time dependent rate of transmission of the disease from susceptible $ \beta(t) $ and the rate of recovery from infectious to recovered $ \gamma $ using Turkey epidemic data. We apply the Taylor matrix and collocation method to the SIR model with $ \gamma $, $ \beta(t) $ and Covid 19 data of Turkey from the date of the first case March 11, 2020 through July 3, 2021. Using this method, we focus on the evolution of the Covid 19 in Turkey. We also show the estimates with the help of graphics and Maple.</p></abstract>

Taylor collocation method for a system of nonlinear Volterra delay integro-differential equations with application to COVID-19 epidemic

The present paper deals with the numerical solution for a general form of a system of nonlinear Volterra delay integro-differential equations (VDIDEs). The main purpose of this work is to provide a current numerical method based on the use of continuous collocation Taylor polynomials for the numerical solution of nonlinear VDIDEs systems. It is shown that this method is convergent. Numerical results will be presented to prove the validity and effectiveness of this convergent algorithm. We apply models to the COVID-19 epidemic in China, Spain, and Italy and one for the Predator–Prey model in mathematical ecology.

Numerical Solution of Second-Order Linear Delay Differential and Integro-Differential Equations by Using Taylor Collocation Method

The main purpose of this work is to provide a numerical approach for linear second-order differential and integro-differential equations with constant delay. An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution [Formula: see text] for approximating the solution of second-order linear DDEs and DIDEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the validity of this algorithm.

Determining constant breadth curve mate of a curve on a surface via Taylor collocation method

Determining constant breadth curve mate of a curve on a surface via Taylor collocation method

Hermite collocation method for fractional order differential equations

This paper focuses on the approximate solutions of the higher order fractional differential equations with multi terms by the help of Hermite Collocation method (HCM). This new method is an adaptation of Taylor's collocation method in terms of truncated Hermite Series. With this method, the differential equation is transformed into an algebraic equation and the unknowns of the equation are the coefficients of the Hermite series solution of the problem. This method appears as a useful tool for solving fractional differential equations with variable coefficients. To show the pertinent feature of the proposed method, we test the accuracy of the method with some illustrative examples and check the error bounds for numerical calculations.

Numerical solution of high-order linear Volterra integro-differential equations by using Taylor collocation method

ABSTRACTThe main purpose of this work is to provide a new direct numerical method for high-order linear Volterra integro-differential equations (VIDEs). An algorithm based on the use of Taylor polynomials is developed for the numerical solution of high-order linear VIDEs. It is shown that this algorithm is convergent. Numerical results are presented and comparisons are made with well-known numerical methods to prove the effectiveness of the presented algorithm.

A reliable algorithm based on the shifted orthonormal Bernstein polynomials for solving Volterra–Fredholm integral equations

ABSTRACTThis paper deals with the numerical solution of Volterra–Fredholm integral equations. In this work, we approximate the unknown functions based on the shifted orthonormal Bernstein polynomials, in conjunction with the least-squares approximation method. The method is using a simple computational manner to obtain a quite acceptable approximate solution. The merits of this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations. On the other hand, the efficiency and accuracy of the method for solving these equations are high. The convergence analysis of proposed method have been discussed through some theorems. Moreover, we will obtain an estimation of error bound for this algorithm. Finally, some examples are given to show the capability of presented method in comparison with four well-known algorithms in the literature namely the Legendre collocation method, Taylor collocation method, Taylor polynomial method and Lagrange collocation method.

A Collocation Method for Solving Fractional Order Linear System

In this paper we propose the collocation method to achieve the numerical solution of fractional order linearsystems where a Fractional Derivative is defined in the Caputo sense.We use Taylor collocation method,which is based on collocation method for solving fractional differential equation. This method is based onfirst taking the truncated Taylor expansions of the vector function’s solution in the Fractional order linearsystem and then substituting their matrix forms into the system. Using collocation points, we have a systemof linear algebraic equation.The method has been tested by some numerical examples.DOI : http://dx.doi.org/10.22342/jims.23.1.257.

A new numerical method for delay and advanced integro-differential equations

A general formulation is constructed for Jacobi operational matrices of integration, product, and delay on an arbitrary interval. The main purpose of this study is to improve Jacobi operational matrices for solving delay or advanced integro–differential equations. Some theorems are established and utilized to reduce the computational costs. All algorithms can be used for both linear and nonlinear Fredholm and Volterra integro-differential equations with delay. An error estimator is introduced to approximate the absolute error when the exact solution of a given problem is not available. The error of the proposed method is less compared to other common methods such as the Taylor collocation, Chebyshev collocation, hybrid Euler–Taylor matrix, and Boubaker collocation methods. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

A Note on Stability Analysis of Taylor Collocation Method

In this study, we investigate the stability of Taylor collocation method for initial value problems in ordinary differential equations. Firstly, we try to show that Taylor collocation method for initial value problem is equivalent to a subset of the implicit Runge-Kutta methods. This equivalence enables us to prove that Taylor collocation method is absolutely stable (A-stable) for the considered equations.

Numerical method for a system of integro-differential equations and convergence analysis by Taylor collocation

Abstract In this paper, we present the Taylor polynomial solutions of system of higher order linear integro-differential Volterra-Fredholm equations (IDVFE). This method transforms IDVFE into the matrix equations which correspond to a system of linear algebraic equations. Some numerical results are also given to illustrate the efficiency of the method.

Free vibration analysis of orthotropic plates by differential transform and Taylor collocation methods based on a refined plate theory

Based on the two-variable refined plate theory, free vibration of orthotropic plates is analyzed using the differential transform method (DTM) and the Taylor collocation method (TCM). The refined plate theory outperforms the classical plate theory, and its formulation is simpler than those of other higher-order theories. Without the need for any shear correction factor, the theory performs reliably. The plates considered have two opposite edges simply supported (Levy plates). The first part of the analysis considers three combinations of clamped, simply supported and free edge conditions for the other two edges, keeping one of them simply supported. Detailed formulations of DTM and TCM for the free vibration analysis are given and, consequently, used to predict the frequency parameters and the effect of various factors ranging from geometric to material parameters. Next, the paper presents analysis of some cases, the multi-span plates and plates with stepped thickness and end rotational springs, whose analytical solutions are not readily available, particularly based on the two-variable refined plate theory. In order to verify the results, formulations of three more plate theories, namely the classical or Kirchhoff plate theory, the first-order shear deformation theory of Mindlin and the high-order shear deformation theory of Sayyad and Gugal, were implemented and solved using the proposed methods.

A numerical approach for solving Volterra type functional integral equations with variable bounds and mixed delays

In this paper, the Taylor collocation method has been used the integro functional equation with variable bounds. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. We have introduced the method to solve the functional integral equations with variable bounds. We have also improved error analysis for this method by using the residual function to estimate the absolute errors. To illustrate the pertinent features of the method numeric examples are presented and results are compared with the other methods. All numerical computations have been performed on the computer algebraic system Maple 15.

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.

Simulation the gas simultaneous adsorption over natural and modified zeolite

Adsorption is of great importance. The unique advantage of adsorption over other separation methods is the higher selectivity that can be achieved by adsorbents. In addition, adsorption phenomena play a vital role in many solid-state reactions and biological mechanisms. In this work, the adsorption process of CO 2 on the clinoptilolite (Skorynskoho field, Transcarpathian region, Ukraine) and SO 2 , NO, and CO 2 adsorption on K 2 CO 3 -modified – γ -alumina in a fixed-bed reactor were theoretical studied and simulated by computer-mathematic methods. The developed mathematical model based on the mass balance in gas and solid phase, the experimental saturation capacities, considering the activity of the adsorbent with respect to the gas by variable coefficients. The model presented by a normal linear system of differential equations with variable coefficients, it was solved by Taylor collocation method. The simulation shows that the data obtained by theoretical study are in agreement with data obtained in the simulation. According to Fisher Criterion the mathematic model adequate in 90 % for modified zeolite and in 75 % for natural zeolite, it can be explained by unordered structure of the natural zeolite. It follows that the offered model adequately describes the dynamics simultaneous adsorption of gases over zeolites. Thus even with a large number of simplifications and assumptions, it is possible to construct efficient mathematical model that can be used in exhaust system. The results indicate that, there is great sense to conduct further researches and simulations to reach the industrial level.