scholarly journals Linear Barycentric Rational Collocation Method for Beam Force Vibration Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Jin Li ◽  
Yu Sang
Keyword(s):  
Convergence Rate ◽  
Collocation Method ◽  
Convergence Rates ◽  
Rational Interpolation ◽  
Vibration Equation ◽  
Numerical Examples ◽  
Space And Time ◽  
Force Vibration ◽  
The Matrix ◽  
Barycentric Rational Interpolation

The linear barycentric rational collocation method for beam force vibration equation is considered. The discrete beam force vibration equation is changed into the matrix forms. With the help of convergence rate of barycentric rational interpolation, both the convergence rates of space and time can be obtained at the same time. At last, some numerical examples are given to validate our theorem.

scholarly journals Linear Barycentric Rational Method for Solving Schrodinger Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Peichen Zhao ◽  
Yongling Cheng
Keyword(s):  
Schrödinger Equation ◽  
Convergence Rate ◽  
Collocation Method ◽  
Schrodinger Equation ◽  
Interpolation Method ◽  
Rational Interpolation ◽  
Barycentric Interpolation ◽  
Rational Polynomial ◽  
The Matrix ◽  
Barycentric Rational Interpolation

A linear barycentric rational collocation method (LBRCM) for solving Schrodinger equation (SDE) is proposed. According to the barycentric interpolation method (BIM) of rational polynomial and Chebyshev polynomial, the matrix form of the collocation method (CM) that is easy to program is obtained. The convergence rate of the LBRCM for solving the Schrodinger equation is proved from the convergence rate of linear barycentric rational interpolation. Finally, a numerical example verifies the correctness of the theoretical analysis.

scholarly journals Linear Barycentric Rational Method for Two-Point Boundary Value Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Qian Ge ◽  
Xiaoping Zhang
Keyword(s):  
Theoretical Analysis ◽  
Convergence Rate ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Rational Method ◽  
Matrix Form ◽  
Point Boundary ◽  
Numerical Examples ◽  
The Matrix

Linear barycentric rational method for solving two-point boundary value equations is presented. The matrix form of the collocation method is also obtained. With the help of the convergence rate of the interpolation, the convergence rate of linear barycentric rational collocation method for solving two-point boundary value problems is proved. Several numerical examples are provided to validate the theoretical analysis.

scholarly journals Numerical Solution for Third-Order Two-Point Boundary Value Problems with the Barycentric Rational Interpolation Collocation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Qian Ge ◽  
Xiaoping Zhang
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Collocation Method ◽  
Boundary Value ◽  
Rational Interpolation ◽  
Point Boundary ◽  
Third Order ◽  
The Third ◽  
The Matrix ◽  
Barycentric Rational Interpolation

The numerical solution for a kind of third-order boundary value problems is discussed. With the barycentric rational interpolation collocation method, the matrix form of the third-order two-point boundary value problem is obtained, and the convergence and error analysis are obtained. In addition, some numerical examples are reported to confirm the theoretical analysis.

scholarly journals The barycentric rational interpolation collocation method for boundary value problems

2018 ◽  
Vol 22 (4) ◽  
pp. 1773-1779 ◽  
Author(s):  
Dan Tian ◽  
Ji-Huan He
Keyword(s):  
Boundary Value Problems ◽  
Collocation Method ◽  
Analytical Methods ◽  
Boundary Value ◽  
Rational Interpolation ◽  
Higher Order ◽  
Barycentric Rational Interpolation

Higher-order boundary value problems have been widely studied in thermal science, though there are some analytical methods available for such problems, the barycentric rational interpolation collocation method is proved in this paper to be the most effective as shown in three examples.

scholarly journals Barycentric Rational Collocation Method for the Incompressible Forchheimer Flow in Porous Media

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Qingli Zhao ◽  
Yongling Cheng
Keyword(s):  
Porous Media ◽  
Numerical Analysis ◽  
Collocation Method ◽  
Numerical Stability ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Incompressible Fluids ◽  
Flow In Porous Media ◽  
The Matrix ◽  
Forchheimer Flow

Barycentric rational collocation method is introduced to solve the Forchheimer law modeling incompressible fluids in porous media. The unknown velocity and pressure are approximated by the barycentric rational function. The main advantages of this method are high precision and efficiency. At the same time, the algorithm and program can be expanded to other problems. The numerical stability can be guaranteed. The matrix form of the collocation method is obtained from the discrete numerical schemes. Numerical analysis and error estimates for velocity and pressure are established. Numerical experiments are carried out to validate the convergence rates and show the efficiency.

scholarly journals COLLOCATION METHOD FOR FUZZY VOLTERRA INTEGRAL EQUATIONS OF THE SECOND KIND

2020 ◽  
Vol 25 (1) ◽  
pp. 146-166 ◽  
Author(s):  
Zahra Alijani ◽  
Urve Kangro
Keyword(s):  
Integral Equation ◽  
Numerical Solution ◽  
Integral Equations ◽  
Collocation Method ◽  
Volterra Integral Equation ◽  
Basis Function ◽  
Convergence Rates ◽  
Numerical Examples ◽  
Convergence Estimates ◽  
Upper And Lower Functions

In this paper we consider fuzzy Volterra integral equation of the second kind whose kernel may change sign. We give conditions for smoothness of the upper and lower functions of the solution. For numerical solution we propose the collocation method with two different basis function sets: triangular and rectangular basis. The smoothness results allow us to obtain the convergence rates of the methods. The proposed methods are illustrated by numerical examples, which confirm the theoretical convergence estimates.

scholarly journals Hermite collocation method for solving Hammerstein integral equations

2018 ◽  
Vol 14 (1) ◽  
pp. 7413-7423 ◽  
Author(s):  
Yasser Amer
Keyword(s):  
Integral Equations ◽  
Collocation Method ◽  
Matrix Equation ◽  
Unknown Function ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Hammerstein Integral Equations ◽  
The Matrix

In this paper, we are presenting Hermite collocation method to solve numer- ically the Fredholm-Volterra-Hammerstein integral equations. We have clearly presented a theory to …nd ordinary derivatives. This method is based on replace- ment of the unknown function by truncated series of well known Hermite expan-sion of functions. The proposed method converts the equation to matrix equation which corresponding to system of algebraic equations with Hermite coe¢ cients. Thus, by solving the matrix equation, Hermite coe¢ cients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

Barycentric Rational Interpolation Iteration Collocation Method for Solving Nonlinear Vibration Problems

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Jian Jiang ◽  
Zhao-Qing Wang ◽  
Jian-Hua Wang ◽  
Bing-Tao Tang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Collocation Method ◽  
Nonlinear Differential Equations ◽  
Rational Interpolation ◽  
Algebraic Equations ◽  
Vibration Problems ◽  
Differential Matrix ◽  
Barycentric Rational Interpolation

In this article, a powerful computational methodology, named as barycentric rational interpolation iteration collocation method (BRICM), for obtaining the numerical solutions of nonlinear vibration problems is presented. The nonlinear vibration problems are governed by initial-value problems of nonlinear differential equations. Given an initial guess value of the unknown function, the nonlinear differential equations can be transformed into linear differential equations. By applying barycentric rational interpolation and differential matrix, the linearized differential equation is discretized into algebraic equations in the matrix form. The latest solution of nonlinear differential equation is obtained by solving the algebraic equations. The numerical solution of nonlinear vibration problem can be calculated by iteration method under given control precision. Then, the velocity and acceleration can be obtained by differential matrix of barycentric rational interpolation, and the period of nonlinear vibration is also computed by BRICM. Some examples of nonlinear vibration demonstrate the proposed methodological advantages of effectiveness, simple formulations, and high precision.

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