scholarly journals On asymptotic convergence of polynomial collocation method for one class of singular integro-differential equations

2020 ◽  
Vol 12 (1) ◽  
pp. 43-55
Author(s):  
Aleksandr Ivanovich Fedotov
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Asymptotic Convergence ◽  
Polynomial Collocation

scholarly journals Piecewise Legendre spectral-collocation method for Volterra integro-differential equations

2015 ◽  
Vol 18 (1) ◽  
pp. 231-249 ◽  
Author(s):  
Zhendong Gu ◽  
Yanping Chen
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Spectral Collocation Method ◽  
Piecewise Polynomial ◽  
Spectral Collocation ◽  
Numerical Errors ◽  
Piecewise Polynomial Collocation Method ◽  
Theoretical Results ◽  
Polynomial Collocation ◽  
Better Than

Our main purpose in this paper is to propose the piecewise Legendre spectral-collocation method to solve Volterra integro-differential equations. We provide convergence analysis to show that the numerical errors in our method decay in$h^{m}N^{-m}$-version rate. These results are better than the piecewise polynomial collocation method and the global Legendre spectral-collocation method. The provided numerical examples confirm these theoretical results.

scholarly journals COLLOCATION APPROXIMATIONS FOR WEAKLY SINGULAR VOLTERRA INTEGRO‐DIFFERENTIAL EQUATIONS

2003 ◽  
Vol 8 (4) ◽  
pp. 315-328 ◽  
Author(s):  
I. Parts ◽  
A. Pedas
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Order Of Convergence ◽  
Piecewise Polynomial ◽  
Volterra Type ◽  
Weakly Singular ◽  
Uniform Grids ◽  
Attainable Order ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

A piecewise polynomial collocation method for solving linear weakly singular integro‐differential equations of Volterra type is constructed. The attainable order of convergence of collocation approximations on arbitrary and quasi‐uniform grids is studied theoretically and numerically.

scholarly journals Modified Chebyshev Collocation Method for Solving Differential Equations

CAUCHY ◽  
2015 ◽  
Vol 3 (4) ◽  
pp. 208
Author(s):  
M Ziaul Arif ◽  
Ahmad Kamsyakawuni ◽  
Ikhsanul Halikin
Keyword(s):  
Exact Solution ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Collocation Method ◽  
Difference Method ◽  
Numerical Scheme ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Polynomial Collocation

This paper presents derivation of alternative numerical scheme for solving differential equations, which is modified Chebyshev (Vieta-Lucas Polynomial) collocation differentiation matrices. The Scheme of modified Chebyshev (Vieta-Lucas Polynomial) collocation method is applied to both Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) cases. Finally, the performance of the proposed method is compared with finite difference method and the exact solution of the example. It is shown that modified Chebyshev collocation method more effective and accurate than FDM for some example given.

scholarly journals DEVIATION OF THE ERROR ESTIMATION FOR SECOND ORDER FREDHOLM-VOLTERRA INTEGRO DIFFERENTIAL EQUATIONS

2016 ◽  
Vol 21 (6) ◽  
pp. 719-740 ◽  
Author(s):  
Reza Parvaz ◽  
Mohammad Zarebnia ◽  
Amir Saboor Bagherzadeh
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Error Estimation ◽  
Final Section ◽  
Second Order ◽  
Numerical Results ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Collocation Method ◽  
Theoretical Results ◽  
Polynomial Collocation

In this paper we study the deviation of the error estimation for the second order Fredholm-Volterra integro-differential equations. We prove that for m degree piecewise polynomial collocation method, our method provides O(hm+1) as the order of the deviation of the error. Also numerical results in the final section are included to confirm the theoretical results.

scholarly journals An Efficient Numerical Method for Solving Volterra-Fredholm Integro-Differential Equations of Fractional Order by Using Shifted Jacobi-Spectral Collocation Method

2018 ◽  
Vol 15 (3) ◽  
pp. 344-351
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Jacobi Polynomial ◽  
Fractional Derivatives ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
Jacobi Spectral Collocation Method ◽  
Polynomial Collocation

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

A Piecewise Nonpolynomial Collocation Method for Fractional Differential Equations

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Shahrokh Esmaeili
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Block Scheme ◽  
Poor Convergence ◽  
Fractional Differential ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

Since the solutions of the fractional differential equations (FDEs) have unbounded derivatives at zero, their numerical solutions by piecewise polynomial collocation method on uniform meshes will lead to poor convergence rates. This paper presents a piecewise nonpolynomial collocation method for solving such equations reflecting the singularity of the exact solution. The entire domain is divided into several small subdomains, and the nonpolynomial pieces are constructed using a block-by-block scheme on each subdomain. The method is applied to solve linear and nonlinear fractional differential equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.

Sign in / Sign up

Export Citation Format

Share Document