scholarly journals The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

1999 ◽  
Vol 68 (227) ◽  
pp. 1079-1096 ◽  
Author(s):  
Hermann Brunner ◽  
Arvet Pedas ◽  
Gennadi Vainikko
Keyword(s):  
Collocation Method ◽  
Volterra Equations ◽  
Piecewise Polynomial ◽  
Weakly Singular ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

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 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 TWO-GRID ITERATION METHOD FOR WEAKLY SINGULAR INTEGRAL EQUATIONS

2000 ◽  
Vol 5 (1) ◽  
pp. 76-85
Author(s):  
K. Hakk
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Iteration Method ◽  
Singular Integral Equations ◽  
Piecewise Polynomial ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
Piecewise Polynomial Collocation Method ◽  
Large Linear Systems ◽  
Polynomial Collocation

For the solution of weakly singular integral equations by the piecewise polynomial collocation method it is necessary to solve large linear systems. In the present paper a two‐grid iteration method for solving such systems is constructed and the convergence of this method is investigated.

scholarly journals NUMERICAL SOLUTION OF VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR KERNELS WHICH MAY HAVE A BOUNDARY SINGULARITY

2009 ◽  
Vol 14 (1) ◽  
pp. 79-89 ◽  
Author(s):  
Marek Kolk ◽  
Arvet Pedas
Keyword(s):  
Integral Equations ◽  
Volterra Integral Equations ◽  
Piecewise Polynomial ◽  
Boundary Singularity ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Convergence Estimates ◽  
Linear Volterra Integral Equations ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with kernels which, in addition to a weak diagonal singularity, may have a weak boundary singularity. Global convergence estimates are derived and a collection of numerical results is 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 PRODUCT INTEGRATION FOR WEAKLY SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS

2011 ◽  
Vol 16 (1) ◽  
pp. 153-172 ◽  
Author(s):  
Arvet Pedas ◽  
Enn Tamme
Keyword(s):  
Differential Equations ◽  
Convergence Rates ◽  
Discrete Version ◽  
Product Integration ◽  
Weakly Singular ◽  
Integration Techniques ◽  
Singular Kernels ◽  
Convergence Estimates ◽  
Piecewise Polynomial Collocation Method ◽  
Polynomial Collocation

On the basis of product integration techniques a discrete version of a piecewise polynomial collocation method for the numerical solution of initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels is constructed. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.

scholarly journals ON THE STABILITY OF PIECEWISE POLYNOMIAL COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRAL EQUATIONS OF THE SECOND KIND

2008 ◽  
Vol 13 (1) ◽  
pp. 29-36 ◽  
Author(s):  
R. Kangro ◽  
I. Kangro
Keyword(s):  
Integral Equations ◽  
Singular Integral Equations ◽  
Collocation Methods ◽  
Piecewise Polynomial ◽  
Nonuniform Grids ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
The Stability ◽  
Implementation Problems ◽  
Polynomial Collocation

Piecewise polynomial collocation methods on special nonuniform grids are efficient methods for solving weakly singular Fredholm and Volterra integral equations but there is a widespread belief that those methods are numerically unstable in the case of large values of the nonuniformity parameter r. We show that this method by itself is stable and discuss some implementation problems that may lead to unstable behavior of numerical results.

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