scholarly journals Superconvergence in the Maximum Norm of a Class of Piecewise Polynomial Collocation Methods for Solving Linear Weakly Singular Volterra Integro-Differential Equations

2003 ◽  
Vol 15 (4) ◽  
pp. 403-421 ◽  
Author(s):  
Raul Kangro ◽  
Inga Parts
Keyword(s):  
Differential Equations ◽  
Collocation Methods ◽  
Maximum Norm ◽  
Piecewise Polynomial ◽  
Weakly Singular ◽  
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 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.

Fully Discrete Galerkin Method For Fredholm Integro-Differential Equations With Weakly Singular Kernels

2008 ◽  
Vol 8 (3) ◽  
pp. 207-222 ◽  
Author(s):  
H. BRUNNER
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Optimal Order ◽  
Piecewise Polynomial ◽  
Fully Discrete ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Discrete Galerkin Method ◽  
Polynomial Collocation

AbstractWe analyze the optimal superconvergence properties of piecewise polynomial collocation solutions on uniform meshes for Volterra integral and integrodifferential equations with multiple (vanishing) proportional delays. It is shown that for delay integro-differential equations the recently obtained optimal order is also attainable. For integral equations with multiple vanishing delays this is no longer true.

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 A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel

1981 ◽  
Vol 22 (4) ◽  
pp. 431-438 ◽  
Author(s):  
G. Vainikko ◽  
P. Uba
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Collocation Methods ◽  
Piecewise Polynomial ◽  
Arbitrary Degree ◽  
Weakly Singular Kernel ◽  
Piecewise Polynomial Approximation ◽  
Collocation Points ◽  
Weakly Singular ◽  
Singular Kernels

AbstractWe construct collocation methods with an arbitrary degree of accuracy for integral equations with logarithmically or algebraically singular kernels. Superconvergence at collocation points is obtained. A grid is used, the degree of non-uniformity of which is in good conformity with the smoothness of the solution and the desired accuracy of the method.

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