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.

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 NUMERICAL SOLUTIONS AND THEIR SUPERCONVERGENCE FOR WEAKLY SINGULAR INTEGRAL EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

1998 ◽  
Vol 3 (1) ◽  
pp. 104-113
Author(s):  
Kristiina Hakk ◽  
Arvet Pedas
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Singular Integral Equations ◽  
Discontinuous Coefficients ◽  
Fredholm Integral Equations ◽  
Collocation Points ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
Singular Kernels ◽  
Piecewise Polynomial Collocation Method

The piecewise polynomial collocation method is discussed to solve second kind Fredholm integral equations with weakly singular kernels K (t, s) which may be discontinuous at s = d, d = const. The main result is given in Theorem 4.1. Using special collocation points, error estimates at the collocation points are derived showing a more rapid convergence than the global uniform convergence in the interval of integration available by piecewise polynomials.

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.

An adapted integration method for Volterra integral equation of the second kind with weakly singular kernel

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ahlem Nemer ◽  
Hanane Kaboul ◽  
Zouhir Mokhtari
Keyword(s):  
Integral Equation ◽  
Integration Method ◽  
New Product ◽  
Linear Interpolation ◽  
Weakly Singular Kernel ◽  
Product Integration ◽  
Weakly Singular ◽  
Product Integration Method ◽  
Singular Kernels ◽  
Linear Volterra Integral Equations

Abstract In this paper, we consider general cases of linear Volterra integral equations under minimal assumptions on their weakly singular kernels and introduce a new product integration method in which we involve the linear interpolation to get a better approximate solution, figure out its effect and also we provide a convergence proof. Furthermore, we apply our method to a numerical example and conclude this paper by adding a conclusion

Solving the Volterra Integral Equations with Weakly Singular Kernel by Taylor Expansion Methods

2012 ◽  
Vol 220-223 ◽  
pp. 2129-2132
Author(s):  
Li Huang ◽  
Yu Lin Zhao ◽  
Liang Tang
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Unknown Function ◽  
Taylor Expansion ◽  
Closed Form Solution ◽  
Form Solution ◽  
Volterra Integral Equations ◽  
Singular Kernel ◽  
Weakly Singular Kernel ◽  
Weakly Singular

In this paper, we propose a Taylor expansion method for solving (approximately) linear Volterra integral equations with weakly singular kernel. By means of the nth-order Taylor expansion of the unknown function at an arbitrary point, the Volterra integral equation can be converted approximately to a system of equations for the unknown function itself and its n derivatives. This method gives a simple and closed form solution for the integral equation. In addition, some illustrative examples are presented to demonstrate the efficiency and accuracy of the proposed method.

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