Weakly singular linear Volterra integral equations: A Nyström method in weighted spaces of continuous functions

2021 ◽  
pp. 114001
Author(s):  
Luisa Fermo ◽  
Donatella Occorsio
Keyword(s):  
Integral Equations ◽  
Continuous Functions ◽  
Volterra Integral Equations ◽  
Weighted Spaces ◽  
Nyström Method ◽  
Nystrom Method ◽  
Weakly Singular ◽  
Spaces Of Continuous Functions ◽  
Linear Volterra Integral Equations

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 Singularly perturbed Volterra integral equations with weakly singular kernels

2002 ◽  
Vol 30 (3) ◽  
pp. 129-143 ◽  
Author(s):  
Angelina Bijura
Keyword(s):  
Integral Equations ◽  
Special Function ◽  
Initial Boundary ◽  
Volterra Integral Equations ◽  
Singularly Perturbed ◽  
Interesting Aspect ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Linear Volterra Integral Equations

We consider finding asymptotic solutions of the singularly perturbed linear Volterra integral equations with weakly singular kernels. An interesting aspect of these problems is that the discontinuity of the kernel causes layer solutions to decay algebraically rather than exponentially within the initial (boundary) layer. To analyse this phenomenon, the paper demonstrates the similarity that these solutions have to a special function called the Mittag-Leffler function.

scholarly journals Numerical Solutions Of Volterra Integral Equations Using Legendre Polynomials

2013 ◽  
Vol 32 ◽  
pp. 29-35
Author(s):  
Md Azizur Rahman ◽  
Md Shafiqul Islam
Keyword(s):  
Integral Equations ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Volterra Integral Equations ◽  
Matrix Formulation ◽  
Numerical Examples ◽  
Piecewise Polynomials ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Linear Volterra Integral Equations

In this paper, Legendre piecewise polynomials are used to approximate the solutions of linear Volterra integral equations. Both second and first kind integral equations with regular as well as weakly singular kernels are considered. A matrix formulation is given for linear Volterra integral equations by the technique of Galerkin method. Numerical examples are considered to verify the accuracy of the proposed derivations, and the numerical solutions in this paper are also compared with the existing methods in the published literature. DOI: http://dx.doi.org/10.3329/ganit.v32i0.13643 GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 32 (2012) 29 – 35  

scholarly journals Superconvergence of interpolated collocation solutions for weakly singular Volterra integral equations of the second kind

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Qiumei Huang ◽  
Min Wang
Keyword(s):  
Integral Equations ◽  
Numerical Experiments ◽  
Volterra Integral Equations ◽  
Least Square ◽  
Convergence Order ◽  
Collocation Points ◽  
Weakly Singular ◽  
Interpolation Postprocessing ◽  
Collocation Solution

AbstractIn this paper, we discuss the superconvergence of the “interpolated” collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution $$u_h$$ u h , two different interpolation postprocessing approximations of higher accuracy: $$I_{2h}^{2m-1}u_h$$ I 2 h 2 m - 1 u h based on the collocation points and $$I_{2h}^{m}u_h$$ I 2 h m u h based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.

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