Comparison of some numerical quadrature formulas for weakly singular periodic Fredholm integral equations

Computing ◽  
1989 ◽  
Vol 43 (2) ◽  
pp. 159-170 ◽  
Author(s):  
Avram Sidi
Keyword(s):  
Integral Equations ◽  
Numerical Quadrature ◽  
Quadrature Formulas ◽  
Fredholm Integral Equations ◽  
Weakly Singular

Numerical Algorithms of Optimal Complexity for Weakly Singular Volterra Integral Equations

2006 ◽  
Vol 6 (4) ◽  
pp. 436-442 ◽  
Author(s):  
A.N. Tynda
Keyword(s):  
Numerical Methods ◽  
Integral Equations ◽  
Numerical Algorithms ◽  
Volterra Integral Equations ◽  
Volterra Equations ◽  
Fredholm Integral Equations ◽  
Optimal Complexity ◽  
Weakly Singular ◽  
Different Types ◽  
Singular Kernels

AbstractIn this paper we construct complexity order optimal numerical methods for Volterra integral equations with different types of weakly singular kernels. We show that for Volterra equations (in contrast to Fredholm integral equations) using the ”block-by-block” technique it is not necessary to employ the additional iterations to construct complexity optimal methods.

scholarly journals ON FULLY DISCRETE COLLOCATION METHODS FOR SOLVING WEAKLY SINGULAR INTEGRAL EQUATIONS

2009 ◽  
Vol 14 (1) ◽  
pp. 69-78 ◽  
Author(s):  
Raul Kangro ◽  
Inga Kangro
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Nonuniform Grids ◽  
Fully Discrete ◽  
Straightforward Application ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations

A popular class of methods for solving weakly singular integral equations is the class of piecewise polynomial collocation methods. In order to implement those methods one has to compute exactly certain integrals that determine the linear system to be solved. Unfortunately those integrals usually cannot be computed exactly and even when analytic formulas exist, their straightforward application may cause unacceptable roundoff errors resulting in apparent instability of those methods in the case of highly nonuniform grids. In this paper fully discrete analogs of the collocation methods, where integrals are replaced by quadrature formulas, are considered, corresponding error estimates are derived.

scholarly journals FAST SOLVERS OF WEAKLY SINGULAR INTEGRAL EQUATIONS OF THE SECOND KIND

2018 ◽  
Vol 23 (4) ◽  
pp. 639-664 ◽  
Author(s):  
Sumaira Rehman ◽  
Arvet Pedas ◽  
Gennadi Vainikko
Keyword(s):  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Fredholm Integral Equations ◽  
Free Term ◽  
Fast Solvers ◽  
Weakly Singular ◽  
Weakly Singular Integral Equations ◽  
Derivatives Of ◽  
Smooth Data

We discuss the bounds of fast solving weakly singular Fredholm integral equations of the second kind with a possible diagonal singularity of the kernel and certain boundary singularities of the derivatives of the free term when the information about the smooth coefficient functions in the kernel and about the free term is restricted to a given number of sample values. In this situation, a fast/quasifast solver is constructed. Thus the complexity of weakly singular integral equations occurs to be close to that of equations with smooth data without singularities. Our construction of fast/quasifast solvers is based on the periodization of the problem.

The economized monic Chebyshev polynomials for solving weakly singular Fredholm integral equations of the first kind

2018 ◽  
Vol 13 (01) ◽  
pp. 2050030 ◽  
Author(s):  
E. S. Shoukralla ◽  
M. A. Markos
Keyword(s):  
Integral Equations ◽  
Chebyshev Polynomials ◽  
Unknown Function ◽  
Asymptotic Expression ◽  
High Order Accuracy ◽  
Computational Time ◽  
Fredholm Integral Equations ◽  
Algebraic Equations ◽  
Weakly Singular ◽  
Parametric Functions

This paper presents a numerical method for solving a certain class of Fredholm integral equations of the first kind, whose unknown function is singular at the end-points of the integration domain, and has a weakly singular logarithmic kernel with analytical treatments of the singularity. To achieve this goal, the kernel is parametrized, and the unknown function is assumed to be in the form of a product of two functions; the first is a badly-behaved known function, while the other is a regular unknown function. These two functions are approximated by using the economized monic Chebyshev polynomials of the same degree, while the given potential function is approximated by monic Chebyshev polynomials of the same degree. Further, the two parametric functions associated to the parametrized kernel are expanded into Taylor polynomials of the first degree about the singular parameter, and an asymptotic expression is created, so that the obtained improper integrals of the integral operator become convergent integrals. Thus, and after using a set of collocation points, the required numerical solution is found to be equivalent to the solution of a linear system of algebraic equations. From the illustrated example, it turns out that the proposed method minimizes the computational time and gives a high order accuracy.

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