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.

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.

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