Nystrom-type techniques for solving electromagnetics integral equations with smooth and singular kernels

2012 ◽  
Vol 25 (5-6) ◽  
pp. 490-511 ◽  
Author(s):  
M. V. Balaban ◽  
E. I. Smotrova ◽  
O. V. Shapoval ◽  
V. S. Bulygin ◽  
A. I. Nosich
Keyword(s):  
Integral Equations ◽  
Singular Kernels

scholarly journals On multi-singular integral equations involving n weakly singular kernels

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1323-1333 ◽  
Author(s):  
Sales Nabavi ◽  
O. Baghani
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Singular Integral ◽  
Applied Mathematics ◽  
Uniform Space ◽  
Singular Integral Equations ◽  
Restrictive Assumption ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels

We deal with some sources of Banach spaces which are closely related to an important issue in applied mathematics i.e. the problem of existence and uniqueness of the solution for the very applicable weakly singular integral equations. In the classical mode, the uniform space (C[a,b], ||.||?) is usually applied to the related discussion. Here, we apply some new types of Banach spaces, in order to extend the area of problems we could discuss. We consider a very general type of singular integral equations involving n weakly singular kernels, for an arbitrary natural number n, without any restrictive assumption of differentiability or even continuity on engaged functions. We show that in appropriate conditions the following multi-singular integral equation of weakly singular type has got exactly a solution in a defined Banach space x(t) = ?p,i=1 ?i/?(^?i) ?t,0 fi(s,x(s)) (tn-tn-1)1-?i,n...(t1-s)1-?i,1 dt + ?(t). In particular we consider the famous fractional Langevin equation and by the method we could extend the region of variations of parameter ?+ ? from interval [0,1) in the earlier works to interval [0,2).

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 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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