Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels

2012 ◽  
Vol 62 (3) ◽  
pp. 201-211 ◽  
Author(s):  
Guangqing Long ◽  
Gnaneshwar Nelakanti ◽  
Xiaohua Zhang
Keyword(s):  
Integral Equations ◽  
Fredholm Integral Equations ◽  
Galerkin Methods ◽  
Weakly Singular Kernels ◽  
Weakly Singular ◽  
Singular Kernels

scholarly journals A note on orthogonal polynomials

1970 ◽  
Vol 17 (1) ◽  
pp. 83-94 ◽  
Author(s):  
D. Kershaw
Keyword(s):  
Orthogonal Polynomials ◽  
Numerical Solution ◽  
Integral Equations ◽  
Fredholm Integral Equations ◽  
Intermediate Result ◽  
Weakly Singular Kernels ◽  
Quadrature Formulae ◽  
Weakly Singular ◽  
Gauss Type ◽  
Singular Kernels

During an investigation into the existence of Gauss-type quadrature formulae for the numerical solution of Fredholm integral equations with weakly singular kernels an intermediate result was found which is of independent interest.

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.

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