Abstract: In this article we have elaborated the numerical schemes of reduction methods for approximate solution of system of singular integro-differential equations when the kernel has a weak singularity. The equations are defined on the arbitrary smooth closed contour of complex plane. We suggest the numerical schemes of the reduction method over the system of Faber-Laurent polynomials for th...

We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an hp-version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal hp-version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near t = 0 caused by the weakly sing...

We employ a hyperbolic tangent function to construct nonlinear transformations which are useful in numerical evaluation of weakly singular integrals and Cauchy principal value integrals. Results of numerical implementation based on the standard Gauss quadrature rule show that the present transformations are available for the singular integrals and, in some cases, give much better approximations...

We present a method to regularize first and second kind integral equations of Fredholm type with singular kernel. By appropriate application of the Poincaré-Bertrand formula we change such integral equations into a second kind Fredholm’s integral equation with at most weakly singular kernel. 2000 Mathematics Subject Classification. 45Exx, 45E05, 45B05.

For a cardinal κ and a model M of cardinality κ let No(M) denote the number of non-isomorphic models of cardinality κ which are L∞κ-equivalent to M. In [She82] Shelah established that when κ is a weakly compact cardinal and μ ≤ κ is a nonzero cardinal, there exists a model M of cardinality κ with No(M) = μ. We prove here that if κ is a weakly compact cardinal, the question of the possible value...

In this paper, we first give a lower and upper bounds for singular values of a 2×2 positive semidefinite block matrices. Then, we give some weakly majorization inequalities of singular values positive semidefinite block matrices. Also, we present inequalities involving the direct sum and sum of positive semidefinite matrices.

We study a class of timelike weakly extremal surfaces in flat Minkowski space R, characterized by the fact that they admit a C parametrization (in general not an immersion) of a specific form. We prove that if the distinguished parametrization is of class C, then the surface is regularly immersed away from a closed singular set of euclidean Hausdorff dimension at most 1 + 1/k, and that this bou...

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

We consider the action of Hecke operators on weakly holomorphic modular forms and a Hecke-equivariant duality between the spaces of holomorphic and weakly holomorphic cusp forms. As an application, we obtain congruences modulo supersingular primes, which connect Hecke eigenvalues and certain singular moduli.

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.