In this work, the convection-diffusion integro-differential equation with a weakly singular kernel is discussed. The  Legendre spectral tau method is introduced for finding the unknown function. The proposed method is based on expanding the approximate solution as the elements of a shifted Legendre polynomials. We reduce the problem to a set of algebraic equations by using operational matrices....

‎this paper presents two main results that the singular values of the hadamard product of normal matrices $a_i$ are weakly log-majorized by the singular values of the hadamard product of $|a_{i}|$ and the singular values of the sum of normal matrices $a_i$ are weakly log-majorized by the singular values of the sum of $|a_{i}|$‎. ‎some applications to these inequalities are also given‎. ‎in addi...

The article is devoted to the approximate solutions of Fredholm integral equations second kind with weak singular kernel that can have additional singularity in numerator. We describe two problems lead such equations. They are problem minimization small deviations and entropy problem. Both them appear when considering a dynamical system involving mixed fractional Brownian motion. In order apply...

One kind of linear integral equation is the weakly singular integral equation; this kind of integral equation is unsolvable by using usual quadrature methods, such as the repeated trapezoid and repeated Simpson quadrature methods. In this article, we intend to present a subtraction method, which is the same as the modified quadrature methods; one can solve linear weakly singular integral equati...

In a recent paper [3], Y. Cao and Y. Xu established the Galerkin method for weakly singular Fredholm integral equations that preserves the singularity of the solution. Their Galerkin method provides a numerical solution that is a linear combination of a certain class of basis functions which includes elements that reflect the singularity of the solution. The purpose of this paper is to extend t...

A collocation method based on optimal nodal splines is presented for the numerical solution of linear Volterra integral equations of the second kind with weakly singular kernel. Since the considered spline operator is a bounded projector we can prove that, for sequences of locally uniform meshes, the approximate solution error converges to zero at exactly the same optimal rate as the spline app...