TOL

Ameur S. SELFADJOINT OPERATORS, NORMAL OPERATORS, AND CHARACTERIZATIONS. Operators and Matrices. 2019;13 (3) :835–842.Abstract

Let B(H) be the C -algebra of all bounded linear operators acting on a complex separable Hilbert space H . We shall show that:

1. The class of all selfadjoint operators in B(H) multiplied by scalars is characterized by ∀X ∈ B(H), S2X +XS2 =>2||SXS||, (S ∈ B(H)).

2. The class of all normal operators in B(H) is characterized by each of the three following properties (where DS = SS−SS , for S ∈ B(H)),

(i) ∀X ∈ B(H), S2X + XS2 =>2||SXS||,(S ∈ B(H)),

(ii) S∗DSS = 0 = SDSS∗,(S ∈ B(H)),

(iii) SDSS=> 0 =>SDSS,(S ∈ B(H)).

 

Mennouni A, Ramdani N-E, Zennir K. A New Class of Fredholm Integral Equations of the Second Kind with Non Symmetric Kernel: Solving by Wavelets Method. Boletim da Sociedade Paranaense de Matem´atica. 2018.Abstract

In this paper, we introduce an efficient modification of the wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel. This method based on orthonormal wavelet basis, as a consequence three systems are obtained, a Toeplitz system and two systems with condition number close to 1. Since the preconditioned conjugate gradient normal equation residual (CGNR) and preconditioned conjugate gradient normal equation error (CGNE) methods are applicable, we can solve the systems in O(2n log(n)) operations, by using the fast wavelet transform and the fast Fourier transform.

CHEBBAH H, Mennouni A, Ramdani N-E. Numerical Solution of Generalized Logarithmic Integral Equations of the Second Kind by Projections. Malaysian Journal of Mathematical Sciences. 2018;12 (3) :349–367.Abstract

In this work, we present a new techniques to solve the integral equations of the second kind with logarithmic kernel. First, we show the existence and uniqueness of the solution for the given problem in a Hilbert space. Next, we discuss a projection method for solving integral equations with logarithmic kernel of the second kind; the present method based on the shifted Legendre polynomials. We examine the existence of the solution for the approximate equation, and we provide a new error estimate for the numerical solutions. At the end, numerical examples are provided to illustrate the theoretical results.

Mennouni A, Zaouia S. Discrete septic spline quasi-interpolants for solving generalized Fredholm integral equation of the second kind via three degenerate kernel methods. Mathematical Sciences. 2017;11 :345–357.Abstract

Three main contributions are presented in this paper. First, the septic quasi-interpolants are calculated with all their coefficients. Second, we explore the results to solve a generalized and broad class of Fredholm integral equations of the second kind. Finally, we present three degenerate kernel methods; the latter is a combination of the two previously established methods in the literature. Moreover, we provide a convergence analysis and we give new error bounds. Finally, we exhibit some numerical examples and compare them with previous results in the literature.

Mennouni A. Piecewise constant Galerkin method for a class of Cauchy singular integral equations of the second kind in L2. Journal of Computational and Applied Mathematics. 2017;326 :268-272.Abstract

In this work, we present piecewise constant Galerkin method for a class of Cauchy singular integral equations of the second kind with constant coefficients in L2([0,1],C), using a sequence of orthogonal finite rank projections. We prove the existence and uniqueness theorems for the Cauchy integral equation and the approximate equation, respectively. We perform the error analysis for which we give new and improved estimates for the rates of convergence. Numerical example illustrates the theoretical results.