In this paper, a numerical implementation of an expansion method is developed for solving Abel's integral equations of the first and second kind. The solution of such equations may demonstrate a singular behaviour in the neighbourhood of the initial point of the interval ofintegration. The suggested method is based on the use of Taylor series expansion to overcome the singularity which leads to approximating the unknown function and it's derivatives in terms of Chebyshev polynomials of the first kind. The proposed method, transforms the Abel's integral equations of the first and second kind into a system of linear algebraic equations which can be solved by Gaussian elimination algorithm. Finally, some numerical examples are included to clarify the accuracy and applicability of the presented method which indicate that proposed method is computationally very attractive. In thispaper, all numerical computations were carried out on a PC executing some programs written in maple software. Manuscript profile

A semigroup S is called a posemigroup if S is equipped with an ordering relation\ ≤ " such that a ≤ b in S implies xa ≤ xb and ax ≤ bx, for all x 2 S. In what followswe study necessary and sufficient conditions that a posemigroup S to be regular, interms of certain conditions of Q∗ S, the semigroup of high-quasi-ideals of S. This studygives us a characterization method of the regular posemigroups. Manuscript profile

In this work, we present a numerical method for solving nonlinear Fredholmand Volterra integral equations of the second kind which is based on the useof Block Pulse functions(BPfs) and collocation method. Numerical examplesshow eciency of the method. Manuscript profile