On the stability of the one-step exact collocation methods for the numerical solution of the second kind Volterra integral equation

The purpose of this paper is to analyze the stability properties of one-step collocation methods for the second kind Volterra integral equation through application to the basic test and the convolution test equation.Stability regions are determined when the collocation parameters are symmetric and when they are zeros of ultraspherical polynomials.     

New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.     

On a new type of singular point for meromorphic functions in the open unit disk

In this note, we will confirm the existence of a new type of singular point of an admissible meromorphic function in the unit disk. This work is a counterpart of the result of a meromorphic function with order in the complex plane studied by W. C. Lin and S. Mori.     

A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

In this paper, a collocation method is given to solve singularly perturbated two‐point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results. Copyright © 2014 JohnWiley & Sons, Ltd.