Assuming linear theory, the phenomenon of scattering of waves by a circular arc shaped barrier with nonuniform porosity is studied. The water region is considered to be of infinite or finite depth. Based on a judicious application of Green’s integral theorem, the corresponding boundary value problem is reduced to a hypersingular integral equation of second kind. The boundary element method and the collocation method are adopted to solve the hypersingular integral equation, and we ensure a good matching of the solutions obtained by the two methods. The reflection coefficient and energy dissipation are evaluated by using the solution of the integral equation which is then studied graphically. Different choices of distributions of pores on the barrier are considered, and we observe that the nonuniform porosity of the barrier has significant effect on the reflected wave and the energy dissipation.

In this article, we have introduced a Taylor collocation method, which is based on collocation method for solving fractional Riccati differential equation. The fractional derivatives are described in the Caputo sense. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional Riccati differential equation and then substituting their matrix forms into the equation. Using collocation points, the systems of nonlinear algebraic equation is derived. We further solve the system of nonlinear algebraic equation using Maple 13 and thus obtain the coefficients of the generalized Taylor expansion. Illustrative examples are presented to demonstrate the effectiveness of the proposed method.

New numerical methods based on collocation methods with the mechanical quadrature rules are proposed to solve some systems of singular integro-differential equations that are defined on arbitrary smooth closed contours of the complex plane. We carry out the convergence analysis in classical Hölder spaces. A numerical example is also presented.

In this paper, we apply the collocation methods to a class of Volterra integral functional equations with multiple proportional delays (VIFEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solutions for this type of equations, and then analyze the convergence orders of the collocation solutions and give corresponding error estimates. The numerical results verify our theoretical analysis.

In this paper, the collocation methods are used to solve the boundary integral equations of the first kind on the polygon. By means of Sidi’s periodic transformation and domain decomposition, the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers (i = 1,...,d), which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations. Numerical experiments are carried out to show that the methods are very efficient.

The effective method of carbon nanotubes antennas' parameters calculation has been developed. The frequency dependencies of input impedance of CNT in dielectric medium have been investigated. It is shown that an increase in the length of a nanotube length does not lead to the appearance of resonances in the centimeter wavelength range.

Spherical radial basis functions are used to define approximate solutions to pseudodifferential equations of negative order on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the collocation method. A salient feature of our approach in this paper is a simple error analysis for the collocation method using the same argument as that for the Galerkin method.