This paper gives an efficient numerical method for solving the nonlinear system of Volterra-Fredholm integral equations. A Legendre-spectral method based on the Legendre integration Gauss points and Lagrange interpolation is proposed to convert the nonlinear integral equations to a nonlinear system of equations where the solution leads to the values of unknown functions at collocation points.

The purpose of this paper is to investigate the use of rational Chebyshev (RC) collocation method for solving high-order linear ordinary differential equations with variable coefficients. Using the rational Chebyshev collocation points, this method transforms the high-order linear ordinary differential equations and the given conditions to matrix equations with unknown rational Chebyshev coeffi...

In this paper, we explore the applicability of the SincCollocation method to a three-dimensional (3D) oceanography model. The model describes a wind-driven current with depth-dependent eddy viscosity in the complex-velocity system. In general, the Sinc-based methods excel over other traditional numerical methods due to their exponentially decaying errors, rapid convergence and handling problems...

This paper proposes a new technique to speed up the computation of the matrix of spectral collocation discretizations of surface single and double layer operators over a spheroid. The layer densities are approximated by a spectral expansion of spherical harmonics and the spectral collocation method is then used to solve surface integral equations of potential problems in a spheroid. With the pr...

Some collocation schemes used to solve mth order ordinary differential equations are known to display superconvergence at the mesh points. Here we show that some such schemes have additional superconvergence points for the approximate solution and all of its derivatives. Using such points, we argue that a mesh selection scheme introduced by Dodson can be expected to perform well under general c...

We present an accurate, efficient numerical analysis for vector modes of dielectric optical waveguide structures with an arbitrary refractive index profile using a quadratic spline collocation method (QSCM). The unknown weights of the polynomials are determined by forcing the errors at the collocation points to be zero. Consequently, the original second order differential equation is converted ...

In this paper, a new collocation method, which is based on Boubaker polynomials, is introduced for the approximate solutions of a class of two-dimensional linear Fredholm integral equationsof the second kind. The properties of two-dimensional Boubaker functions are presented. The fundamental matrices of integration with the collocation points are utilized to reduce the solution of the integral ...

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to ...

Local projection methods which yield c'm_1) piecewise polynomials of order m + k as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functional at which the residual error in each partition interval is required to vanish on. We develop a condition on these k f unctionals which implies breakpoint superconvergence (of...