the main purpose of this article is to present an approximate solution for the two-dimensional nonlinear volterra integral equations using legendre orthogonal polynomials. first, the two-dimensional shifted legendre orthogonal polynomials are defined and the properties of these polynomials are presented. the operational matrix of integration and the product operational matrix are introduced. th...

<abstract><p>We present a novel direct integral pseudospectral (PS) method (a IPS method) for solving class of continuous-time infinite-horizon optimal control problems (IHOCs). The transforms the IHOCs into finite-horizon in their forms by means certain parametric mappings, which are then approximated finite-dimensional nonlinear programming (NLPs) through rational collocations bas...

properties of the hybrid of block-pulse functions and lagrange polynomials based on the legendre-gauss-type points are investigated and utilized to define the composite interpolation operator as an extension of the well-known legendre interpolation operator. the uniqueness and interpolating properties are discussed and the corresponding differentiation matrix is also introduced. the applicabili...

In this report, we study in a detailed way higher order variances and quadrature Gauss Jacobi. Recall that the variance of order j measures the concentration of a probability close to j points x j,s with weight λ j,s which are determined by the parameters of the quadrature Gauss Jacobi. We shall study many example in which these measures specify adequately the distribution of probabilities. We ...

This paper presents a Gauss Legendre quadrature method for numerical integration over the standard triangular surface: {(x, y) | 0 , 1, 1} x y x y ≤ ≤ + ≤ in the Cartesian two-dimensional (x, y) space. Mathematical transformation from (x, y) space to (ξ, η) space map the standard triangle in (x, y) space to a standard 2-square in (ξ, η) space: {(ξ, η)|–1 ≤ ξ, η ≤ 1}. This overcomes the difficul...

Gauss–Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss–Legendre weights is derived. Together, these two expansions provide a practical ...

The problem of counting lattice points on a hyperboloid of two sheets is Gauss’ circle problem in hyperbolic geometry. The problem of counting lattice points on a hyperboloid of one sheet does not have the same geometric interpretation, and in general, the solution(s) to Gauss’ circle problem gives a lower bound, but not an upper bound. In this paper, we describe an exception. Given an ample he...

A common type of integral to solve numerically in computational room acoustics and other applications is the diffraction integral. Various formulations are encountered but they are usually of the Fourier-type, which means an oscillating integrand which becomes increasingly expensive to compute for increasing frequencies. Classical asympotic solution methods, such as the stationary-phase method,...