A novel element with arbitrary domain shape by using decoupled scaled boundary finite element (DSBFEM) is proposed for eigenvalue analysis of 2D vibrating rods with different boundary conditions. Within the proposed element scheme, the mode shapes of vibrating rods with variable boundary conditions are modelled and results are plotted. All possible conditions for the rods ends are incorporated ...

We have implemented in Matlab a Gauss-like cubature formula over arbitrary bivariate domains with a piecewise regular boundary, which is tracked by splines of maximum degree p (spline curvilinear polygons). The formula is exact for polynomials of degree at most 2n− 1 using N ∼ cmn2 nodes, 1 ≤ c ≤ p, m being the total number of points given on the boundary. It does not need any decomposition of ...

This paper concerns a class of deferred correction methods recently developed for initial value ordinary differential equations; such methods are based on a Picard integral form of the correction equation. These methods divide a given timestep [tn, tn+1] into substeps, and use function values computed at these substeps to approximate the Picard integral by means of a numerical quadrature. The m...

Superconvergence phenomenon of the Legendre spectral collocation method and the p-version finite element method is discussed under the one dimensional setting. For a class of functions that satisfy a regularity condition (M): ‖u‖L∞ ≤ cMk on a bounded domain, it is demonstrated, both theoretically and numerically, that the optimal convergent rate is supergeometric. Furthermore, at proper Gaussia...

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 appli...

Gauss–Legendre quadrature formulas have excellent convergence properties when applied to integrals ∫ 1 0 f(x) dx with f ∈ C∞[0, 1]. However, their performance deteriorates when the integrands f(x) are in C∞(0, 1) but are singular at x = 0 and/or x = 1. One way of improving the performance of Gauss–Legendre quadrature in such cases is by combining it with a suitable variable transformation such ...

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as points and derived differentiation matrices for derivatives. With matrices, were transformed into linear systems, which are easier solve. Two types of numerical simulations, results demo...

In this paper we present explicit formulas for discrete orthogonal polynomials over the so-called Gauss-Lobatto Chebyshev points. We also give the “three-term recurrence relation” to construct such polynomials. As a numerical application, we apply our formulas to the least-squares problem.