The spectral method with discrete spherical harmonics transform plays an important role in many applications. In spite of its advantages, the spherical harmonics transform has a drawback of high computational complexity, which is determined by that of the associated Legendre transform, and the direct computation requires time of O(N3) for cut-off frequency N . In this paper, we propose a fast a...

We extend the application of Legendre-Galerkin algorithms for sixth-order elliptic problems with constant coefficients to sixth-order elliptic equations with variable polynomial coefficients. The complexities of the algorithm are O(N) operations for a one-dimensional domain with N − 5 unknowns. An efficient and accurate direct solution for algorithms based on the LegendreGalerkin approximations...

In paper [4], transformation matrices mapping the Legendre and Bernstein forms of a polynomial of degree n into each other are derived and examined. In this paper, we derive a matrix of transformation of Chebyshev polynomials of the first kind into Bernstein polynomials and vice versa. We also study the stability of these linear maps and show that the Chebyshev–Bernstein basis conversion is rem...

We extend a collocation method for solving a nonlinear ordinary differential equation ODE via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth, see the works of Doha and Bhrawy 2006, Guo 2000, and Guo et al. 2002 . Choosing the optimal polynomial for solving every ODEs problem depends on many f...

The present study included 3,358 observations of 675 bulls and heifers from the Iowa State University beef cattle breeding project. Data were collected over a 3-yr period between 1998 and 2000. Each year, cattle were scanned four to six times for ultrasound-predicted percentage of intramuscular fat (UPFAT) and other ultrasound traits, starting at a minimum age of 28 wk. The objective of the cur...

Abstract: The Legendre group of lowpass filters are designed and compared against several known types of filters at dB c f 3 = 2.5GHz in lumped elements, microstrip and DGS environment. In microstrip format the Legendre Polynomial Product (LPP)-312 and 213 filters have reduced group delay variation in the passband, sharper cut-off and wider impedance matching BW. In the DGS format the measured ...

Motivated by an expression Persson and Strang on integral involving Legendre polynomials, stating that the square of $P_{2n+1}(x)/x$ integrated over $[-1,1]$ is always $2$, we present analog results for Hermite, Chebyshev, Laguerre Gegenbauer polynomials as well original polynomial with even index.