[1] A superposed epoch analysis of 100–300 substorms is performed to determine the median size and shape of the substorm-associated VLF chorus, magnetic bay, and Pi2 pulsation burst observed at the near-auroral Halley research station, Antarctica, and at the midlatitude Faraday station at three different local times (2230, 2330, 0130 MLT). The spatial and temporal properties of the magnetic bay...

1. Introduction and Summary. Clenshaw [1, 2] has described a simple and effective method for summing a Chebyshev series based on the recurrence relation between the Chebyshev coefficients. He has further pointed out that the same method could be used to sum a series of any set of functions which are generated by a linear recurrence relation [1]. Such a set of functions is a set of orthogonal po...

By using the integral operational matrix and the product operation matrix of the Chebyshev wavelet, a class of nonlinear fractional integral-differential equations of Bratu-type is transformed into nonlinear algebraic equations, which makes the solution process and calculation more simple. At the same time, reliable approaches for uniqueness and convergence of the Chebyshev wavelet method are d...

In this paper, we use the spectral collocation method based on Chebyshev polynomials for spatial derivatives and fourth order Runge-Kutta (RK) method for time integration to solve the generalized Zakharov equation (GZE). Firstly, theory of application of Chebyshev spectral collocation method on the GZE is presented. This method yields a system of ordinary differential equations (ODEs). Secondly...

Through a geometncal approach of the blossoming pnnciple, we achieve a dimension élévation process for extended Chebyshev spaces Applied to a nested séquence ofsuch spaces included in a polynomial one, this allows to compute the Bézier points from the initial Chebyshev-Bézier points This method leads to interesting shape effects © Elsevier, Paris

Problem statement: Not all differential equations can be solved analytically, to overcome this problem, there is need to search for an accurate approximate solution. Approach: The objective of this study was to find an accurate approximation technique (scheme) for solving linear differential equations. By exploiting the Trigonometric identity property of the Chebyshev polynomial, we developed a...

In this note, we provide simple convergence analysis for the algebraic multilevel iteration methods [37, 51]. We consider two examples of AMLI methods with different polynomial acceleration. The first one is based on shifted and scaled Chebyshev polynomial and the other on the polynomial of best approximation to x−1 on a finite interval [λmin , λmax ], 0 < λmin < λmax in the ‖ ·‖∞ norm. The con...

I apply Fourier and Chebyshev spectral methods to derive accurate and efficient algorithms for velocity continuation. As expected, the accuracy of the spectral methods is noticeably superior to that of the finite-difference approach. Both methods apply a transformation of the time axis to squared time. The Chebyshev method is slightly less efficient than the Fourier method, but has less problem...

A conjecture for the projection norm (or Lebesgue constant) of a weighted interpolation method based on the zeros of Chebyshev polynomials of the third and fourth kinds is resolved. This conjecture was made in a paper by J. C. Mason and G. H. Elliott in 1995. The proof of the conjecture is achieved by relating the projection norm to that of a weighted interpolation method based on zeros of Cheb...

Experimentation and the evaluation of algorithms have a long history in algebra. In this paper we follow single test example over more than 250 years. 1685, John Wallis published A treatise algebra, both historical practical, containing solution Colonel Titus’ problem that was proposed to him around 1650. The Titus consists three algebraic quadratic equations unknowns, which transformed into fi...