330 research outputs found
SOME MODIFICATIONS OF CHEBYSHEV-HALLEY’S METHODS FREE FROM SECOND DERIVATIVE WITH EIGHTH-ORDER OF CONVERGENCE
The variant of Chebyshev-Halley’s method is an iterative method used for solving a nonlinear equation with third order of convergence. In this paper, we present some new variants of three steps Chebyshev-Halley’s method free from second derivative with two parameters. The proposed methods have eighth-order of convergence for and and require four evaluations of functions per iteration with index efficiency equal to . Numerical simulation will be presented by using several functions to show the performance of the proposed methods
Numerical methods for calculating poles of the scattering matrix with applications in grating theory
Waveguide and resonant properties of diffractive structures are often
explained through the complex poles of their scattering matrices. Numerical
methods for calculating poles of the scattering matrix with applications in
grating theory are discussed. A new iterative method for computing the matrix
poles is proposed. The method takes account of the scattering matrix form in
the pole vicinity and relies upon solving matrix equations with use of matrix
decompositions. Using the same mathematical approach, we also describe a
Cauchy-integral-based method that allows all the poles in a specified domain to
be calculated. Calculation of the modes of a metal-dielectric diffraction
grating shows that the iterative method proposed has the high rate of
convergence and is numerically stable for large-dimension scattering matrices.
An important advantage of the proposed method is that it usually converges to
the nearest pole.Comment: 9 pages, 2 figures, 4 table
Numerical Methods for Solving Fractional Differential Equations
Department of Mathematical SciencesIn this thesis, several efficient numerical methods are proposed to solve initial value problems and boundary value problems of fractional di???erential equations.
For fractional initial value problems, we propose a new type of the predictorevaluate-corrector-evaluate method based on the Caputo fractional derivative operator.
Furthermore, we propose a new type of the Caputo fractional derivative operator that does not have a di???erential form of a solution. However, with some fractional orders, there are problems that a solution blows up and the scheme has a low convergence. Thus, we identify new treatments for these values. Then, we can expect a significant improvement for all fractional orders. The advantages and improvements are shown by testing various numerical examples.
For fractional BVPs, we propose an explicit method that dramatically reduces the computational time for solving a dense matrix system. Moreover, by adopting
high-order predictor-corrector methods which have uniform convergence rates O(h2) or O(h3) for all fractional orders [8], we propose a second-order method and a third-order method by using the Newton???s method and the Halley method, respectively. We show its advantage by testing various numerical examples.clos
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