Improvement of Fourier Series Convergence on the Basis of Splines and Its Application

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Mechanics, Materials Science & Engineering, July 2016

ISSN 2412-5954

Improvement of Fourier Series Convergence on the Basis of Splines and Its Application for Numerical Inversion of Lapl Tanya Solyar 1,a 1 Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, 3-B Naukova St., 79060 Lviv, Ukraine a

tanya@iapmm.lviv.ua DOI 10.13140/RG.2.1.4069.2727

Keywords: Fourier series, improvement of series convergence, piecewise-continuous polynomials, conformal mapping, numerical inversion of Laplace transform.

ABSTRACT. The method of approximation of functions by piecewise continuous polynomials of second degree by means of least squares method is proposed. At that, the finding of functions in the nodal points is reduced to solving the system of linear algebraic equations. The developed approach is used for functions given by Fourier series for which this system is solved in closed form. Thus, the formula for finding the functions in nodal points through modified Fourier series is obtained. There is illustrated the effectiveness of proposed formulas for numerically-analytical finding the original based on an improved approach of Prudnikov, which in general is reduced to calculation of the slowly convergent Fourier series.

1. Introduction. The solution of a wide class of applied problems of mathematical physics is written in the form of Fourier series. In addition, on solving a system of differential equations in partial derivatives the method of Fourier series is used jointly with other approaches. In particular, on additional applying the method of boundary elements the series coefficients are found by means of solution the one-or two-dimensional integral equations by numerical method [1-6]. In addition, determination of the series coefficients demands a lot of calculations and these coefficients are found with some errors what, in turn, can cause the loss of calculation accuracy. Due to this fact, at the studying these problems actual will be the problem of development the approaches to improve the series convergence which would enable one to sum these series with controlled accuracy on the basis of relatively small number of terms. To improve the series convergence in literature [7] are used widely the approaches, obtained on the basis of consideration the problem of series summation, which is incorrect problem [8]. Since the functions, which are described by the series with improved convergence, are found at that as some averaging of the initial [7], then the fast-charging values in the vicinity of localized actions will be found with the largest error. Other methods of improving the series convergence [7, 8] are also efficient only for separate classes of functions. The article offers a way to improve the Fourier series convergence for the functions in the assumption that they can be approximated with given accuracy by piecewise-continuous polynomials of the second degree. The constructed formula is applied to increase the accuracy of numerical inversion of Laplace transform. Note that in [9] way of improving the Fourier series convergence is given for the functions in the assumption that they can be approximated with given accuracy by piecewise-continuous polynomials of the first degree. 2. Basic Ratio. First present the formulas for approximation the functions by piecewise continuous polynomials using the least squares method. Consider the function f(x) continuous on the interval c x d . We will describe it on this interval approximately by the function of the form MMSE Journal. Open Access www.mmse.xyz

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