Journal of Approximation Theory and Applied Mathematics - 2015 Vol. 5 E-Book

• Herausgeber: Books on Demand
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Deutsch
• Veröffentlichungsjahr: 2016

Seitenzahl: 70

Beschreibung

Journal of Approximation Theory and Applied Mathematics (ISSN 2196-1581) is a journal which started in 2013. Themes of our journal are: Approximation theory (with a focus on wavelets) and applications in mathematics like numerical analysis, statistics or financial mathematics.

Leseprobe

Contents

New Methods of Approximation of Step Functions

Resolution of Nonlinear Partial Differential Equations by Elzaki Transform Decomposition Method

Testing an Algorithm for an Adaptive Wavelet Based Method in Mathematica and Comparing the Method with NDSolve

New Methods of Approximation of Step Functions

S. V. Aliukov

South Ural State University, Chelyabinsk, Russia

e-mail: [email protected]

Abstract—New methods of approximation of step functions with an estimation of the error of the approximation are suggested. The suggested methods do not have any of the disadvantages of traditional approximations of step functions by means of Fourier series and can be used in problems of mathematical modeling of a wide range of processes and systems.

Keywords: step functions, mathematical modeling, approximation, convergence, estimation of error, examples of application.

1. INTRODUCTION

Step functions are widely applied in various areas of scientific research. Technical and mathematical disciplines, such as automatic control theory, electrical and radio engineering, information and signal transmission theory, equations of mathematical physics, theory of vibrations, and differential equations are traditional fields of application [1–3].

Systems with step parameters and functions are considered highly nonlinear structures to emphasize the complexity of obtaining solutions for such structures. Despite the simplicity of step functions in segments, the construction of solutions in problems with step functions on the whole domain of definition requires using special mathematical methods, such as the alignment method [4] with the coordination of the solution by segments and switching surfaces. Generally, application of the alignment method requires overcoming substantial mathematical difficulties, and intricate solutions represented by complex expressions are obtained rather often.

(1)

the maximum point of the partial sum Sn(f0) of the trigonometric Fourier series [6] with

i.e., the absolute error value . It should be noted that .

The graph of the partial sum S20( f0 ) of the trigonometric series on the interval [-π,π], which illustrates the presence of the Gibbs phenomenon is presented in Fig. 1.

Fig. 1. Presence of the Gibbs phenomenon

The function Δ(d) is an infinitely large value, as . Such expression as , where [A] is the integral part of the number A, may be taken as d*.

It should be noted that it is not necessary for the Fourier series to converge at each point even on the set of continuous functions C [–π, π], which is commonly known.

The presence of the Gibbs phenomenon leads to extremely negative consequences of the use of the partial sum of a trigonometric series as an approximating function in fields such as radio engineering and signal transmission.

2. DESCRIPTION OF THE METHOD

In order to eliminate the mentioned disadvantages, new methods of approximation of step functions based on the use of trigonometric expressions represented by recursive functions are suggested in the present paper.

For example, consider the step function (1) in more detail. This function is often used as an example of the application of Fourier series, and, therefore, it is convenient to take this function for comparative analysis of a traditional Fourier series expansion and the suggested method.

Expansion of (1) into Fourier series has all the above mentioned disadvantages. In order to eliminate them, it is proposed to approximate the initial step function by a sequence of recursive periodic functions

(2)

Graphs of the initial function (a thickened line) and its five successive approximations for this case are presented in Fig. 2. It can be seen that, even when n values are relatively small in the iterative procedure (2), the graph of the approximating functions approximates the initial function (1) rather well. In addition, approximating functions obtained using the suggested method do not have any of the disadvantages of Fourier series expansion. There is absolutely no sign of the Gibbs phenomenon.

Fig. 2 Graphs of the initial function and its successive approximation

Certain peculiarities of the proposed approximating iterative procedure are to be mentioned.

It should be noted that functions fn(x) and f0(x) are uneven and periodic ones with a period of 2π. Functions fn(x + π/2) and f0(x + π/2) are even and periodic. Therefore, it is sufficient to consider the sequence of approximating functions (2) on the interval [0, π/2].

Let {fn(x)}⊂ L2 [0, π/2] and f0(x) ∈ L2 [0, π / 2]. As (due to the boundedness of functions fn(x)) and (due to the monotonicity of functions fn(x) on the interval [0,π/2]), then, a subsequence converging at each point of [0, π/2] to a certain function f with may be extracted from the sequence {fn(x)} based on Helly’s theorem. The possibility of taking the initial function f0(x) as such function f will be shown below.

Theorem 1. A sequence of functions fn(x) converges to the initial function fo(x), with the convergence being point -by-point, though not uniform.

Theorem 2. The sequence of approximating functions fn (x) converges along the norm towards the initial function f0(x) in Banach L1[0,π/2] and Hilbert spaces of measurable functions L2[0,π/2].

Then, with functions fn (x) and ηn (x) being non -negative terms and limited on the considered interval, we obtain the following in the space L1[0, π/2]:

As .

Similarly it may be proved that the sequence fn(x) converges along the norm towards the function f0 (x) in the space L2[0,π/2].

Thus, the sequence of approximating functions fn (x) in spaces L1[–π, π] and L2[–π, π] fundamental. Whereas, the sequence fn (x) is not fundamental in the space C[–π,π].

The number π/2 was used in the sequence of approximating functions (3) as a constant factor; however, it is possible to take another factor, which may be variable as well. Cosine and