Piecewise Polynomial Algorithms for the Analysis of Processes in Inhomogeneous Media
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PIECEWISE POLYNOMIAL ALGORITHMS FOR THE ANALYSIS OF PROCESSES IN INHOMOGENEOUS MEDIA V. I. Bilenko,1† K. V. Bozhonok,1‡ S. Yu. Dzyadyk,2 and O. B. Stelya3
UDC 519.622; 517.5
Abstract. The authors propose, theoretically substantiate, and programmatically implement high-precision numerical-analytical algorithms for approximation of problems solutions in inhomogeneous media on the basis of linear polynomial operators.
Keywords: piecewise polynomial approximation, inhomogeneous medium, unsaturation, best approximation, algebraic-nonlinear equations, optimal algorithms, computational optimization, parabolic splines of special kind.
INTRODUCTION The paper considers solution of the problem of increasing the accuracy of the quantitative research (analysis and prognosis) of processes in multicomponent media and inhomogeneous complex engineering objects, as well as finding their dynamic characteristics. This problem was formulated and solved on the basis of difference methods by Sergienko and Deineka, Academicians of the NAS of Ukraine [1, 2], and by their students and followers. The purpose of the study is to construct and theoretically substantiate two complementary algorithms on the basis of the well-known Dzyadyk’s approximation method and parabolic splines for the solution of problems in inhomogeneous media whose models are parabolic equations with initial–boundary-value conditions. Such algorithms have important properties of unsaturation with respect to the accuracy and optimality in the sense of the best polynomial approximation in quadratic and uniform metrics. Importance of further development and application of Dzyadyk’s approximation method [3, 4] is due to the increasing requirements to the three main characteristics of computing algorithms: accuracy, high-speed performance, and information complexity [2, 5] in the solution of modern problems of mathematical and computer simulation. To solve such problems, powerful finite-difference methods, finite element methods, spline functions, integro-interpolation methods, etc. [6–9] are usually used. The main disadvantage of these methods is the phenomenon of saturation (the well-known Favard–Kolmogorov problem in the theory of approximation of functions and saturation in the numerical analysis), which can result in an “explosion” of errors [3, 10, 11]. Note also that these algorithms are convenient for computer implementation in systems of computer algebra [12]. The issues related to applying the developed algorithms to applied problems of environmental protection and adjacent fields, in particular, for information-mathematical modeling and forecasting of the ecological state of ground waters [13] are investigated.
1
National Pedagogical Dragomanov University, Kyiv, Ukraine, †[email protected]; ‡[email protected]. State University of Telecommunications, Kyiv, Ukraine. 3Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2018, pp. 135–141. Original article su
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