Construction of interpolation method for numerical solution of the Cauchy's problem
DOI:
https://doi.org/10.15587/1729-4061.2017.108327Keywords:
Newton's minorant, differential equations, Cauchy's problem, Newton's diagram, convex functionAbstract
An interpolation numerical method is developed in order to solve the Cauchy's problem for ordinary first order differential equations using the apparatus of non-classical minorants and diagrams of Newton's functions, assigned in a tabular form. We have proven computational stability of the method, that is, an error of the initial data is not piled up. It is also shown that the method possesses a second order of accuracy and in the case of a convex function produces more accurate results than the Euler's method. The advantages also include simplicity and visual clarity of the method. Given this, it could gain widespread use in many areas, in particular mathematics, physics and mechanics. We also give an example of solving the Cauchy's problem applying the new method, the Euler's method, and the Runge-Kutta fourth order method, with the results compared. The proposed method does not require solving the systems of linear algebraic equations because we do not employ the Bernstein polynomials, and it is not required to superimpose additional conditions, in contrast to the method that applies the Haar functionsReferences
- Verzhbitskiy, V. M. (2001). Chislennyye metody (matematicheskiy analiz i obyknovennyye differentsialnyye uravneniya). Moscow: Vyschaya shkola, 382.
- Süli, E., Mayers, D. (2003). An Introduction to Numerical Analysis. Cambridge University Press, 435.
- Zadachyn, V. M. (2014). Chyselni metody. Kharkiv: Vyd. KhNEU im. S. Kuznetsya, 180.
- Abdulkawi, M. (2015). Solution of Cauchy type singular integral equations of the first kind by using differential transform method. Applied Mathematical Modelling, 39 (8), 2107–2118. doi: 10.1016/j.apm.2014.10.003
- Setia, A. (2014). Numerical solution of various cases of Cauchy type singular integral equation. Applied Mathematics and Computation, 230, 200–207. doi: 10.1016/j.amc.2013.12.114
- De Bonis, M. C., Laurita, C. (2012). Numerical solution of systems of Cauchy singular integral equations with constant coefficients. Applied Mathematics and Computation, 219 (4), 1391–1410. doi: 10.1016/j.amc.2012.08.022
- Lukomskii, D. S., Lukomskii, S. F., Terekhin, P. A. (2016). Solution of Cauchy Problem for Equation First Order Via Haar Functions. Izvestiya of Saratov University. New Series. Series: Mathematics. Mechanics. Informatics, 16 (2), 151–159. doi: 10.18500/1816-9791-2016-16-2-151-159
- Lyashenko, M. Ya.‚ Holovan, M. S. (1996). Chysel'ni metody. Kyiv: "Lybid'"‚ 285.
- Varenych, I. I. (2008). Vyshcha matematyka: matematychnyy analiz, dyferentsialni rivnyannya. Kyiv: DiaSoft, 267.
- Bihun, R., Tsehelyk, G. (2017). Construction of a numerical method for finding the zeros of both smooth and nonsmooth functions. Eastern-European Journal of Enterprise Technologies, 2 (4 (86)), 58–64. doi: 10.15587/1729-4061.2017.99273
- Bihun, R. R., Tsehelyk, G. G. (2017). Formulas of minorant type to approximate computing definite integrals. Scientific light, 1 (7), 76–79.
- Bihun, R. R., Tsehelyk, G. G. (2015). Numerical Method for Finding All Points of Extremum of Random as Smooth and NonSmooth Functions of One Variable. Global Journal of Science Frontier Research: F Mathematics & Decision Sciences, 15 (2), 87–93.
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