Title: Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions
Authors: Maria S. Filipkovska
Addresses: Department of Mathematical Physics, B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine; Department of Higher Mathematics and Computer Sciences, V.N. Karazin Kharkiv National University, 4 Svobody Sq., Kharkiv, 61022, Ukraine
Abstract: Two combined numerical methods for solving implicit semilinear differential equations are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the equation. Also, the obtained methods enable to compute approximate solutions of the equations on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of the corresponding mathematical models. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations.
Keywords: implicit differential equation; differential-algebraic equation; DAE; combined method; regular pencil; spectral projector; global dynamics.
International Journal of Computing Science and Mathematics, 2022 Vol.15 No.1, pp.1 - 29
Received: 03 Dec 2018
Accepted: 04 Jun 2019
Published online: 11 Apr 2022 *