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A new compact finite difference quasilinearization method for nonlinear evolution partial differential equations

100%
Dlamini P. G., Khumalo M.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1450-1462
EN
This article presents a new method of solving partial differential equations. The method is an improvement of the previously reported compact finite difference quasilinearization method (CFDQLM) which is a combination of compact finite difference schemes and quasilinearization techniques. Previous applications of compact finite difference (FD) schemes when solving parabolic partial differential equations has been solely on discretizing the spatial variables and another numerical technique used to discretize temporal variables. In this work we attempt, for the first time, to use the compact FD schemes in both space and time. This ensures that the rich benefits of the compact FD schemes are carried over to the time variable as well, which improves the overall accuracy of the method. The proposed method is tested on four nonlinear evolution equations. The method produced highly accurate results which are portrayed in tables and graphs.
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Periodic Solutions for Nonlinear Evolution Equations with Non-instantaneous Impulses

88%
Fečkan M., Wang J., Zhou Y.
Nonautonomous Dynamical Systems
|
2014
|
tom 1
|
nr 1
EN
In this paper, we consider periodic solutions for a class of nonlinear evolution equations with non-instantaneous impulses on Banach spaces. By constructing a Poincaré operator, which is a composition of the maps and using the techniques of a priori estimate, we avoid assuming that periodic solution is bounded like in [1-4] and try to present new sufficient conditions on the existence of periodic mild solutions for such problems by utilizing semigroup theory and Leray-Schauder's fixed point theorem. Furthermore, existence of a global compact connected attractor for the Poincaré operator is derived.
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