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EditorialRecent Advances in Numerical Methods and Analysis forNonlinear Differential Equations
Dongfang Li ,1 JinmingWen,2 and Jiwei Zhang3
1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China2Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S3G4, Canada3Beijing Computational Science Research Center, Beijing 100193, China
Correspondence should be addressed to Dongfang Li; [email protected]
Received 26 November 2018; Accepted 26 November 2018; Published 1 January 2019
Copyright © 2019 Dongfang Li et al. is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
e construction and analysis of numerical schemes fornonlinear differential equations are very important. A well-designed numerical scheme not only may save computationalcosts for long-time simulations in the real-world problems,but also can give a better performance in the prediction ofthe mathematical models. erefore, it is highly desirable todevelop effective and efficient numerical schemes and theirnumerical analysis for nonlinear differential equations.
e paper authored by J. H. Al-Smail et al. studies theexistence and uniqueness of weak solutions for the nonlinearparabolic equations and then presents a linearized finiteelement Galerkin method to numerically solve the modelequations. e decay property of solutions is confirmed bythe numerical results.
e paper by H. Qin et al. deals with the stability analysisof additive Runge-Kutta methods for delay-integrodifferen-tial equations arising from some spatially discretized time-dependent partial differential equations. Such equations o enown different stiff terms. In such cases, it is more effective inapplying some split numerical schemes to approximate thesystems. e given results imply that if the additive Runge-Kutta methods are algebraically stable, the perturbationsof the numerical solutions are controlled by the initialperturbations from the system and the methods.
e paper of F. Wu et al. is concerned with the analysisof the two fully discrete numerical schemes for solvingdelay reaction-diffusion equation. Solvability and conver-gence of the fully discrete numerical schemes are studied.Besides, asymptotical stability of the fully discrete schemes
is investigated extensively. Several examples are presented atlast to confirm the theoretical results.
e paper by A. F. Nindjin et al. studies continuousnonlinear economic dynamics with a continuous delay. It isshown that themodel is bounded and admits an attractor unit(set). In addition, the delay term can justify the bifurcation ofan economic model of the stationary growth towards cyclicgrowth. Several numerical results are given to illustrate thegiven results.
e paper of M. A. Khan et al. presents an approximatesolution for somewell-known linear and nonlinear two-pointboundary value problems by using the optimal homotopyasymptotic method. Numerical results are provided to showthe effectiveness and reliability of the proposed method, andsome comparisons are also given to show the advantages ofthe methods.
Conflicts of Interest
e authors declare that they have no conflicts of interest.
Acknowledgments
We are grateful to all the authors who have made a contribu-tion to this special issue.
Dongfang LiJinmingWenJiwei Zhang
HindawiInternational Journal of Differential EquationsVolume 2019, Article ID 3243510, 1 pagehttps://doi.org/10.1155/2019/3243510
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