Summary of doctoral thesis: Development of nonstandard finite difference methods for some classes of differential equations

Although the research direction on NSFD schemes for differential equations have achieved a lot ofresults shown by both quantity and quality of existing research works, real-world situations have always posednew complex problems in both qualitative study and numerical simulation aspects. On the other hand, there aremany differential models that have been established completely in the qualitative aspect but their correspondingdynamically consistent discrete models have not yet been proposed and studied
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Summary of doctoral thesis: Development of nonstandard finite difference methods for some classes of differential equations MINISTRY OF EDUCATION AND VIETNAM ACADEMY TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY Hoang Manh Tuan DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS Major: Applied Mathematics Code: 9 46 01 12 SUMMARY OF DOCTORAL THESIS HANOI - 2021 This thesis has been completed at: Graduate University of Science and Technology – Vietnam Academy of Science and Technology. Supervisor 1: Prof. Dr. Dang Quang A Supervisor 2: Assoc. Prof. Dr. Habil. Vu Hoang Linh Reviewer 1: Reviewer 2: Reviewer 3: The thesis will be defended at the Board of Examiners of Graduate University of Science and Technology – Vietnam Academy of Science and Technology at ............................ on.............................. The thesis can be explored at: - Library of Graduate University of Science and Technology - National Library of Vietnam INTRODUCTION 1. Overview of research situation Many essential phenomena and processes arising in fields of science and technology are mathematically modeled by ODEs of the form: dy(t) y(t0 ) = y0 ∈ Rn ,
= f y(t) , (0.0.1) dt  T where y(t) denotes the vector-function y1 (t), y2 (t), . . . , yn (t) , and the function f satisfies appropriate condi- tions which guarantee that solutions of the problem (0.0.1) exist and are unique. The problem (0.0.1) is called an initial value problem (IVP) or also a Cauchy problem. The problem (0.0.1) has always been playing an essential role in both theory and practice. Theoretically, it is not difficult to prove the existence, uniqueness and continuous dependence on initial data of the solutions of the problem (0.0.1) thanks to the standard methods of mathematical analysis. However, it is very challenging, even impossible, to solve the problem (0.0.1) exactly in general. In common real-world situations, the problem of finding approximate solutions is almost inevitable. Consequently, the study of numerical methods for solving ODEs has become one of the fundamental and practically important research challenges (see, for example, Ascher and Petzold 1998; Burden and Faires 2011; Hairer, Nørsset and Wanner 1993, Hairer and Wanner 1996, Stuart and Humphries 1998). Due to requirements of practice as well as the development of mathematical theory, many numerical methods, typically finite difference methods have been constructed and developed. It is safe to say that the general theory of the finite difference methods for the problem (0.0.1) has been developed thoroughly in many monographs. These methods will be called the standard finite difference (SFD) methods to distinguish them from the nonstandard finite difference (NSFD) schemes that will be presented in the remaining parts. Except for key requirements such as the convergence and stability, numerical schemes must correctly preserve essential properties of corresponding differential equations. In other words, differential models must be transformed into discrete models with the preservation of essential properties. However, in many problems, the SFD schemes revealed a serious drawback called ”numerical instabilities”. To describe this, Mickens, the creator of the concept of NSFD methods, wrote: ”numerical instabilities are an indication that the discrete models are not able to model the correct mathematical properties of the solutions to the differential equations of interest” (Mickens 1994, 2000, 2005, 2012). In a large number of works, Mickens discovered and analyzed numerous examples related to the numerical instabilities occurring when using SFD methods. In 1980, Mickens proposed the concept of NSFD schemes to overcome numerical instabilities. According to the Mickens’ methodology, NSFD schemes are those constructed following a set of basic rules derived from the analysis of the numerical instabilities that occur when using SFD schemes (Mickens 1994, 2000, 2005, 2012). Over the past four decades, the research direction on NSFD schemes has attracted the attention of many researchers in many different aspects and gained a great number of interesting and significant results. All of the works confirmed the usefulness and advantages of NSFD schemes. An advantage of NSFD schemes 1 over standard ones is that they can correctly preserve essential properties (positivity, boundedness, asymptotic stability, periodicity, etc.). In major surveys Mickens (2012) and Patidar (2005, 2016) and several monographs Mickens (1994, 2000, 2005), Mickens and Padidar systematically presented results on NSFD methods in recent decades as well as directions of the development in the future. Nowadays, NSFD methods have been and will continue to be widely used as a powerful and effective approach to solve ODEs, PDEs, delay differential equations (DDEs) and fractional differential equations (FDEs) (see, for instance, Arenas, Gonzalez-Parra and Chen-Charpentier 2016; Garba et al. 2015; Ehrardt and Mickens 2013; Mickens 1994, 2000, 2005, 2012; Modday, Hashim and Momani 2011; Patidar 2005, 2016). 2. The necessity of the research Although the research direction on NSFD schemes for differential eq ...