TAILIEUCHUNG - 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 | 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 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 exist and are unique. The problem is called an initial value problem IVP or also a Cauchy problem. The problem 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 thanks to the standard methods of mathematical analysis. However it is very challenging even impossible to solve the problem 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 .

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• Mathemati al analysis
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