1. Full name:     Nguyen Duy Truong                               2. Sex: Male

3. Date of birth: 11 – 11 – 1980                                      4. Place of birth: Hai Phong

5. Admission decision number: Decision 4982/QĐ-ĐHKHTN, dated 27 November 2013 by the Rector of VNU University of Science.

6. Changes in academic process: Decision 1033/QĐ-ĐHKHTN, dated 25 April 2017  by Rector of Hanoi University of Science on extending the study up to 31 December 2017. Decision 597/QĐ-ĐHKHTN, dated 06 March 2018  by Rector of Hanoi University of Science on extending study to 31 December 2018.

7. Thesis title: Some efficient methods for solving structured nonlinear differential-algebraic equations

8. Major: Applied Mathematics                                       9. Code: 62460112

10. Supervisors: Assoc.Prof.Dr.habil. Vu Hoang Linh     

11. Summary of new findings of the thesis

In this thesis, Runge-Kutta and linear multistep methods for a class of structured nonlinear DAEs have been proposed and investigated.  In addition, we have extended the use of  linear multistep methods with an interpolation and half-explicit Runge-Kutta methods with continuous extension to a class of structured nonlinear DDAEs with a constant delay.

For a class of structured nonlinear DAEs:

+  Algorithm, stability and convergence of numerical methods (Runge-Kutta and linear multistep methods)  have been analyzed.

+ These methods preserve the stability and convergence as if they were applied to ordinary differential equations. For semi-explicit linear problems, half-explicit methods are  much cheaper than implicit methods.

+ Numerical experiments have been presented to illustrate the theoretical results.    

For a class of structured nonlinear DDAEs with a constant delay:

+ We have analyzed and classified the structured nonlinear DDAEs with a constant delay.

+ Algorithm description and convergence analysis for numerical solutions of a class of structured nonlinear DDAEs by the linear multistep methods with interpolation and half-explicit Runge-Kutta methods with continuous extension have been presented.

+ Numerical experiments have been presented to illustrate the theoretical results.

12. Practical applicability, if any: The methods can be used to solve real-life problems which arise in many scientific and engineering areas such as multibody mechanics, electrical circuit, optimal control, chemical reactions, …

13. Further research directions, if any:

+ Algorithms with error control and automatic stepsize selection should be designed.

+ Application of the methods in the computation of Lyapunov exponent and  and Sacker-Sell spectral interval for DAEs.

 + A complete theory and numerical analysis for nonlinear DDAEs with non-constant delay would be an interesting project, as well.

14. Thesis-related publications:

[1] V.H. Linh, N.D. Truong (2018), "Runge-Kutta methods revisited for a class of

structured strangeness-free differential-algebraic equations", Electr. Trans. Num. Anal.,

48: 131-155 (SCIE).

[2] V.H. Linh and N.D. Truong, "Stable numerical solution for a class of structured differential-algebraic equations by linear multistep methods", Acta. Math. Vietnamica., published online 29 January 2019, https://doi.org/10.1007/s40306-018-00310-5, (ESCI/SCOPUS).

[3] V.H. Linh, N.D. Truong and M.V. Bulatov  (2018), "Convergence analysis of linear multistep methods for a class of delay differential-algebraic equations",  Bull. South Ural State Univ., Series Math. Model., Prog & Software,  11 (4): 78-93, (ESCI/SCOPUS).

[4] V.H. Linh and N.D. Truong, "On convergence of continuous Runge-Kutta methods for a class of delay differential-algebraic equations", (submitted for publication).