Phát triển phương pháp sai phân khác thường giải một số lớp phương trình vi phân

MINISTRY OF EDUCATION AND VIETNAM ACADEMY TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCES AND TECHNOLOGY Hoàng Mạnh Tuấn DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2021 MINISTRY OF EDUCATION AND VIETNAM ACADEMY TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCES AND TECHNOLOGY Hoàng Mạnh Tuấn DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS Speciality: Applied Mathematics Speciality Code: 46 01 12 DOCTOR OF PHILOSOPHY IN MATHEMATICS SUPERVISORS: Prof Dr Đặng Quang Á Assoc Prof Dr Habil Vũ Hoàng Linh HANOI - 2021 BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM HỌC VIỆN KHOA HỌC VÀ CƠNG NGHỆ Hồng Mạnh Tuấn PHÁT TRIỂN PHƯƠNG PHÁP SAI PHÂN KHÁC THƯỜNG GIẢI MỘT SỐ LỚP PHƯƠNG TRÌNH VI PHÂN LUẬN ÁN TIẾN SĨ TOÁN HỌC HÀ NỘI - 2021 BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VIỆT NAM HỌC VIỆN KHOA HỌC VÀ CƠNG NGHỆ Hồng Mạnh Tuấn PHÁT TRIỂN PHƯƠNG PHÁP SAI PHÂN KHÁC THƯỜNG GIẢI MỘT SỐ LỚP PHƯƠNG TRÌNH VI PHÂN Chuyên ngành: Toán ứng dụng Mã số: 46 01 12 LUẬN ÁN TIẾN SĨ TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: GS TS Đặng Quang Á PGS TSKH Vũ Hoàng Linh HÀ NỘI - 2021 Lời cam đoan Luận án hoàn thành Học viện Khoa học Công nghệ, Viện Hàn lâm Khoa học công nghệ Việt Nam hướng dẫn khoa học GS TS Đặng Quang Á PGS TSKH Vũ Hoàng Linh Những kết nghiên cứu trình bày luận án mới, trung thực chưa cơng bố cơng trình khác Các kết công bố chung cán hướng dẫn cho phép sử dụng luận án Hà Nội, tháng 01 năm 2021 Nghiên cứu sinh Hoàng Mạnh Tuấn i Declaration This thesis has been completed at Graduate University of Science and Technology (GUST), Vietnam Academy of Science and Technology (VAST) under the supervision of Prof Dr Đặng Quang Á and Assoc Prof Dr Habil Vũ Hoàng Linh I hereby declare that all the results presented in this thesis are new, original and have never been published fully or partially in any other work The author Hoàng Mạnh Tuấn ii Lời cảm ơn Trước hết, xin bày tỏ lòng biết ơn chân thành sâu sắc tới cán hướng dẫn, GS TS Đặng Quang Á GS TSKH Vũ Hoàng Linh Luận án khơng thể hồn thành khơng có hướng dẫn giúp đỡ tận tình Thầy Tôi vô biết ơn giúp đỡ mà Thầy dành cho không thời gian thực luận án mà suốt thời gian học Đại học Cao học Sự quan tâm giúp đỡ Thầy công việc lẫn sống giúp vượt qua những khó khăn thất vọng để hồn thiện cơng trình nghiên cứu hồn thành luận án Tơi xin gửi lời cảm ơn tới Học viện Khoa học Công nghệ, Viện Hàn lâm Khoa học Công nghệ Việt Nam, nơi học tập, nghiên cứu hoàn thành luận án Luận án hoàn thành cách thuận lợi thời hạn nhờ vào công tác quản lý đào tạo chuyên nghiệp, môi trường học tập nghiên cứu khoa học lý tưởng với giúp đỡ nhiệt tình cán Học viện Tôi xin chân thành cảm ơn Lãnh đạo đồng nghiệp Viện Công nghệ Thông tin, Viện Hàn lâm Khoa học Công nghệ Việt Nam, nơi tơi cơng tác, dàng điều kiện thuận lợi cho suốt nhiều năm qua nói chung thời gian thực luận án nói riêng Tơi xin gửi cảm ơn tới Thầy Cô, anh chị bạn bè đồng nghiệp Seminar "Toán ứng dụng" GS Đặng Quang Á chủ trì, đặc biệt cá nhân TS Nguyễn Cơng Điều, ý kiến sâu sắc, có chất lượng cao mặt học thuật buổi trao đổi chun mơn Những điều giúp tơi hồn thiện tốt cơng trình nghiên cứu Tơi xin chân thành cảm ơn các anh, chị đồng nghiệp Bộ mơn Tốn học, trường ĐH FPT, giúp đỡ động viên suốt trình thực luận án Điều tạo cho tơi nhiều cảm hứng nghiên cứu khoa học thực luận án Đặc biệt, Tôi xin gửi lời biết ơn sâu sắc tới GS TSKH Phạm Kỳ Anh, người Thầy giảng dạy hướng dẫn tận tình tơi suốt thời gian học Đại học Cao học Những giảng thầy mơn học Giải tích số Tốn ứng dụng từ thời Đại học có ảnh hưởng to lớn tới lựa chọn sau đường iii nghiên cứu khoa học Đặc biệt, Thầy có nhiều góp ý sâu sắc quan trọng giúp cho luận án hoàn thiện tốt Tôi xin gửi lời cảm ơn chân thành tới GS R E Mickens (Clark Atlanta University), GS M Ehrhardt (Bergische Universitat Wuppertal), GS A J Arenas (Universidad de Córdoba), GS J Cresson (Université de Pau et des Pays de l’Adour) nhiều đồng nghiệp nước khác dành nhiều thời gian đọc cho tơi nhiều ý kiến giá trị nội dung lẫn hình thức trình bày luận án Tơi xin chân thành cảm nhiều Giáo sư, Thầy Cô nhiều bạn bè đồng nghiệp khác dành nhiều thời gian đọc cho nhiều ý kiến giá trị hình thức trình bày luận án Tơi xin gửi lời cảm ơn chân thành tới Ths Đặng Quang Long (Viện CNTT) góp ý giá trị quan trọng cho nội dung hình thức trình bày luận án Tôi xin gửi lời cảm ơn tới tất bạn bè đồng nghiệp, người dành cho nhiều quan tâm động viên sống lẫn nghiên cứu khoa học Cuối cùng, luận án khơng thể hồn thành khơng có giúp đỡ, động viên khích lệ mặt gia đình Tơi khơng thể diễn đạt hết lời biết ơn gia đình Với tất lịng biết ơn sâu sắc, luận án nói riêng tất điều tốt đẹp mà cố gắng thực để gửi tới Bố Mẹ, vợ con, anh, chị, em người thân gia đình, người với yêu thương, đức kiên nhẫn lịng vị tha khích lệ động viên theo đuổi đường nghiên cứu khoa học suốt năm qua Hà Nội, tháng 01 năm 2021 Nghiên cứu sinh Hoàng Mạnh Tuấn iv Acknowledgments Firstly, I would like to thank my two supervisors Prof Dr Habil Vũ Hoàng Linh and especially Prof Dr Đặng Quang Á for the continuous support of my PhD study and related research; for their patience, motivation and immense knowledge Without their help I could not have overcome the difficulties in research and study The wonderful research environment of the Graduate University of Sciences and Technology, Vietnam Academy of Science and Technology, and the excellence of its staff have helped me to complete this work within the schedule I would like to thank all the staff at the Graduate University of Sciences and Technology for their help and support during the years of my PhD studies I would like to thank my big family for their endless love and unconditional support Last but not least, I would like to thank my colleagues and many other people beside me for their love, motivation and constant guidance Thanks all for your encouragement! The author Hoàng Mạnh Tuấn v List of notations and abbreviations N The set of natural numbers N+ The set of non-negative nature numbers R The set of real numbers R+ The set of non-negative real numbers Rn Real coordinate space of n-dimension Rn+ The set of all the n-tuples with non-negative real numbers σ(A) The set of the eigenvalues of the matrix A |z| The modulus of the complex number z x The norm of the vector x y(t), ˙ y (t), dy(t)/dt The first derivative of the function y(t) DDE Delay differential equation EEFD Explicit exact finite difference EFD Exact finite difference ENRK Explicit nonstandard Runge-Kutta ESRK Explicit standard Runge-Kutta FD Finite difference FDE Fractional differential equation GAS Global asymptotic stability/Globally asymptotically stable IEFD Implicit exact finite difference IVP Initial value problem HBV Hepatitis B virus LAS Local asymptotic stability/Locally asymptotically stable NSFD Nonstandard finite difference ODE Ordinary differential equation PDE Partial differential equation RK2 The second order Runge-Kutta method RK4 The classical four stage Runge-Kutta method SFD Standard finite difference w.r.t with respect to T r(J) The trace of the matrix J vi GENERAL CONCLUSIONS In this thesis, we have successfully developed the Mickens’ methodology to construct nonstandard finite difference (NSFD) methods for solving some important classes of differential equations arising in fields of science and technology The proposed NSFD schemes are not only dynamically consistent with the differential equation models, but also easy to be implemented; furthermore, they can be used to solve a large class of mathematical problems in both theory and practice The validity of the theoretical results and the superiority of the NSFD schemes have been confirmed and supported by many numerical simulations The results have indicated that there is a good agreement between the theoretical aspect and experimental one In the first part, we have successfully constructed NSFD schemes for some mathematical models described by systems of ODEs including two metapopulation models, one predator-prey model and two computer virus propagation models It is worth noting that all of the models possess at least one of the following characteristics: (i) having large dimensions (ii) having non-hyberbolic equilibrium points (iii) having the GAS Firstly, we have investigated the GAS of the constructed NSFD schemes for the metapopulation model formulated in [97] by using the standard techniques of mathematical analysis Secondly, we have used the Lyapunov stability theorem to study the GAS of proposed NSFD schemes for the computer virus propagation model and the general predator-prey model constructed in [12] and [105], respectively Lastly, we have proposed two novel approaches to establish the stability properties of the NSFD schemes for the metapopulation model and the propagation model of computer viruses formulated in [98] and [16], respectively The first approach is based on the extension of the classical Lyapunov stability theorem, and the second one is based on the Lyapunov stability theorem and its extensions in combination with a theorem on the GAS of discrete-time nonlinear cascade systems Both approaches lead the study of the stability of the proposed NSFD schemes to the study of the stability of 158 discrete models with smaller dimension, and therefore, complicated calculations and transforms are limited significantly In the second part, we have constructed EFD schemes and high order NSFD schemes for a class of general dynamical systems based on the general Runge-Kutta methods The obtained results can be considered as a generalization of the results formulated in [28, 29, 42] Firstly, implicit and explicit EDS schemes for systems of three linear ODEs with constant coefficients are constructed Importantly, the obtained results not only answer the open question posted by Roeger [40] but also can be extended to design EFD schemes for general n-dimensional systems of linear ODEs with constant coefficients Next, we have constructed and analyzed high order ENRK methods preserving two important properties of general autonomous dynamical systems, namely, the positivity and LAS The main result resolved the contradiction between the dynamics consistency and high order of accuracy of NSFD schemes Additionally, two important applications of the constructed ENRK methods to the predator-prey model and the vaccination model are also presented In the near future, the established results in this thesis will be developed to construct highly effective NSFD schemes for PDEs, DDEs, FDEs and stochastic differential equations Also, we intent to study the combination of the Mickens’ methodology and other existing approaches to create new numerical methods with high performance for both differential equations and integro-differential equations 159 THE LIST OF THE WORKS OF THE AUTHOR RELATED TO THE THESIS [A1] Quang A Dang, Manh Tuan Hoang, Dynamically consistent discrete metapopulation model, Journal of Difference Equations and Applications, 2016, 22, 1325-1349, (SCI-E) [A2] Quang A Dang, Manh Tuan Hoang, Lyapunov direct method for investigating stability of nonstandard finite difference schemes for metapopulation models, Journal of Difference Equations and Applications, 2018, 24, 15-47, (SCI-E) [A3] Quang A Dang, Manh Tuan Hoang, Complete global stability of a metapopulation model and its dynamically consistent discrete models, Qualitative Theory of Dynamical Systems, 2019, 18, 461-475, (SCI-E) [A4] Quang A Dang, Manh Tuan Hoang, Numerical dynamics of nonstandard finite difference schemes for a computer virus propagation model, International Journal of Dynamics and Control, 2020, 8, 772-778, (SCOPUS) [A5] Quang A Dang, Manh Tuan Hoang, Nonstandard finite difference schemes for a general predator-prey system, Journal of Computational Science, 2019, 36, 101015, (SCI-E) [A6] Quang A Dang, Manh Tuan Hoang, Positivity and global stability preserving NSFD schemes for a mixing propagation model of computer viruses, Journal of Computational and Applied Mathematics 2020, 374, 112753, (SCI) [A7] Manh Tuan Hoang, On the global asymptotic stability of a predator-prey model with Crowley-Martin function and stage structure for prey, Journal of Applied Mathematics and Computing, 2020, 64, 765-780, (SCI-E) [A8] Quang A Dang, Manh Tuan Hoang, Exact finite difference schemes for three-dimensional linear systems with constant coefficients, Vietnam Journal of Mathematics, 2018, 46, 471-492, (ESCI, SCOPUS) 160 [A9] Quang A Dang, Manh Tuan Hoang, Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems, International Journal of Computer Mathematics, 2020, 97, 2036-2054, (SCI-E) 161 Bibliography R P Agarwal, An Introduction to Ordinary Differential Equations, Springer, 2000 L.J.S Allen, An Introduction to Mathematical Biology, Prentice Hall, 2007, 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PHÁT TRIỂN PHƯƠNG PHÁP SAI PHÂN KHÁC THƯỜNG GIẢI MỘT SỐ LỚP PHƯƠNG TRÌNH VI PHÂN LUẬN ÁN TIẾN SĨ TOÁN HỌC HÀ NỘI - 2021 BỘ GIÁO DỤC VÀ ĐÀO TẠO VI? ??N HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VI? ??T NAM HỌC VI? ??N... VI? ??N KHOA HỌC VÀ CƠNG NGHỆ Hồng Mạnh Tuấn PHÁT TRIỂN PHƯƠNG PHÁP SAI PHÂN KHÁC THƯỜNG GIẢI MỘT SỐ LỚP PHƯƠNG TRÌNH VI PHÂN Chuyên ngành: Toán ứng dụng Mã số: 46 01 12 LUẬN ÁN TIẾN SĨ TOÁN HỌC NGƯỜI... SUPERVISORS: Prof Dr Đặng Quang Á Assoc Prof Dr Habil Vũ Hoàng Linh HANOI - 2021 BỘ GIÁO DỤC VÀ ĐÀO TẠO VI? ??N HÀN LÂM KHOA HỌC VÀ CÔNG NGHỆ VI? ??T NAM HỌC VI? ??N KHOA HỌC VÀ CƠNG NGHỆ Hồng Mạnh Tuấn PHÁT