luận án tiến sĩ phát triển phương pháp sai phân khác thường giải một số lớp phương trình vi phân

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 mathem

MINISTRY OF EDUCATION AND VIETNAM ACADEMY

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

GRADUATE UNIVERSITY OF SCIENCES AND TECHNOLOGY

Hoàng Mạnh Tuấn

DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS

FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS

Speciality: Applied MathematicsSpeciality Code: 9 46 01 12

DOCTOR OF PHILOSOPHY IN MATHEMATICS

SUPERVISORS:

1 Prof Dr Đặng Quang Á

2 Assoc Prof Dr Habil Vũ Hoàng Linh

BỘ GIÁO DỤC VÀ ĐÀO TẠO VIỆN HÀN LÂM KHOA HỌC

VÀ CÔNG NGHỆ VIỆT NAM

HỌC VIỆN KHOA HỌC VÀ CÔNG NGHỆ

Hoàng 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 CÔNG NGHỆ

LUẬN ÁN TIẾN SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC:

Lời cam đoan

Luận án này được hoàn thành tại Học viện Khoa học và Công nghệ,Viện Hàn lâm Khoa học và công nghệ Việt Nam dưới sự hướng dẫn khoahọc của GS TS Đặng Quang Á và PGS TSKH Vũ Hoàng Linh Những kếtquả nghiên cứu được trình bày trong luận án là mới, trung thực và chưatừng được ai công bố trong bất kỳ công trình nào khác Các kết quả đượccông bố chung đã được cán bộ hướng dẫn cho phép sử dụng trong luậnán

Hà Nội, tháng 01 năm 2021Nghiên cứu sinh

Hoàng Mạnh Tuấn

i

This thesis has been completed at Graduate University of Science andTechnology (GUST), Vietnam Academy of Science and Technology (VAST)under the supervision of Prof Dr Đặng Quang Á and Assoc Prof Dr Habil VũHoàng Linh I hereby declare that all the results presented in this thesis arenew, original and have never been published fully or partially in any other work

The author

Hoàng Mạnh Tuấn

ii

Lời cảm ơn

Trước hết, tôi xin bày tỏ lòng biết ơn chân thành và sâu sắc tới các cán bộhướng dẫn, GS TS Đặng Quang Á và GS TSKH Vũ Hoàng Linh Luận ánnày sẽ không thể được hoàn thành nếu không có sự hướng dẫn và giúp đỡ tậntình của các Thầy Tôi vô cùng biết ơn những giúp đỡ mà các Thầy đã dànhcho tôi không chỉ trong thời gian thực hiện luận án mà còn cả trong suốt thờigian học Đại học và Cao học Sự quan tâm và giúp đỡ của các Thầy trong cảcông việc lẫn cuộc sống đã giúp tôi vượt qua được những những khó khăn vàthất vọng để hoàn thiện các công trình nghiên cứu và hoàn thành luận án

Tôi xin gửi lời cảm ơn tới Học viện Khoa học và Công nghệ, Viện Hàn lâmKhoa học và Công nghệ Việt Nam, nơi tôi học tập, nghiên cứu và hoàn thành luận

án Luận án này đã được hoàn thành một cách thuận lợi và đúng thời hạn là nhờvào công tác quản lý đào tạo chuyên nghiệp, môi trường học tập và nghiên cứukhoa học lý tưởng cùng với sự giúp đỡ nhiệt tình của các cán bộ Học viện

Tôi xin chân thành cảm ơn Lãnh đạo cùng các đồng nghiệp ở ViệnCông nghệ Thông tin, Viện Hàn lâm Khoa học và Công nghệ Việt Nam, nơitôi đang công tác, vì đã dàng mọi điều kiện thuận lợi nhất cho tôi trong suốtnhiều năm qua nói chung và thời gian thực hiện luận án nói riêng

Tôi cũng xin được gửi cảm ơn tới các Thầy Cô, các anh chị và bạn bèđồng nghiệp trong Seminar "Toán ứng dụng" do GS Đặng Quang Á chủ trì,đặc biệt là cá nhân TS Nguyễn Công Điều, vì những ý kiến sâu sắc, có chấtlượng cao về mặt học thuật trong các buổi trao đổi chuyên môn Những điều

đó đã giúp tôi hoàn thiện tốt hơn các công trình nghiên cứu của mình

Tôi cũng xin chân thành cảm ơn các các anh, chị và đồng nghiệp ở

Bộ môn Toán học, trường ĐH FPT, vì những giúp đỡ và động viên trongsuốt quá trình thực hiện luận án Điều đó đã tạo cho tôi nhiều cảm hứngtrong nghiên cứu khoa học và thực hiện luận án

Đặc biệt, Tôi cũng xin gửi lời biết ơn sâu sắc tới GS TSKH Phạm Kỳ Anh, người Thầy đã giảng dạy và hướng dẫn tận tình tôi trong suốt thời gian học Đại học và Cao học Những bài giảng của thầy về môn học Giải tích số và Toán ứng dụng từ thời Đại học đã

có ảnh hưởng to lớn tới những lựa chọn sau này của tôi trên con đường

nghiên cứu khoa học Đặc biệt, Thầy cũng có rất nhiều góp ý sâu sắc vàquan trọng giúp cho luận án này được hoàn thiện tốt hơn.

Tôi cũng xin gửi lời cảm ơn chân thành tới các 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) cùng nhiều đồng nghiệp nước ngoài khác vì đã dành nhiều thời gian đọc và cho tôi nhiều ý kiến giá trị về cả nội dung lẫn hình thức trình bày của luận án.

Tôi xin chân thành cảm nhiều Giáo sư, Thầy Cô cùng nhiều bạn bèđồng nghiệp khác vì đã dành nhiều thời gian đọc và cho tôi nhiều ý kiến giátrị về hình thức trình bày của luận án

Tôi xin gửi lời cảm ơn chân thành tới Ths Đặng Quang Long (Viện CNTT) vì những góp ý giá trị và quan trọng cho nội dung và hình thức trình bày của luận án.

Tôi xin gửi lời cảm ơn tới tất cả bạn bè và đồng nghiệp, những người

đã dành cho tôi nhiều sự quan tâm và động viên trong cuộc sống lẫn trongnghiên cứu khoa học

Cuối cùng, luận án này sẽ không thể được hoàn thành nếu như không có sự giúp đỡ, động viên và khích lệ về mọi mặt của gia đình Tôi không thể diễn đạt được hết bằng lời sự biết ơn của mình đối với gia đình Với tất cả lòng biết ơn sâu sắc, luận

án này nói riêng cùng tất cả những điều tốt đẹp mà tôi đã và đang cố gắng thực hiện

là để gửi tới Bố Mẹ, vợ con, các anh, chị, em và những người thân trong gia đình, những người với sự yêu thương, đức kiên nhẫn và lòng vị tha đã khích lệ và động viên tôi theo đuổi con đường nghiên cứu khoa học trong suốt những năm qua.

Hà Nội, tháng 01 năm 2021Nghiên cứu sinh

Hoàng Mạnh Tuấn

iv

Firstly, I would like to thank my two supervisors Prof Dr Habil Vũ Hoàng Linhand especially Prof Dr Đặng Quang Á for the continuous support of my PhDstudy 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 Sciencesand Technology, Vietnam Academy of Science and Technology, and theexcellence of its staff have helped me to complete this work within the schedule Iwould like to thank all the staff at the Graduate University of Sciences andTechnology for their help and support during the years of my PhD studies

I would like to thank my big family for their endless love andunconditional support

Last but not least, I would like to thank my colleagues and many otherpeople beside me for their love, motivation and constant guidance

Thanks all for your encouragement!

The author

Hoàng Mạnh Tuấn

List of notations and abbreviations

The set of non-negative real numbersReal coordinate space of n-dimensionThe set of all the n-tuples with non-negative real numbersThe set of the eigenvalues of the matrix A

The modulus of the complex number z

The norm of the vector x

The first derivative of the function y(t)Delay differential equation

Explicit exact finite differenceExact finite difference

Explicit nonstandard Runge-KuttaExplicit standard Runge-KuttaFinite difference

Fractional differential equationGlobal asymptotic stability/Globally asymptotically stableImplicit exact finite difference

Initial value problemHepatitis B virus

Nonstandard finite differenceOrdinary differential equationPartial differential equationThe second order Runge-Kutta methodThe classical four stage Runge-Kutta methodStandard finite difference

with respect toThe trace of the matrix J

vi

List of Figures

2.1 The RK4 scheme with h = 6:5, x(0) = 0:1; y(0) = 0:8 46

2.2 The RK2 scheme with h = 5, x(0) = 0:1; y(0) = 0:4 46

2.3 The explicit Euler scheme with h = 5, x(0) = 0:4; y(0) = 0:2 47

2.4 The solutions generated by the scheme (2.1.5)-(i) for h = 10 48

2.5 The solutions computed by the scheme (2.1.5)-(ii) for h = 10

49 2.6 The solutions computed by the scheme (2.1.5)-(iii) for h = 10 49

2.7 The numerical solutions computed by the scheme (2.1.5)-(i) for h = 10 50 2.8 The numerical solutions computed by the scheme (2.1.5)-(ii) for h = 10 51

2.9 The numerical solutions computed by the explicit nonstandard Euler’s scheme (2.1.5)-(iii) for h = 10 51

2.10 Solutions obtained by the RK4 method with (I(0); S(0); L(0); R(0)) = (0:25; 0:1; 0:2; 0:45) and h = 2 64

2.11 Solutions obtained by the Euler method with (I(0); S(0); L(0); R(0)) = (0:25; 0:1; 0:2; 0:45) and h = 1:6 64

2.12 Solutions generated by the scheme (2.2.11) with (I(0); S(0); L(0); R(0)) = (0:25; 0:1; 0:2; 0:45) and h = 5 in Example 2.3 65

2.13 Solutions obtained by the scheme (2.2.11) with (I(0); S(0); L(0); R(0)) = (0:25; 0:1; 0:2; 0:45) and h = 5 in Example 2.3 66

2.14 Graphs of the functions i(t) 67

2.15 Numerical solutions obtained by the scheme (2.2.11) with h = 1 and ’ = (1 e 1:1h)=1:1 in Example 2.4 68

2.16 Numerical solutions obtained by the scheme (2.2.11) with h = 5 and ’ = (1 e 2:5h)=2:5 in Example 2.4 69 2.17 Numerical solutions obtained by the scheme (2.2.31) with h = 0:1 in Example 2.5 74 2.18 Numerical solutions obtained by the scheme (2.2.31) with h = 0:1 in Example 2.6 75

2.19 Numerical solutions obtained by the Euler scheme, RK4 scheme and

NFSD scheme (2.2.31) in Example 2.7 (t 2 [0; 2100]) 762.20 Phase portrait obtained by the scheme (2.3.7) for h = 0:01 in Example 2.8 842.21 Phase portrait obtained by the scheme (2.3.7) for h = 0:01 in Example 2.9 852.22 Numerical solutions (Lk; Bk; Sk) obtained by the RK4 scheme for

h = 2000=812 in Example 2.10 862.23 Numerical solutions (Lk; Bk; Sk) obtained by the Euler scheme for

h = 1:75 in Example 2.10 862.24 Numerical solutions (Lk; Bk; Sk) obtained by the NSFD schemes for

h = 2:5 in Example 2.10 872.25 The x-component obtained by the explicit Euler scheme for (x0; y0) =

(100; 160), h = 1:111 after 180 iterations 952.26 The phase portrait obtained by the explicit Euler scheme for (x0; y0) =

(100; 160), h = 1:111 after 180 iterations 962.27 The x-component obtained by the RK4 scheme for (x0; y0) = (100; 160),

h = 1:429 after 140 iterations 962.28 The phase potrait obtained by the RK4 scheme for (x0; y0) = (100; 160),

h = 1:429 after 140 iterations 972.29 The phase portrait obtained by the scheme (2.4.3)-(i) for h = 2:5,

t 2 [0; 2000] 982.30 The phase portrait obtained by the scheme (2.4.3)-(ii) for h = 2:5,

t

2 [0; 2000] and P = (0; 0:4406) 9922.31 The phase portrait obtained by the scheme (2.4.3)-(iii) for h = 2:5,

t

2 [0; 2000] and P = (0; 0:4406) 9922.32 The phase portrait obtained by the scheme (2.4.3)-(iv) for h = 2:5,

t

2 [0; 2000] and P = (0:7575; 0:4422) 10032.33 The phase portrait obtained by the scheme (2.4.3)-(v) for h = 2:5,

t

2 [0; 2000] and P = (39:996; 0:0143) 10032.34 The phase portrait obtained by the scheme (2.4.3)-(vi) for h = 2:5,

t

2 [0; 2000] and P = (0:8696; 0) 1011

viii

2.35 The numerical solutions obtained by the numerical schemes in ple 2.12 102

Exam-2.36 Computational time of the numerical schemes in seconds with h = 0:8

in Example 2.12 1042.37 Numerical solutions obtained the ode45 and NSFD scheme The ode45requires 2236 grid points with hmin = 0:0230 and hmax = 0:0550

and the computational time is 0:0753 seconds NSFD scheme 3-(i) use

’(h) = h and h = 1, the computational time is 1:0330e 04 seconds 1052.38 The required step sizes for the ode45 1062.39 The numerical solutions (S-components) obtained by the RK4 schemewith t 2 [0; 145] and h = 1:45, the Euler scheme with t 2 [0; 147]

and h = 1:05 and NSFD scheme (2.5.3) with t 2 [0; 150], ’(h) =

1 e 2h =2 and h = 2 in Case (i) 117 2.40 The numerical solutions (I-components) obtained by the RK4 scheme

with t 2 [0; 145] and h = 1:45, the Euler scheme with t 2 [0; 147]

and h = 1:05 and NSFD scheme (2.5.3) with t 2 [0; 150], ’(h) =

1 e 2h =2, and h = 2 in Case (i) 118 2.41 The numerical solutions (C-components) obtained by the RK4 scheme

with t 2 [0; 145] and h = 1:45, the Euler scheme with t 2 [0; 147]

and h = 1:05 and NSFD scheme (2.5.3) with t 2 [0; 150], ’(h) =

1 e 2h =2, and h = 2 in Case (i) 119 2.42 The numerical solutions obtained by the RK4 scheme with t 2 [0; 1450]

and h = 1:45 in Case (i) 1192.43 Phase portrait obtained by the scheme (2.5.3) with t 2 [0; 50] in Case (ii) 1202.44 Phase portrait obtained by the scheme (2.5.3) with t 2 [0; 50] in Case (iii) 1212.45 Phase portrait obtained by the scheme (2.5.3) with t 2 [0; 50] in Case (iv) 1212.46 Phase portrait obtained by the scheme (2.5.3) with t 2 [0; 500] in Case (v) 122

3.1 The exact solution and the solution generated by the EFD scheme 1333.2 Exact solutions and Exact difference scheme 134

3.3 Graphs of the functions ’i(h) in two cases of the paramerters.

In the upper figure: ’ = 1, ’1 = 1 e h, ’2 = he 0:12h4 , ’3 = (1 e

h3 )’1 + e h3 ’2 In the lower figure: ’ = 1=1:2, ’1 = (1

e 1:2h)=1:2, ’2 = he 0:2h5 , ’3 = (1 e h4 )’1 + e h4 ’2 149 3.4 Phase planes for the model (3.3.1) with some different inital data

obtained by ENRK54 method with ’3(h) and h = 4 154 3.5 Phase portrait for the vaccination model with some different initial

data obtained by ENRK54 method for ’3(h) = e h6 he 0:5h4 + (1

1:6h)=1:6 and h = 2 156

x

e h6 )(1 e

List of Tables

1.1 The coefficients of an ERK method 30

1.2 Some popular ERK methods .31

1.3 Number of order conditions 31

1.4 Some examples of implicit R-K methods 32

1.5 EFD schemes and SFD schemes for some ODEs 38

2.1 The preserved properties of the difference schemes 45

2.2 The sufficient conditions for dynamic consistency 94

2.3 The errors of the numerical schemes 102

2.4 The time of the schemes in seconds 103

2.5 The dynamical properties of the NSFD scheme (2.5.3) under the con-dition (2.5.4) 115

2.6 Parameters in numerical simulations 116

3.1 Error of the methods 137

3.2 Error of the methods 139

3.3 The values opti and the denominator functions ’i(h) (i = 1; 2; 3) of the ENRK methods 151

3.4 The errors and rates of ENRK1 methods 151

3.5 The errors and rates of ENRK2 methods 152

3.6 The errors and rates of ENRK43 methods 152

3.7 The errors and rates of ENRK54 methods 152

3.8 The errors and rates of ENRK4 methods 152

3.9 The errors and rates of the Wood and Kojouharov methods 155

3.10 Positivity and elementary stability thresholds for ENRK 156

Lời cam đoan i

Declaration ii

Lời cảm ơn . iii

Acknowledgments v

List of notations and abbreviations vi

List of Figures vi

List of Tables xi

INTRODUCTION 1

Chapter 1 PRELIMINARIES

18 1.1 Continuous-time dynamical systems 18

1.1.1 Initial value problems

18 1.1.2 Stability theory of continuous-time dynamical systems 20

1.2 Discrete-time dynamical systems

24 1.2.1 Difference equations 24

1.2.2 Stability theory of discrete-time dynamical systems

25 1.3 Runge-Kutta methods for solving ODEs 29

1.3.1 Explicit Runge-Kutta methods

29 1.3.2 Implicit Runge-Kutta methods

31 1.3.3 Positivity of Runge-Kutta methods 33

1.4 Nonstandard finite difference methods 36

1.4.1 Exact finite difference schemes

36 1.4.2 Nonstandard finite difference schemes

38 Chapter 2 NONSTANDARD FINITE DIFFERENCE SCHEMES FOR SOME CLASSES OF ORDINARY DIFFERENTIAL EQUATIONS 40

2.1 Dynamically consistent NSFD schemes for a metapopulation model

402.1.1 Dynamical properties of the metapopulation model 41

2.1.2 The construction of NSFD schemes

42 2.1.3 Numerical experiments 46 2.2 A novel approach for studying stability of NSFD schemes for two metapopula-tion models

52 2.2.1 Complete global stability of the Amarasekare and Possingham’s metapop-ulation model

53 2.2.2 Semi-implicit NSFD schemes for metapopulation model (2.2.1)

56 2.2.3 Explicit NSFD schemes for metapopulation model (2.2.1) 69 2.2.4 An improvement to the stability analysis of NSFD schemes for the metapopulation model (2.2.1) 76 2.3 Numerical dynamics of NSFD schemes for a computer virus propagation model 79 2.3.1 Dynamics of a computer virus model with graded cure rates 79

2.3.2 Nonstandard finite difference schemes for the full model 81

2.3.3 Numerical simulation 84

2.4 NSFD schemes for a general predator-prey model 87

2.4.1 Continuous model and its properties 88

2.4.2 Construction of NSFD scheme

90 2.4.3 Stability analysis

91 2.4.4 Dynamically consistent NSFD schemes

93 2.4.5 Numerical simulation 94

2.5 A novel approach for studying global stability of NSFD schemes for a mixing propagation model of computer viruses 106

2.5.1 Mathematical model and its dynamics 107

2.5.2 Positive NSFD schemes for Model (2.5.1) 109 2.5.3 GAS analysis for NSFD schemes and dynamically consistent NSFD schemes 110 2.5.4 Numerical simulations 115

xiii

2.5.5 A note on the global asymptotic stability of a predator-prey model

122 2.6 Conclusions

123

Chapter 3 HIGH ORDER NONSTANDARD FINITE DIFFERENCE SCHEMES

FOR SOME CLASSES OF GENERAL AUTONOMOUS CALSYSTEMS

129 3.1.3 Explicit EFD schemes

131 3.1.4 Perturbation analysis

132 3.1.5 Numerical simulations

1453.2.3 The choice of the denominator function 1463.3 Some applications of the ENRK methods 1493.3.1 ENRK methods for a predator-prey system 149

3.3.2 ENRK methods for a vaccination model with multiple endemic states 155

3.4 Conclusions

157

GENERALCONCLUSIONS 158

THE LIST OF THE WORKS OF THE AUTHOR RELATED TO THETHESIS 160 BIBLIOGRAPHY 162

1 Overview of research situation

Many essential phenomena and processes arising in real-worldsituations are mathematically modeled by ODEs of the form:

dy(t)

= f y(t) ; y(t0) = y0 2 Rn;dt

where y(t) denotes the vector-function y1(t); y2(t); : : : ; yn(t) T , and the function

f satisfies appropriate conditions which guarantee that solutions of the problem(0.0.1) exist and are unique (see, for example, [1–10]) The problem (0.0.1) is

called an initial value problem (IVP) or also a Cauchy problem.

Problem (0.0.1) has always been playing an essential role in both theoryand practice By using appropriate definitions for the right-hand side function f, wecan obtain a large class of essential mathematical models in real-world situations,for instance, the Logistic differential equation, the classical Lotka-Volterra models,predator-prey models, epidemic models, vaccination models, biological systems[2, 4, 5, 7, 10], computer virus propagation models [11–16], etc The study andanalysis of the problem (0.0.1) have become one of the prominent and mostimportant topics in both theoretical and applied mathematics over the past severaldecades This topic has attracted the attention of mathematicians and engineers invarious aspects, such as, the existence and uniqueness of solutions, qualitativestudy for solutions, asymptotic stability properties, methods of finding solutions and

so on (see [1, 3, 6, 8, 9, 17] and references therein) It is safe to say that most

of the general theory on the qualitative study for the problem (0.0.1) has beendeveloped thoroughly in many books that have become classical today

Theoretically, it is not difficult to prove the existence, uniqueness and tinuous dependence on initial data of the solutions of the problem (0.0.1) thanks to the

con-standard methods of mathematical analysis However, it is very challenging, even

impossible, to solve the problem (0.0.1) exactly 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 fundamen-tal and practically important research challenges [3, 17–20], and many numerical

1

methods for the problem (0.0.1), typically the finite difference methods have been constructed and strongly developed Nowadays, the finite difference methods are still implemented widely to numerically solve ODEs [3, 17–20] 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 NSFD schemes that will be presented in the remaining parts Note that the Runge-Kutta and Taylor methods can be considered as the most typical and general standard one-step difference methods.

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" [21–24] In a large number of works, Mickens discovered and analyzed numerous examples regarding the numerical instabilities occurring when using the SFD methods for differential equations (see, for instance, [21–24]) In 1980, Mickens proposed the concept of NSFD schemes to overcome the numerical instabilities and to compensate for shortcoming of the SFD schemes 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 [21–24] In particular, by using the basic rules, some authors proposed definitions of NSFD schemes as follows.

Consider a one-step numerical scheme with a step size h, thatapproximates the solution y(tk) of the problem (0.0.1) in the form:

Dh(yk) = Fh(f; yk);

where Dh(yk) dy=dt; Fh(f; yk) f(y), and tk = t0 + kh

Definition 0.1 (see [25, Definition 1], [26, Definition 3.3], [27, Definition 3])

The one-step finite-difference scheme (0.0.2) for solving System (0.0.1) is

an NSFD method if at least one of the following conditions is satisfied:

2

where ’(h) = h + O(h2) is a non-negative function;

;h), where g(yk; yk+1; h) is a non-local

approximation of System (0.0.1).

Definition 0.2 (see [28, Definition 4]) The finite-difference method is called "weakly"

nonstandard if the traditional denominator h in the first-order discrete derivative Dh(yk)

is replaced by a nonnegative function ’(h) such that ’(h) = h + O(h2).

It is important to note that NSFD schemes for PDEs, DDEs and FDEscan be defined similarly to Definitions 0.1 and 0.2 For many years, NSFDmethods have been strongly developed to compensate for shortcomings of theSFD methods and become one of the most effective and powerful methods forsolving differential equations nowaday This fact is proved convincingly inseveral monographs [21–23] and a great number of publications in prestigiousjournals (see the works related to NSFD schemes in References) All of theseworks confirmed the usefulness and advantages of NSFD methods Nowadays,NSFD methods have also been widely used for PDEs, DDEs and FDEs

In general, there are many non-local approximations for a givendifferential equation depending on its properties and its right-hand sidefunction Similarly, there are many denominator functions satisfying ’(h) = h +O(h2) for a nonstandard scheme, typically ’(h) = (1 e h)= , where > 0 (see [21–24]) Note that this function is bounded from above by 1 The derivation of theabove function ’(h) in particular and the nonstandard denominator functions ingeneral were first introduced and explained by Mickens (see [21–24] Moregenerally, the first derivative can be discretized by (see [21–24]

dy

k ;

where (h) = 1 + O(h2) This formula appeared in NSFD schemes for systems

of linear ODEs and some oscillating problems (see [21–24])

If the traditional denominator function ’(h) = h and the local approximation Fhare used simultaneously for the numerical scheme (0.0.1), we obtain the classical explicit Euler scheme Generally, the use of the traditional denominator function and

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local approximations can generate the classical Runge-Kutta and Taylor schemes It should be emphasized that the main advantage of NSFD schemes over the SFD ones is that they are able to correctly preserve essential properties of corresponding differential

models for all finite step sizes h > 0 These properties appear in most of important

mathematical models arising in the real world, typically positivity, boundedness, monotonicity, periodicity and asymptotic stability To make it easier to follow, we now recall some important concepts regarding properties of NSFD schemes.

Definition 0.3 ( [25, Defintion 2]) Assume that the solutions of Eq (0.0.1)

satisfy some property P The numerical scheme (0.0.2) is called (qualitatively) stable with respect to property P (or P-stable), if for every value of h > 0 the set of solutions of (0.0.2) satisfies property P.

In practice, properties P are diverse, typically the positivity and theasymptotic stability Regarding NSFD schemes preserving these properties,

we have the following concepts

Definition 0.4 (see [28, Definition 3], [29, Definition 3]) The finite-difference

method (0.0.2) is called elementary stable if, for any value of the step size h, the linear stability of each equilibrium y of System (0.0.1) is the same as the stability of y as a fixed point of the discrete method (0.0.1).

Definition 0.5 ( [27, Definition 1]) The finite difference method (0.0.2) is

In general, if the corresponding difference equations possess the samedynamical behavior as the continuous equations, such as local stability,bifurcations, and/or chaos, then they are said to be dynamically consistent [30].More specifically, Mickens [23] defined dynamic consistency as the following:

Definition 0.6 Consider the differential equation y0 = f(y) Let a finite difference

scheme for the equation be yk+1 = F (yk; h) Let the differential equation and/or its

solutions have property P The discrete mode equation is dynamically consistent with the differential equation if it and/or its solutions also have property P

It should be emphasized that Definitions 0.3-0.6 were stated for all finite step sizes, i.e., properties of NSFD schemes are independent of selected step sizes Meanwhile, the SFD schemes can only preserving essential properties of differential equations if selected step sizes are small enough, i.e., properties of the SFD schemes depend on step sizes However, when studying dynamical systems over a long period, the use of small step sizes will lead to a very large volume of computations, and hence, the SFD schemes are not efficient in this case Furthermore, in many cases, the SFD methods fail to preserve properties of differential equations for any finite step size, for instance, for problems having periodic or invariant properties (see [21–24]).

A special case of NSFD schemes is EFD schemes The original definition ofEFD schemes was first introduced by Mickens [21–24] More clearly, a scheme issaid to be exact if its solution coincides with the exact solution of thecorresponding differential equations at all grid nodes Obviously, EFD schemesare the best schemes for a differential equation Theoretically, Mickens provided amethod for constructing exact schemes for a given differential equation based onits general solution [21–24] Until now, there have been many results on EFDschemes for special differential equations including linear differential equationsand some scalar nonlinear equations (see [21–24, 31–37]) In general, an NSFDscheme is not an EFD one but an EFD scheme should be an NSFD one

Over the past four decades, the research direction on NSFD schemes hasat-tracted the attention of many researchers in many different aspects and gained

a great number of interesting and significant results All of the works confirmed theusefulness and advantages of NSFD schemes In major surveys [24, 38, 39] aswell as several monographs [21–23], the authors have systematically presentedresults on NSFD meth-ods in recent decades as well as directions of thedevelopment 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,DDEs and FDEs For convenience, we review some important topics as follows

Topic 1 NSFD schemes for ordinary differential equations

To the best of our knowledge, this is the most exciting topic with most published works among the topics on NSFD schemes Here, the essential properties of the ODEs

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under consideration are mainly the positivity and LAS.

For scalar differential equations, in 2003, Anguelov and Lubuma proposed a method for constructing NSFD schemes by non-local approximations [25] This method allows us to construct NSFD schemes preserving the monotonic properties and the LAS of hyperbolic equilibrium points of ODEs Then, in 2009, Roeger extended the result to construct general NSFD schemes for ODEs with three fixed points [40] Previously, in 2007, Roeger and Mickens had constructed EFD schemes for ODEs of this type [33] Next, NSFD schemes for ODEs with n + 1 distinct fixed points had been also introduced in another work [41] Note that ODEs with three and n + 1 fixed-points mentioned above can be considered as a special case of differential equations with polynomial right-hand sides For equations of this type, NSFD schemes were also constructed by Mickens and Roeger in 2009 [42].

Additionally, EFD schemes for the ODE with the right-hand side function f(y)

= y were formulated in 2011 [34] NSFD methods having second-order accuracyfor ODEs with polynomial right-hand sides were designed in 2006 [43] In 2004,nonstandard discrete approximations preserving stability properties of con-tinuousmathematical models of the form (0.0.1) were studied by Solis and Chen-Charpentier [44] After that, in 1998, Mickens demonstrated that by usingnonstandard schemes, the appearance of spurious solutions when using Runge–Kutta schemes for first-order ODEs can be eliminated, and that qualitativelycorrect numerical solutions are obtained for all values of the step size [45]

For systems of ODEs, in 1994, Mickens and Ramadhani constructed a class of finite-difference schemes for two coupled first-order ODEs such that the differ-ence equations have the correct linear stability properties for all finite values of the step-size [46] A major consequence of such schemes is the absence of elementary numerical instabilities In 2005, Dimitrov and Kojouharov proposed elementary stable NSFD methods based on the explicit and implicit Euler methods, and the RK2 method for general two-dimensional autonomous dynamical systems [28] Later, in 2007, the result was extended for the general n-dimensional dynamical systems [29] Here, the constructed NSFD schemes are based on the -method and the RK2 method It should

be emphasized that the above-mentioned NSFD schemes only preserve the LAS of

hyperbolic equilibria, and hence, equilibria must be assumed to be hyperbolic In

2015, Wood and Kojouharov [27] designed a class of NSFD schemes preservingthe positivity of solutions and the local behavior of dynamical systems nearequilibria These schemes are formulated by novel non-local approximations incombination with suitable nonstandard denominator functions Recently, Cressonand Pierre obtained NSFD schemes preserving the positivity and LAS of a generalclass of two dimen-sional ODEs including several models in population dynamicsusing the Mickens’s methodology [26] Besides, NSFD schemes for some classes

of second-order ODEs were also considered [47–49]

Along with the general differential equation models mentioned above, alarge number of important mathematical models in the real world were transformed

to dynamically consistent discrete models It is possible to mention typical results

in this topic of Mickens and Roeger on NSFD schemes for the Lotka-Volterrasystems [50–55] In 2006 and 2008, Dimitrov and Kojouharov created positive andelementary stable nonstandard numerical methods for predator-prey models [56,57] Many other results on NSFD for important mathematical models in biology,epidemiology and pharmacology are also noteworthy [58–65] NSFD results foroscillating problems were also studied and developed, typically results of Mickensand his colleagues in Journal of Sound and Vibration [31, 32, 66–68]

In 2015, Wood’s doctoral thesis studied NSFD schemes for some classes of ODEs including productive-destructive systems and autonomous dynamical systems with positive solutions [69] The constructed NSFD schemes preserve two essential properties of ODEs, which are the positivity and LAS Recently, Egbelowo’s doctoral thesis successfully applied NFSD methods for pharmacokinetic models described by systems of ODEs including both linear and nonlinear cases [70] These results indicate that NSFD schemes are both computationally efficient and easy to implement and can be used to solve a broad range of problems in science and technology.

The improvement of the accuracy for NSFD schemes is also a significant problem and was investigated by some authors [43, 71–73] It is well-known that most

of the constructed NSFD schemes for ODEs have only the first order of accuracy This can be considered as a common drawback of NSFD schemes In recent years,

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some authors have proposed some different approaches, such as, the combination of EFD schemes and NSFD schemes [43], the Richardson’s extrapolation technique [71], extrapolation techniques in combination with NSFD schemes [72, 73], etc in order to build highly accurate and dynamically consistent NSFD schemes for ODEs.

On the other hand, EFD schemes for systems of ODEs have also attractedthe attention of some authors, especially for linear ODEs with constantcoefficients Some notable works in this topic can be found in [21–24, 31–37]

Topic 2 NSFD methods for partial differential equations

In recent years, the study of NSFD schemes for PDEs is also of interest to many researchers (see [21–24, 38, 39]) The classes of equations under consideration arise in many areas of science and technology and satisfy several important physical properties Some typical results in this subject can be listed as NSFD schemes for the diffusionless Burgers equation with logistic reaction [74], NSFD schemes for a Fisher PDE having nonlinear diffusion [75], NSFD schemes for a PDE modeling combustion with nonlinear advection and diffusion [76], NSFD schemes for a nonlinear PDE having diffusive shock wave solutions [77] Some other results, for instance, positivity- preserving NSFD schemes for cross-diffusion equations in biosciences [78], NSFD schemes for a nonlinear Black-Scholes equation [79], NSFD schemes for convection- diffusion equations having constant coefficients [80] and NSFD schemes for a diffusive within-host virus dynamics model [81] are also very important and worthy.

Topic 3 NSFD methods for fractional differential equations

In recent years, NSFD schemes for some classes of FDEs have beenstudied by many researchers with very important applications (see, forinstance, [82–85] and references therein) To the best of our knowledge, this

is a quite new research direction with few published works, and especially,there are many important issues and new problems that were posted but notyet resolved In general, the research direction on NSFD schemes for FDEshas not been developed commensurately with the qualitative study

Topic 4 NSFD methods for delay differential equations

Recently, some authors have been interested in NSFD schemes for DDEs (see, for example, [86–88]) To the best of our knowledge, this is a quite new research

direction with very few published works However, the existing works allconfirmed that NSFD schemes were also effective for DDEs.

2 The necessity of the research

Although the research direction on NSFD schemes for differential equationshave achieved a lot of results shown by both quantity and quality of existingresearch works, real-world situations always pose new problems having complexproperties in both qualitative study and numerical simulation On the other hand,there are many differential models that have been established completely in thequalitative aspect but their corresponding dynamically consistent discrete modelshave not yet been studied Therefore, the construction of discrete models thatcorrectly preserve essential properties of differential models is truly necessary, hasscientific significance and needs to be studied

On the other hand, the construction of NSFD schemes for ODEmodels still faces many difficulties and has not been completely resolved,especially for models with at least one of the following characteristics:

(i) Having higher dimensions and many parameters

(ii) Having non-hyberbolic equilibrium points

(iii) Having GAS property

Generally, most of the previous results only focus on differential models havinghyper-bolic equilibrium points with the LAS property, and there are no effectiveapproaches for problems possessing non-hyperbolic equilibrium points and/orhaving the GAS property On the other hand, the study of the LAS of NSFDschemes for models having large dimensions is still a big challenge, andtherefore, effective approaches are needed for these models Furthermore, theimprovement of the accuracy of NSFD schemes and the construction of EFDschemes for ODE models are also essential with many important applications

From the above reasons as well as the ones mentioned in Section 1, we believe that the following research subjects are timely, have great scientific and practical significance, and therefore, need to be studied That is why we set the aim of developing NSFD schemes for important mathematical models described by systems of ODEs,

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which arise in applied fields.

Subject 1 NSFD schemes for some classes of ODEs arising in applied fields

So far, ODEs have continued to play an especially important role in boththeory and practice Because of this reason, the study of NSFD schemes forODEs is still of special interest to mathematicians and engineers The ODEmodels under consideration often possess a number of characteristicproperties, typically the positivity and the asymptotic stability (the LAS andGAS) It is important to note that the existing NSFD schemes mainly focus onthe preservation of the LAS of continuous models Here, the main approach isthe Lyapunov stability theorem in combination with the Schur-Hurwitz criteria.However, the approach has the following weaknesses and limitations:

(i) All equilibria must be assumed to be hyperbolic, whereas, there are many models having non-hyperbolic equilibrium points

(ii) Only conclude the LAS of discrete models, whereas, there are many models being GAS

(iii) Lyapunov indirect method requires the computation of the eigenvalues of the Jacobian matrix of discrete models at equilibria and the determination of whether each eigenvalue is inside or outside the unit ball Theoretically, it is possible to use the Jury stability criterion to perform the second requirement However, this condition requires the determination of the coefficients of the characteristic polynomial, and hence, it is difficult to use the criterion for systems with higher dimension Especially, for systems with higher dimensions, the approach is no longer effective because we may not have the explicit expressions of equilibria as well as Jacobian matrices In addition, it is not easy to apply this criterion because the coefficients of the characteristic polynomial can be very complex.

Subject 2: EFD schemes for some ODEs and their applications

EFD schemes for ODEs have many important applications in scientific puting and numerical analysis This topic has been studied by many researchers, but there are still many problems that need to be addressed Especially, in 2008, after constructing EFD schemes for two-dimensional linear systems with constant coeffi-

com-cients, Roeger posted an interesting question on the construction of EDFschemes for three-dimensional linear systems, however, this question hadnot been answered until we resolved it in 2017.

Subject 3: High order NSFD schemes for some classes of general autonomous

dynamical systems and their applications

The construction of NSFD schemes of high accuracy is very essentialwith many important applications but not a simple task Although someresearchers have successfully undertook the problem by some differentapproaches, there is no general approach for general dynamical systems

3 Objectives and contents of the thesis

The aim of the thesis is to study NSFD methods for solving some differential equations arising in fields of science and technology In case when qualitative proper-ties

of the differential models have not been established completely, we will perform qualitative research before proposing NSFD schemes and investigating their properties.

The thesis intends to study the following contents:

Content 1: NSFD schemes for some ODE models arising in biology, epidemiology, environment, physics and computer science.

Content 2: EFD schemes for systems of linear ODEs and their applications.

Content 3: High order NSFD schemes for general autonomous dynamical systems

and their applications

4 Approach and research method

We will approach to the proposed contents of the thesis from boththeoretical and practical points of view, which are the qualitative study andthe construction of NSFD schemes From the theoretical point of view,differential models under consideration should be completely established onthe qualitative aspect On this basis, we propose appropriate NSFDschemes for the differential models and study their properties

In order to perform the above research, we will use a combination oftools, namely:

• To establish the qualitative aspects of the differential models, qualitative theory

of ODEs and stability theory of continuous-time dynamical systems will be used.

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• To propose NSFD schemes and analyze their properties, we will usethe Mickens’ NSFD methodology, theory of numerical methods and finitedifference schemes, stability theory of discrete-time dynamical systems and theLyapunov stability theory.

• The experimental method should be used to examine and demonstrate

theoretical results, especially in the case theoretical proofs are not completed.

5 The new contributions of the thesis

In this thesis, we have successfully developed the Mickens’ methodology toconstruct nonstandard finite difference (NSFD) methods for solving someimportant classes of differential equations arising in fields of science andtechnology The new contributions are as follows:

1 Proposing and analyzing NSFD schemes for some important classes ofdiffer-ential equations, which are mathematical models of processes and phenomenaarising in science and technology The proposed NSFD schemes are not onlydynamically consistent with the differential equation models, but also easy to beimplemented; furthermore, they can be used to solve a large class of mathematicalproblems in both theory and practice

2 Proposing novel efficient approaches and techniques to study

asymptotic stability of the constructed NSFD schemes

3 Constructing high-order NSFD methods for some classes of general dynamical systems; consequently, the contradiction between dynamic consistency and high order of accuracy of NSFD methods has been resolved.

4 Proposing exact finite difference schemes for linear systems of differential tions with constant coefficients This result not only resolves some open questions related

equa-to exact schemes but also generalizes some existing works.

5 Performing many numerical experiments to confirm the theoreticalresults and to demonstrate the advantages and superiority of the proposedNSFD schemes over the standard numerical schemes

Below, we analyze in detail the obtained results of the thesis.

Result 1 NSFD schemes for some classes of ordinary differential equations

For the problems mentioned in Content 1, Section 3, the thesis has focused onODE models with at least one of the following characteristics:

(i) Having higher dimensions and many parameters

(ii) Having non-hyperbolic equilibria

(iii) Having GAS property

Consequently, we have achieved the following results

Firstly, we have successfully constructed NSFD schemes for someclasses of ODE models arising in biology, ecology and computer science It

is worth noting that the constructed NSFD schemes preserve the GASproperty of equilibria even when they are non-hyperbolic

In [A1], we have formulated NSFD schemes preserving all dynamical properties

of a metapopulation model proposed by Keymer et al in 2000 Theseproperties include monotone convergence, boundedness, stability and non-periodicity of solutions It should be emphasized that the continuous model hasequilibria which are not only LAS but also GAS By using standard methods ofmathematical analysis, we prove that the GAS of the continuous model ispreserved by the proposed NSFD schemes for all finite step sizes

In [A4], we have constructed NSFD schemes preserving essential qualitative properties of

a computer virus propagation model These properties of the model include positivity and boundedness of solutions, equilibria and their stability properties Importantly, the GAS of the model is investigated by using an appropriate Lyapunov function.

In [A5], we have transformed a continuous-time predator-prey system with general functional response and recruitment for both species into a discrete-time model by NSFD scheme We prove theoretically and confirm by numerical simulations that the constructed NSFD schemes preserve the essential qualitative properties including positivity and stability of the continuous model for any finite step size We also show that some typical SFD schemes such as the Euler scheme, the RK2 scheme and the RK4 scheme cannot preserve the properties of the continuous model for large step sizes.

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They can generate the numerical solutions which are completely different from the solutions of the continuous model Especially, the GAS of a non-hyperbolic equilibrium point of the constructed NSFD schemes is proved by the Lyapunov stability theorem.

Secondly, we have successfully designed NSFD schemes for differential models having higher dimensions and containing many parameters To overcome difficulties when considering the models of this type, we have proposed two novel approaches to investigate the stability of the proposed NSFD schemes The first approach is based on extensions of the classical Lyapunov stability theorem In [A2, A3], this approach has been used to establish the asymptotic stability of NSFD schems for two metapopulation models The obtained results are an important improvement for the results constructed in [A1] The second approach is based on the classical Lyapunov stability theorem and its extension and a well-know theorem on the GAS of Cascade nonlinear systems In [A6], we have used this approach to study the GAS of NSFD schemes for a mixing propagation model of computer viruses [A6] Thanks to two approaches, we only need to consider the reduced models with smaller dimensions rather than the original models, and consequently, complex transformations and calculations were limited significantly Especially, in [A7], the second approach has been also utilized to establish the complete GAS of a continuous- time predator-prey model The results indicate that our approaches are effective for both discrete and continuous models and can be applied to a large class of other models.

Result 2 EFD schemes for linear systems of ODEs and their applications

To handle the problems stated in Content 2, Section 3, we have proposed anew approach based on the classical Runge-Kutta methods to construct EFDschemes for systems of three-dimensional linear systems of ODEs with constantcoefficients In [A8], we have successfully resolved Roeger’s open question on theconstruction of EFD schemes for 3-D linear systems of linear ODEs with constantcoefficients Importantly, the result not only resolves Roeger’s open equations butalso can be extended to con-struct EFD schemes for general n-dimensionalsystems Some important applications showing the advantages of the constructedEFD schemes were also presented and analyzed in details

Result 3 High order NSFD schemes general autonomous dynamical systems and

their applications

To resolve the problems mentioned in Content 3, Section 3, we haveintroduced a new approach which is different from some existing approaches toconstruct high order NSFD schemes for a class of autonomous dynamical systems[A9] The approach is based on the classical Runge-Kutta method in combinationwith novel nonstandard denominator functions The proposed NSFD schemes notonly preserve the positivity and LAS of continuous model but also resolve thecontradiction between the dynamic consistency and high order of accuracy ofNSFD schemes As an important conse-quence of this, high order NSFD schemesfor some important biological systems have been formulated

The new results of the thesis have been published in (see also the works [A9] in "List of the works of the author related to the thesis", pages 160-161)

[A1]-1 Quang A Dang, Manh Tuan Hoang, Dynamically consistent discrete

metapop-ulation model, Journal of Difference Equations and Applications22(2016) 1325-1349, (SCIE)

2. 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 24(2018) 15-47, (SCIE).

3 Quang A Dang, Manh Tuan Hoang, Complete global stability of a

metapopu-lation model and its dynamically consistent discrete models,Qualitative Theory of Dynamical Systems 18(2019) 461-475, (SCIE)

4 Quang A Dang, Manh Tuan Hoang, Numerical dynamics of nonstandard

finite difference schemes for a computer virus propagation model, InternationalJournal of Dynamics and Control 8(2020) 772-778, (SCIE)

5 Quang A Dang, Manh Tuan Hoang, Nonstandard finite difference

schemes for a general predator-prey system, Journal of Computational Science36(2019), 101015, (SCIE)

6. Quang A Dang, Manh Tuan Hoang, Positivity and global stability preserving

NSFD schemes for a mixing propagation model of computer viruses, Journal of

15

Computational and Applied Mathematics 374(2020), 112753, (SCI).

7 Quang A Dang, Manh Tuan Hoang, Exact finite difference schemes

for three-dimensional linear systems with constant coefficients, Vietnam Journal

of Math-ematics 46(2018) 471-492, (ESCI)

8. 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 97(2020) 2036-2054 , (SCIE).

9 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 64(2020) 765-780, (SCIE)

and also presented at

1 The 9th, 10th, 11th, 12th National Conferences on Fundamental and Applied Information Technology

2 Seminar for Applied Mathematics, Institute of Information Technology, Vietnam Academy of Science Technology

3 Seminar for Applied Mathematics, Center for Informatics and

Computing, Viet-nam Academy of Science Technology

4 The 9th Vietnam Mathematical Congress, 2018

5 The 4th Congress on Mathematical Applications, Hanoi, 2015

6 Seminar for Computational and Applied Mathematics, VNU University

of Sci-ence

7 Conference on Applied Mathematics and Informatics, HUST, 2016

6 Structure of the thesis

In addition to "Introduction", "General conclusions" and "References",the contents of this thesis are presented in there chapters, among which themain results are in Chapters 2 and 3

• Chapter 1: In this chapter, we recall some preliminaries and important concepts related to continuous-time and discrete-time continuous dynamical

systems, numerical methods for solving ODEs, EFD and NSFD schemes fordifferential equations, positivity of Runge-Kutta methods These results will beused in Chapters 2 and 3

• Chapter 2: In this chapter, we propose and analyze NSFD schemes for solving

some classes of ODEs that describe various phenomena and processes in the real world The ODE models under consideration include metapopulation models, a general predator- prey model and computer virus propagation models Along with the construction of NSFD schemes, numerical simulations are performed to support the theoretical results as well as

to show the advantages of the constructed NSFD schemes over the standard ones.

• Chapter 3: In this chapter, we consider some classes of general autonomous

dynamical systems described by systems of linear and nonlinear ODEs We proposegeneral approaches to construct EFD schemes for three-dimensional linear systemswith constant coefficients and high order NSFD schemes for a class of generaldynamical systems Additionally, numerical simulations are performed to confirm thevalidity of the obtained theoretical results as well as the effectiveness of the proposedmethods

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CHAPTER 1 PRELIMINARIES

In this chapter, we recall some preliminaries and important concepts lated to continuous-time and discrete-time continuous dynamical systems, numerical

re-methods for solving ODEs, EFD and NSFD re-methods for differential equations and

positivity of Runge-Kutta methods, which will be used in the next chapters

1.1 Continuous-time dynamical systems

1.1.1 Initial value problems

Let Rn be an open, nonempty set A mapping f : ! Rn is called a

vector field on , usually denoted by

f(y) := f1(y1; y2; : : : ; yn); : : : ; fn(y1; y2; : : : ; yn) T :

We will assume throughout that the function f is continuous Additionally, if f is

differentiable, then its derivative is given by

det(J(y)) is called the Jacobian

A vector field may also depend on an additional parameter t, i.e., f := f(t; y)

This parameter t will play the role of an independent parameter, while y will be

assumed to depend on t The set from which y(t) has its values is called

the state