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 instabilitiesare an indication that the discrete models are not able to model the correct mathema

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

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

HANOI - 2021

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

Hoàng 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: 9 46 01 12

LUẬN ÁN TIẾN SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC:

1 GS TS Đặng Quang Á

2 PGS TSKH Vũ Hoàng Linh

HÀ NỘI - 2021

Lời cam đoan

Luận án này được hoàn thành tại Học viện Khoa học và Công nghệ, Viện Hànlâm Khoa học và công nghệ Việt Nam dưới sự hướng dẫn khoa học của GS TS ĐặngQuang Á và PGS TSKH Vũ Hoàng Linh Những kết quả nghiên cứu được trình bàytrong luận án là mới, trung thực và chưa từng được ai công bố trong bất kỳ công trìnhnào khác Các kết quả được công bố chung đã được cán bộ hướng dẫn cho phép sửdụng trong luận án

Hà Nội, tháng 01 năm 2021Nghiên cứu sinh

Hoàng Mạnh Tuấn

The author

Hoàng Mạnh Tuấn

Lời cảm ơn

Trước hết, tôi xin bày tỏ lòng biết ơn chân thành và sâu sắc tới các cán bộ hướngdẫn, GS TS Đặng Quang Á và GS TSKH Vũ Hoàng Linh Luận án này sẽ không thểđược hoàn thành nếu không có sự hướng dẫn và giúp đỡ tận tình của các Thầy Tôi vôcùng biết ơn những giúp đỡ mà các Thầy đã dành cho tôi không chỉ trong thời gianthực hiện luận án mà còn cả trong suốt thời gian học Đại học và Cao học Sự quan tâm

và giúp đỡ của các Thầy trong cả công việc lẫn cuộc sống đã giúp tôi vượt qua đượcnhững những khó khăn và thất vọng để hoàn thiện các công trình nghiên cứu và hoànthành 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 nhờ vàocông tác quản lý đào tạo chuyên nghiệp, môi trường học tập và nghiên cứu khoa học lýtưởng 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 Viện Công nghệThông tin, Viện Hàn lâm Khoa học và Công nghệ Việt Nam, nơi tôi đang công tác, vì

đã dàng mọi điều kiện thuận lợi nhất cho tôi trong suốt nhiều năm qua nói chung vàthời gian thực hiện luận án nói riêng

Tôi cũng xin được gửi cảm ơn tới các Thầy Cô, các anh chị và bạn bè đồngnghiệp trong Seminar "Toán ứng dụng" do GS Đặng Quang Á chủ trì, đặc biệt là cánhân TS Nguyễn Công Điều, vì những ý kiến sâu sắc, có chất lượng cao về mặt họcthuật trong các buổi trao đổi chuyên môn Những điều đó đã giúp tôi hoàn thiện tốthơn các công trình nghiên cứu của mình

Tôi cũng xin chân thành cảm ơn các các anh, chị và đồng nghiệp ở Bộ mônToán học, trường ĐH FPT, vì những giúp đỡ và động viên trong suốt quá trình thựchiện luận án Điều đó đã tạo cho tôi nhiều cảm hứng trong nghiên cứu khoa học vàthực 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 từthời 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 và quan trọnggiúp 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 AtlantaUniversity), 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 tôinhiều 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 bè đồng nghiệpkhác vì đã dành nhiều thời gian đọc và cho tôi nhiều ý kiến giá trị về hình thức trìnhbày của luận án

Tôi xin gửi lời cảm ơn chân thành tới Ths Đặng Quang Long (Viện CNTT) vìnhững 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 dànhcho tôi nhiều sự quan tâm và động viên trong cuộc sống lẫn trong nghiên cứu khoahọc

Cuối cùng, luận án này sẽ không thể được hoàn thành nếu như không có sựgiúp đỡ, động viên và khích lệ về mọi mặt của gia đình Tôi không thể diễn đạt đượchết 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 thựchiện là để gửi tới Bố Mẹ, vợ con, các anh, chị, em và những người thân trong giađình, những người với sự yêu thương, đức kiên nhẫn và lòng vị tha đã khích lệ vàđộng 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

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 PhD studyand related research; for their patience, motivation and immense knowledge Withouttheir 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 the excellence

of its staff have helped me to complete this work within the schedule I would like tothank all the staff at the Graduate University of Sciences and Technology for their helpand support during the years of my PhD studies

I would like to thank my big family for their endless love and unconditionalsupport

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

Thanks all for your encouragement!

The author

Hoàng Mạnh Tuấn

List of notations and abbreviations

N+ The set of non-negative nature 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 numbersσ(A) The set of the eigenvalues of the matrix A

|z| The modulus of the complex number z

˙

y(t), y0(t), dy(t)/dt The first derivative of the function y(t)

DDE Delay differential equation

EEFD Explicit exact finite difference

ENRK Explicit nonstandard Runge-Kutta

ESRK Explicit standard Runge-Kutta

FDE Fractional differential equation

GAS Global asymptotic stability/Globally asymptotically stableIEFD Implicit exact finite difference

LAS Local asymptotic stability/Locally asymptotically stableNSFD 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

T r(J ) The trace of the matrix J

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 λi(t) 67

2.15 Numerical solutions obtained by the scheme (2.2.11) with h = 1 and ϕ = (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 ϕ = (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 ∈ [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 ∈ [0, 2000] 982.30 The phase portrait obtained by the scheme (2.4.3)-(ii) for h = 2.5,

t ∈ [0, 2000] and P2∗ = (0, 0.4406) 992.31 The phase portrait obtained by the scheme (2.4.3)-(iii) for h = 2.5,

t ∈ [0, 2000] and P2∗ = (0, 0.4406) 992.32 The phase portrait obtained by the scheme (2.4.3)-(iv) for h = 2.5,

t ∈ [0, 2000] and P3∗ = (0.7575, 0.4422) 1002.33 The phase portrait obtained by the scheme (2.4.3)-(v) for h = 2.5,

t ∈ [0, 2000] and P3∗ = (39.996, 0.0143) 1002.34 The phase portrait obtained by the scheme (2.4.3)-(vi) for h = 2.5,

t ∈ [0, 2000] and P1∗ = (0.8696, 0) 101

2.35 The numerical solutions obtained by the numerical schemes in

Exam-ple 2.12 1022.36 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 ode45

requires 2236 grid points with hmin = 0.0230 and hmax = 0.0550

and the computational time is 0.0753 seconds NSFD scheme 3-(i) use

ϕ(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 scheme

with t ∈ [0, 145] and h = 1.45, the Euler scheme with t ∈ [0, 147]

and h = 1.05 and NSFD scheme (2.5.3) with t ∈ [0, 150], ϕ(h) =

1 − e−2h
/2 and h = 2 in Case (i) 1172.40 The numerical solutions (I-components) obtained by the RK4 scheme

with t ∈ [0, 145] and h = 1.45, the Euler scheme with t ∈ [0, 147]

and h = 1.05 and NSFD scheme (2.5.3) with t ∈ [0, 150], ϕ(h) =

1 − e−2h
/2, and h = 2 in Case (i) 1182.41 The numerical solutions (C-components) obtained by the RK4 scheme

with t ∈ [0, 145] and h = 1.45, the Euler scheme with t ∈ [0, 147]

and h = 1.05 and NSFD scheme (2.5.3) with t ∈ [0, 150], ϕ(h) =

1 − e−2h
/2, and h = 2 in Case (i) 1192.42 The numerical solutions obtained by the RK4 scheme with t ∈ [0, 1450]

and h = 1.45 in Case (i) 1192.43 Phase portrait obtained by the scheme (2.5.3) with t ∈ [0, 50] in Case (ii) 1202.44 Phase portrait obtained by the scheme (2.5.3) with t ∈ [0, 50] in Case (iii) 1212.45 Phase portrait obtained by the scheme (2.5.3) with t ∈ [0, 50] in Case (iv) 1212.46 Phase portrait obtained by the scheme (2.5.3) with t ∈ [0, 500] in Case (v) 1223.1 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 ϕi(h) in two cases of the paramerters In

the upper figure: ϕ∗ = 1, ϕ1 = 1 − e−h, ϕ2 = he−0.12h4, ϕ3 =

(1 − e−h3)ϕ1+ e−h3ϕ2 In the lower figure: ϕ∗ = 1/1.2, ϕ1 = (1 −

e−1.2h)/1.2, ϕ2 = he−0.2h5, ϕ3 = (1 − e−h4)ϕ1+ e−h4ϕ2 1493.4 Phase planes for the model (3.3.1) with some different inital data

obtained by ENRK54 method with ϕ3(h) and h = 4 1543.5 Phase portrait for the vaccination model with some different initial

data obtained by ENRK54 method for ϕ3(h) = e−h6he−0.5h4 + (1 −

e−h6)(1 − e−1.6h)/1.6 and h = 2 156

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 τopti and the denominator functions ϕi(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 40

2.1.1 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

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 DYNAMI-CAL SYSTEMS

125 3.1 EDF schemes for three-dimensional linear systems with constant coefficients 125 3.1.1 Construction of exact finite difference schemes 126

3.1.2 Implicit EFD schemes 129

3.1.3 Explicit EFD schemes 131

3.1.4 Perturbation analysis 132

3.1.5 Numerical simulations 132

3.2 Nonstandard Runge-Kutta methods for a class of autonomous dynamical systems 139 3.2.1 Elementary stable ENRK methods 142

3.2.2 Positive ENRK methods 145

3.2.3 The choice of the denominator function 146

3.3 Some applications of the ENRK methods 149

3.3.1 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

GENERAL CONCLUSIONS 158

THE LIST OF THE WORKS OF THE AUTHOR RELATED TO THE THESIS 160

BIBLIOGRAPHY 162

1 Overview of research situation

Many essential phenomena and processes arising in real-world situations aremathematically modeled by ODEs of the form:

dy(t)

dt = f y(t)
, y(t0) = y0 ∈ Rn, (0.0.1)where y(t) denotes the vector-functiony1(t), y2(t), , yn(t)T, and the function fsatisfies 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 theory andpractice 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 and analysis ofthe problem (0.0.1) have become one of the prominent and most important topics inboth theoretical and applied mathematics over the past several decades This topic hasattracted the attention of mathematicians and engineers in various aspects, such as,the existence and uniqueness of solutions, qualitative study for solutions, asymptoticstability properties, methods of finding solutions and so on (see [1, 3, 6, 8, 9, 17] andreferences therein) It is safe to say that most of the general theory on the qualitativestudy for the problem (0.0.1) has been developed thoroughly in many books that havebecome 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

con-the standard methods of macon-thematical 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, thestudy of numerical methods for solving ODEs has become one of the fundamen-tal and practically important research challenges [3, 17–20], and many numerical

methods for the problem (0.0.1), typically the finite difference methods have beenconstructed and strongly developed Nowadays, the finite difference methods are stillimplemented widely to numerically solve ODEs [3, 17–20] The general theory of thefinite 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 theremaining parts Note that the Runge-Kutta and Taylor methods can be considered asthe most typical and general standard one-step difference methods

Except for key requirements such as the convergence and stability, numericalschemes must correctly preserve essential properties of corresponding differentialequations In other words, differential models must be transformed into discrete modelswith 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 instabilitiesare an indication that the discrete models are not able to model the correct mathematicalproperties of the solutions to the differential equations of interest" [21–24] In a largenumber of works, Mickens discovered and analyzed numerous examples regarding thenumerical 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 theSFD schemes According to the Mickens’ methodology, NSFD schemes are thoseconstructed following a set of basic rules derived from the analysis of the numericalinstabilities that occur when using SFD schemes [21–24] In particular, by using thebasic rules, some authors proposed definitions of NSFD schemes as follows

Consider a one-step numerical scheme with a step size h, that approximates the solutiony(tk) of the problem (0.0.1) in the form:

Dh(yk) = Fh(f ; yk), (0.0.2)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:

• Dh(yk) = yk+1− yk

ϕ(h) , where ϕ(h) = h + O(h

2) is a non-negative function;

• F (f, yk) = g(yk, yk+1, h), where g(yk, yk+1, h) is a non-local approximation of

the right-hand side 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 ϕ(h) such that ϕ(h) = h + O(h2).

It is important to note that NSFD schemes for PDEs, DDEs and FDEs can bedefined similarly to Definitions 0.1 and 0.2 For many years, NSFD methods have beenstrongly developed to compensate for shortcomings of the SFD methods and becomeone of the most effective and powerful methods for solving differential equationsnowaday This fact is proved convincingly in several monographs [21–23] and a greatnumber of publications in prestigious journals (see the works related to NSFD schemes

in References) All of these works confirmed the usefulness and advantages of NSFDmethods Nowadays, NSFD methods have also been widely used for PDEs, DDEs andFDEs

In general, there are many non-local approximations for a given differentialequation depending on its properties and its right-hand side function Similarly, thereare many denominator functions satisfying ϕ(h) = h + O(h2) for a nonstandardscheme, typically ϕ(h) = (1 − e−τ h)/τ , where τ > 0 (see [21–24]) Note that thisfunction is bounded from above by τ−1 The derivation of the above function ϕ(h) inparticular and the nonstandard denominator functions in general were first introducedand explained by Mickens (see [21–24] More generally, the first derivative can bediscretized by (see [21–24]

If the traditional denominator function ϕ(h) = h and the local approximation

Fh are used simultaneously for the numerical scheme (0.0.1), we obtain the classicalexplicit Euler scheme Generally, the use of the traditional denominator function and

local approximations can generate the classical Runge-Kutta and Taylor schemes Itshould be emphasized that the main advantage of NSFD schemes over the SFD ones isthat 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 nowrecall 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 the asymptoticstability Regarding NSFD schemes preserving these properties, we have the followingconcepts

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 called

positive, if, for any value of the step size h, and y0 ∈ Rn

+its solution remains positive, i.e., yk ∈ Rn

+for all k ∈ N.

In general, if the corresponding difference equations possess the same dynamicalbehavior 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 finitestep sizes, i.e., properties of NSFD schemes are independent of selected step sizes.Meanwhile, the SFD schemes can only preserving essential properties of differentialequations if selected step sizes are small enough, i.e., properties of the SFD schemesdepend 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 SFDmethods fail to preserve properties of differential equations for any finite step size, forinstance, 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 the correspondingdifferential equations at all grid nodes Obviously, EFD schemes are the best schemesfor a differential equation Theoretically, Mickens provided a method for constructingexact schemes for a given differential equation based on its general solution [21–24].Until now, there have been many results on EFD schemes for special differentialequations including linear differential equations and some scalar nonlinear equations(see [21–24, 31–37]) In general, an NSFD scheme is not an EFD one but an EFDscheme should be an NSFD one

Over the past four decades, the research direction on NSFD schemes has tracted the attention of many researchers in many different aspects and gained a greatnumber of interesting and significant results All of the works confirmed the usefulnessand advantages of NSFD schemes In major surveys [24, 38, 39] as well as severalmonographs [21–23], the authors have systematically presented results on NSFD meth-ods 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 andeffective approach to solve ODEs, PDEs, DDEs and FDEs For convenience, we reviewsome important topics as follows

at-Topic 1 NSFD schemes for ordinary differential equations

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

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] Thismethod allows us to construct NSFD schemes preserving the monotonic properties andthe LAS of hyperbolic equilibrium points of ODEs Then, in 2009, Roeger extendedthe 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 ofthis type [33] Next, NSFD schemes for ODEs with n + 1 distinct fixed points had beenalso introduced in another work [41] Note that ODEs with three and n + 1 fixed-pointsmentioned above can be considered as a special case of differential equations withpolynomial right-hand sides For equations of this type, NSFD schemes were alsoconstructed by Mickens and Roeger in 2009 [42]

Additionally, EFD schemes for the ODE with the right-hand side function

f (y) = −λyα were formulated in 2011 [34] NSFD methods having second-orderaccuracy for ODEs with polynomial right-hand sides were designed in 2006 [43]

In 2004, nonstandard discrete approximations preserving stability properties of tinuous mathematical models of the form (0.0.1) were studied by Solis and Chen-Charpentier [44] After that, in 1998, Mickens demonstrated that by using nonstandardschemes, the appearance of spurious solutions when using Runge–Kutta schemes forfirst-order ODEs can be eliminated, and that qualitatively correct numerical solutionsare obtained for all values of the step size [45]

con-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 ence equations have the correct linear stability properties for all finite values of thestep-size [46] A major consequence of such schemes is the absence of elementarynumerical instabilities In 2005, Dimitrov and Kojouharov proposed elementary stableNSFD methods based on the explicit and implicit Euler methods, and the RK2 methodfor general two-dimensional autonomous dynamical systems [28] Later, in 2007, theresult was extended for the general n-dimensional dynamical systems [29] Here, theconstructed NSFD schemes are based on the θ-method and the RK2 method It should

differ-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 preserving thepositivity of solutions and the local behavior of dynamical systems near equilibria.These schemes are formulated by novel non-local approximations in combination withsuitable nonstandard denominator functions Recently, Cresson and Pierre obtainedNSFD schemes preserving the positivity and LAS of a general class of two dimen-sional ODEs including several models in population dynamics using the Mickens’smethodology [26] Besides, NSFD schemes for some classes of second-order ODEswere also considered [47–49]

Along with the general differential equation models mentioned above, a largenumber of important mathematical models in the real world were transformed todynamically consistent discrete models It is possible to mention typical results inthis topic of Mickens and Roeger on NSFD schemes for the Lotka-Volterra systems[50–55] In 2006 and 2008, Dimitrov and Kojouharov created positive and elementarystable nonstandard numerical methods for predator-prey models [56, 57] Many otherresults on NSFD for important mathematical models in biology, epidemiology andpharmacology are also noteworthy [58–65] NSFD results for oscillating problemswere also studied and developed, typically results of Mickens and his colleagues inJournal of Sound and Vibration [31, 32, 66–68]

In 2015, Wood’s doctoral thesis studied NSFD schemes for some classes ofODEs including productive-destructive systems and autonomous dynamical systemswith positive solutions [69] The constructed NSFD schemes preserve two essentialproperties of ODEs, which are the positivity and LAS Recently, Egbelowo’s doctoralthesis successfully applied NFSD methods for pharmacokinetic models described bysystems of ODEs including both linear and nonlinear cases [70] These results indicatethat 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 significantproblem 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,

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

On the other hand, EFD schemes for systems of ODEs have also attracted theattention of some authors, especially for linear ODEs with constant coefficients Somenotable 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 tomany researchers (see [21–24, 38, 39]) The classes of equations under considerationarise in many areas of science and technology and satisfy several important physicalproperties Some typical results in this subject can be listed as NSFD schemes for thediffusionless Burgers equation with logistic reaction [74], NSFD schemes for a FisherPDE having nonlinear diffusion [75], NSFD schemes for a PDE modeling combustionwith nonlinear advection and diffusion [76], NSFD schemes for a nonlinear PDEhaving diffusive shock wave solutions [77] Some other results, for instance, positivity-preserving NSFD schemes for cross-diffusion equations in biosciences [78], NSFDschemes for a nonlinear Black-Scholes equation [79], NSFD schemes for convection-diffusion equations having constant coefficients [80] and NSFD schemes for a diffusivewithin-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 been studied

by many researchers with very important applications (see, for instance, [82–85] andreferences therein) To the best of our knowledge, this is a quite new research directionwith few published works, and especially, there are many important issues and newproblems that were posted but not yet resolved In general, the research direction onNSFD schemes for FDEs has not been developed commensurately with the qualitativestudy

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 all confirmedthat 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 existing researchworks, real-world situations always pose new problems having complex properties inboth qualitative study and numerical simulation On the other hand, there are manydifferential models that have been established completely in the qualitative aspectbut their corresponding dynamically consistent discrete models have not yet beenstudied Therefore, the construction of discrete models that correctly preserve essentialproperties of differential models is truly necessary, has scientific significance andneeds to be studied

On the other hand, the construction of NSFD schemes for ODE models stillfaces many difficulties and has not been completely resolved, especially for modelswith 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 having bolic equilibrium points with the LAS property, and there are no effective approachesfor problems possessing non-hyperbolic equilibrium points and/or having the GASproperty On the other hand, the study of the LAS of NSFD schemes for models havinglarge dimensions is still a big challenge, and therefore, effective approaches are neededfor these models Furthermore, the improvement of the accuracy of NSFD schemesand the construction of EFD schemes for ODE models are also essential with manyimportant applications

hyper-From the above reasons as well as the ones mentioned in Section 1, we believethat the following research subjects are timely, have great scientific and practicalsignificance, and therefore, need to be studied That is why we set the aim of developingNSFD schemes for important mathematical models described by systems of ODEs,

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 both theoryand practice Because of this reason, the study of NSFD schemes for ODEs is still ofspecial interest to mathematicians and engineers The ODE models under considerationoften possess a number of characteristic properties, typically the positivity and theasymptotic stability (the LAS and GAS) It is important to note that the existingNSFD schemes mainly focus on the preservation of the LAS of continuous models.Here, the main approach is the Lyapunov stability theorem in combination with theSchur-Hurwitz criteria However, the approach has the following weaknesses andlimitations:

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

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

(iii) Lyapunov indirect method requires the computation of the eigenvalues of theJacobian matrix of discrete models at equilibria and the determination of whethereach 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 characteristicpolynomial, and hence, it is difficult to use the criterion for systems with higherdimension Especially, for systems with higher dimensions, the approach is nolonger 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 criterionbecause 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, butthere are still many problems that need to be addressed Especially, in 2008, afterconstructing EFD schemes for two-dimensional linear systems with constant coeffi-

com-cients, Roeger posted an interesting question on the construction of EDF schemes forthree-dimensional linear systems, however, this question had not been answered until

3 Objectives and contents of the thesis

The aim of the thesis is to study NSFD methods for solving some differentialequations arising in fields of science and technology In case when qualitative proper-ties of the differential models have not been established completely, we will performqualitative 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 both theoreticaland practical points of view, which are the qualitative study and the construction

of NSFD schemes From the theoretical point of view, differential models underconsideration should be completely established on the qualitative aspect On this basis,

we propose appropriate NSFD schemes for the differential models and study theirproperties

In order to perform the above research, we will use a combination of tools,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

• To propose NSFD schemes and analyze their properties, we will use the Mickens’NSFD methodology, theory of numerical methods and finite difference schemes,stability theory of discrete-time dynamical systems and the Lyapunov stabilitytheory.

• The experimental method should be used to examine and demonstrate theoreticalresults, 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 to constructnonstandard finite difference (NSFD) methods for solving some important classes ofdifferential equations arising in fields of science and technology The new contributionsare as follows:

1 Proposing and analyzing NSFD schemes for some important classes of 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

differ-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 dynamicalsystems; consequently, the contradiction between dynamic consistency and highorder 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 questionsrelated to exact schemes but also generalizes some existing works

equa-5 Performing many numerical experiments to confirm the theoretical results and

to demonstrate the advantages and superiority of the proposed NSFD schemesover 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 some classes ofODE models arising in biology, ecology and computer science It is worth noting thatthe constructed NSFD schemes preserve the GAS property of equilibria even whenthey are non-hyperbolic

In [A1], we have formulated NSFD schemes preserving all dynamical properties of ametapopulation model proposed by Keymer et al in 2000 These properties includemonotone convergence, boundedness, stability and non-periodicity of solutions Itshould be emphasized that the continuous model has equilibria which are not onlyLAS but also GAS By using standard methods of mathematical analysis, we provethat the GAS of the continuous model is preserved by the proposed NSFD schemes forall 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 positivityand boundedness of solutions, equilibria and their stability properties Importantly, theGAS of the model is investigated by using an appropriate Lyapunov function

In [A5], we have transformed a continuous-time predator-prey system with generalfunctional response and recruitment for both species into a discrete-time model byNSFD scheme We prove theoretically and confirm by numerical simulations thatthe constructed NSFD schemes preserve the essential qualitative properties includingpositivity and stability of the continuous model for any finite step size We also showthat some typical SFD schemes such as the Euler scheme, the RK2 scheme and the RK4scheme cannot preserve the properties of the continuous model for large step sizes

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

Secondly, we have successfully designed NSFD schemes for differential modelshaving higher dimensions and containing many parameters To overcome difficultieswhen considering the models of this type, we have proposed two novel approaches toinvestigate the stability of the proposed NSFD schemes The first approach is based onextensions of the classical Lyapunov stability theorem In [A2, A3], this approach hasbeen used to establish the asymptotic stability of NSFD schems for two metapopulationmodels The obtained results are an important improvement for the results constructed

in [A1] The second approach is based on the classical Lyapunov stability theorem andits 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 mixingpropagation model of computer viruses [A6] Thanks to two approaches, we onlyneed to consider the reduced models with smaller dimensions rather than the originalmodels, and consequently, complex transformations and calculations were limitedsignificantly Especially, in [A7], the second approach has been also utilized to establishthe complete GAS of a continuous-time predator-prey model The results indicatethat our approaches are effective for both discrete and continuous models and can beapplied 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 a newapproach based on the classical Runge-Kutta methods to construct EFD schemes forsystems of three-dimensional linear systems of ODEs with constant coefficients In[A8], we have successfully resolved Roeger’s open question on the construction of EFDschemes for 3-D linear systems of linear ODEs with constant coefficients Importantly,the result not only resolves Roeger’s open equations but also can be extended to con-struct EFD schemes for general n-dimensional systems Some important applicationsshowing the advantages of the constructed EFD schemes were also presented andanalyzed 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 have introduced

a new approach which is different from some existing approaches to construct highorder NSFD schemes for a class of autonomous dynamical systems [A9] The approach

is based on the classical Runge-Kutta method in combination with novel nonstandarddenominator functions The proposed NSFD schemes not only preserve the positivityand LAS of continuous model but also resolve the contradiction between the dynamicconsistency and high order of accuracy of NSFD schemes As an important conse-quence of this, high order NSFD schemes for some important biological systems havebeen formulated

The new results of the thesis have been published in (see also the works [A1]-[A9]

in "List of the works of the author related to the thesis", pages 160-161)

1 Quang A Dang, Manh Tuan Hoang, Dynamically consistent discrete

metapop-ulation model, Journal of Difference Equations and Applications 22(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, International Journal

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 Science 36(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

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

Math-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 AppliedMathematics and Computing 64(2020) 765-780, (SCIE)

and also presented at

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

2 Seminar for Applied Mathematics, Institute of Information Technology, VietnamAcademy of Science Technology

3 Seminar for Applied Mathematics, Center for Informatics and Computing, nam Academy of Science Technology

Viet-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 ence

Sci-7 Conference on Applied Mathematics and Informatics, HUST, 2016

6 Structure of the thesis

In addition to "Introduction", "General conclusions" and "References", thecontents of this thesis are presented in there chapters, among which the main resultsare 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 for differentialequations, positivity of Runge-Kutta methods These results will be used inChapters 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 realworld The ODE models under consideration include metapopulation models,

a general predator-prey model and computer virus propagation models Alongwith the construction of NSFD schemes, numerical simulations are performed tosupport the theoretical results as well as to show the advantages of the constructedNSFD 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 Wepropose general approaches to construct EFD schemes for three-dimensionallinear systems with constant coefficients and high order NSFD schemes for aclass of general dynamical systems Additionally, numerical simulations areperformed to confirm the validity of the obtained theoretical results as well asthe effectiveness of the proposed methods

CHAPTER 1 PRELIMINARIES

In this chapter, we recall some preliminaries and important concepts lated to continuous-time and discrete-time continuous dynamical systems, numericalmethods for solving ODEs, EFD and NSFD methods for differential equations andpositivity of Runge-Kutta methods, which will be used in the next chapters

re-1.1 Continuous-time dynamical systems

1.1.1 Initial value problems

Let Ω ⊂ Rn be an open, nonempty set A mapping f : Ω → Rn is called avector field on Ω, usually denoted by

This derivative is also called the Jacobian matrix or functional matrix Its determinantdet(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 beassumed to depend on t The set Ω from which y(t) has its values is called the statespace or phase space If t takes its values from an interval I, we call I ×Ω the time-statespace

For a given f (t, y) defined on I × Ω, the associated IVP has the form:

dy

dt = f (t, y), y(t0) = y0 (given). (1.1.1)

Without loss of generality, we can assume that the starting point for t is t0 = 0 Theproblem (1.1.1) is also called the evolution problem We now present a theorem on theexistence, uniqueness and continuous dependence on the data of the IVP (1.1.1).

Theorem 1.1 ( [3, Theorem 1]) Let f (t, y) be continuous for all (t, y) in a region

D =0 ≤ t ≤ T, |y| < ∞} Moreover, assume Lipschitz continuity in y: there exists

a constant L such that for all (t, y) and (t, ˆ y) in D,

|f (t, y) − f (t, ˆy)| ≤ L|y − ˆy|

Then

(i) For any y0 ∈ Rn there exists a unique solution y(t) throughout the interval [0, T ] for IVP (1.1.1) This solution is differentiable.

(ii) The solution y(t) depends continuously on the initial data: if ˆ y also satisfies IVP

(but not the same initial values) then

|y(t) − ˆy(t)| ≤ eLt|y(0) − ˆy(0)|

(iii) If ˆ y satisfies, more generally, a perturbed ODE

ˆ

y0 = f (t, ˆy) + r(t, ˆy),

where r is bounded on D, krk ≤ M , then

|y(t) − ˆy(t)| ≤ eLt|y(0) − ˆy(0)| + M

L(e

Lt− 1)

Remark 1.1 ( [3, Chapter 1]) If the function f is differentiable in y, then the constant

L can be taken as a bound on the first derivatives of f with respect to y Namely,

L = sup(t,y)∈D

∂f

∂y(t, y) .The prototype IVP (1.1.1) is called autonomous if the function f does notdepend explicitly on t, i.e,

dy

dt = f (y), y(0) = y0 ∈ Rn, (1.1.2)

It should be emphasized that the problem (1.1.2) describes a great number of importantphenomena and processes arising in the real world In this thesis, we will mainly focus

on the autonomous system (1.1.2) under appropriate assumptions guaranteeing thatthe solutions of the problem exist and are unique The following theorem provide such

an assumption

Theorem 1.2 ( [2, Theorem 5.1]) Suppose f and ∂f /∂yi for i = 1, 2, , n are continuous functions of (y1, y2, , yn) on Rn Then, a unique solution exists to the IVP (1.1.2) for any initial value y0 ∈ Rn.

Remark 1.2 ( [2, Chapter 5]) Although the unique solution exists, the maximal

interval of existence (0, t) may be finite, i.e., T < ∞, unless the solution is bounded.

The solution y(t) = (y1(t), y2(t), , yn(t))T describes parametrically acurve lying in Rn This curve is called a trajectory (orbit or path) of the system Theregion Rn, where the solution is graphed, is called (phase space) when n = 3, phaseplane when n = 2, and phase line when n = 1

Next, we recall some basic concepts regarding continuous dynamical systemsdefined by the autonomous system (1.1.2)

Definition 1.1 ( [17, Definition 2.1.2]) Equation (1.1.2) is said to define a continuous

dynamical sytem on a subset Ω ⊂ Rn if, for every y0 ∈ Ω, there exists a unique solution

of (1.1.2) which is defined for all t ∈ [0, ∞) and remaining in Ω for all t ∈ [0, ∞).

Theorem 1.3 [17, Theorem 2.13] Let f : Rn → Rn

be globally Lipschitz on Rn Then there exists a unique solution y(t) to the problem (1.1.2) for all t ∈ R Hence, (1.1.2) defines a continuous dynamical system on Rn.

1.1.2 Stability theory of continuous-time dynamical systems

We now recall some important concepts and results on the asymptotic stability

of continuous dynamical systems

Definition 1.2 ( [17, Definition 2.1.1]) A point y∗ ∈ Rn such that f (y∗) = 0 is called

an equilibrium point of (1.1.2).

Remark 1.3 ( [2,6,17]) Such a point is also known as a fixed point, constant solution,

steady state, critical point or a steady solution.

Remark 1.4 ( [2,6,17]) Any equilibrium point can be shifted to the origin by a suitable

change of variables Therefore, we are able to state all definitions and theorems for the case when the equilibrium point is at the origin.

Definition 1.3 ( [17, Definition 4.1]) The equilibrium point y∗ = 0 of (1.1.2) is:

(i) stable if, for each  > 0, there is δ = δ() > 0 such that

ky(0)k < δ =⇒ ky(t)k < , ∀t ≥ 0

(ii) unstable if it is not stable.

(iii) LAS if it is stable and δ can be chosen such that

Theorem 1.4 ( [6, Theorem 4.1]) Let y∗ = 0 be an equilibrium point for (1.1.2)

and D ⊂ Rn be a domain containing y∗ = 0 Let V : D → R be a continuously

differentiable function such that

Theorem 1.5 ( [6, Theorem 4.2]) Let y∗ = 0 be an equilibrium point for (1.1.2) Let

V : Rn → R be a continuously differentiable function such that

V (0) = 0 and V (y) > 0, ∀y 6= 0,

kyk → ∞ =⇒ V (y) → ∞,

˙

V (y) < 0, ∀y 6= 0

Then, y∗ = 0 is GAS.

Remark 1.5 Theorem 1.5 is known as Barbashin-Krasovskii theorem A function V

satisfying the second condition of Theorem 1.5 is said to be radially unbounded.

The following result is known as LaSalle’s invariant theorem

Theorem 1.6 ( [6, Theorem 4.4]) Let Ω ⊂ D be a compact set that is positively

invariant with respect to (1.1.2) Let V : D → R be a continuously differentiable

function such that ˙ V (y) ≥ 0 in Ω Let E be the set of all points in Ω where ˙ V (y) = 0.

Let M be the largest invariant set in E Then every solution starting in Ω approaches

to M as t → ∞.

Theorem 1.4 is known as Lyapunov’s direct method The following theorem iscalled Lyapunov’s indirect method, which is a direct consequence of Lyapunov’s directmethod

Theorem 1.7 ( [6, Theorem 4.7]) Let y∗ = 0 be an equilibrium point for the

non-linear system (1.1.2), where f : D → Rn is continuously differentiable and D is a neighborhood of the origin Let

A = ∂f

∂y(y)

y ∗ =0

Then,

(i) the origin is LAS if Reλi < 0 for all eigenvalues of A.

(ii) the origin is unstable if Reλi > 0 for one of more the eigenvalues of A.

The equation y0 = Ay is known as the linearization about y∗of (1.1.2) Clearly,Theorem 1.7 is only applicable if none of the eigenvalues of the matrix A lies on theimaginary axis, i.e., Reλi 6= 0 for all eigenvalues of A In this case, the origin is said

to be hyperbolic

Definition 1.4 ( [17, Definition 2.3.6]) An equilibrium point y∗of (1.1.2) is said to

be hyperbolic if none of the eigenvalues of the matrix A lies on the imaginary axis.

to be hyperbolic

Definition 1.4 ( [17, Definition 2.3.6]) An equilibrium point y∗of (1.1.2) is said to

be