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

MINISTRY OF EDUCATION AND

VIETNAM ACADEMY

TRAINING

OF SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY OF SCIENCES AND TECHNOLOGY

Hoàng Mạnh Tuấn

DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS
FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS

DOCTOR OF PHILOSOPHY IN MATHEMATICS

HANOI - 2021


MINISTRY OF EDUCATION AND

VIETNAM ACADEMY

TRAINING

OF SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY OF SCIENCES AND TECHNOLOGY

Hoàng Mạnh Tuấn

DEVELOPMENT OF NONSTANDARD FINITE DIFFERENCE METHODS
FOR SOME CLASSES OF DIFFERENTIAL EQUATIONS

Speciality: Applied Mathematics
Speciality Code: 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 CƠNG NGHỆ VIỆT NAM

HỌC VIỆN KHOA HỌC VÀ CƠNG NGHỆ

Hồng Mạnh Tuấn

PHÁT TRIỂN PHƯƠNG PHÁP SAI PHÂN KHÁC THƯỜNG
GIẢI MỘT SỐ LỚP PHƯƠNG TRÌNH VI PHÂN

LUẬN ÁN TIẾN SĨ TỐN HỌC

HÀ NỘI - 2021


BỘ GIÁO DỤC VÀ ĐÀO TẠO

VIỆN HÀN LÂM KHOA HỌC
VÀ CƠNG NGHỆ VIỆT NAM

HỌC VIỆN KHOA HỌC VÀ CƠNG NGHỆ

Hồng Mạnh Tuấn

PHÁT TRIỂN PHƯƠNG PHÁP SAI PHÂN KHÁC THƯỜNG
GIẢI MỘT SỐ LỚP PHƯƠNG TRÌNH VI PHÂN

Chun ngành: Tốn 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 Hàn
lâm Khoa học và công nghệ Việt Nam dưới sự hướng dẫn khoa học của GS. TS. Đặng

Quang Á và PGS. TSKH. Vũ Hồng Linh. Những kết quả nghiên cứu được trình bày
trong luận án là mới, trung thực và chưa từng được ai cơng bố trong bất kỳ cơng trình
nào khác. Các kết quả được công bố chung đã được cán bộ hướng dẫn cho phép sử
dụng trong luận án.
Hà Nội, tháng 01 năm 2021
Nghiên cứu sinh

Hoàng Mạnh Tuấn

i


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

Hoàng Mạnh Tuấn

ii


Lời cảm ơn
Trước hết, tơi xin bày tỏ lịng biết ơn chân thành và sâu sắc tới các cán bộ hướng
dẫn, GS. TS. Đặng Quang Á và GS. TSKH. Vũ Hồng Linh. Luận án này sẽ khơng thể
được hồn thành nếu khơng có sự hướng dẫn và giúp đỡ tận tình của các Thầy. Tơi vơ
cùng biết ơn những giúp đỡ mà các Thầy đã dành cho tôi không chỉ trong thời gian

thực 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 được
những những khó khăn và thất vọng để hồn thiện các cơng trình nghiên cứu và hồn
thành luận án.
Tơi xin gửi lời cảm ơn tới Học viện Khoa học và Công nghệ, Viện Hàn lâm
Khoa học và Công nghệ Việt Nam, nơi tơi học tập, nghiên cứu và hồn thành luận
án. Luận án này đã được hoàn thành một cách thuận lợi và đúng thời hạn là nhờ vào
công tác quản lý đào tạo chuyên nghiệp, môi trường học tập và nghiên cứu khoa học lý
tưởng cùng với sự giúp đỡ nhiệt tình của các cán bộ Học viện.
Tơi xin chân thành cảm ơn Lãnh đạo cùng các đồng nghiệp ở Viện Công nghệ
Thông tin, Viện Hàn lâm Khoa học và Cơng nghệ Việt Nam, nơi tơi đang cơng tác, vì
đã dàng mọi điều kiện thuận lợi nhất cho tôi trong suốt nhiều năm qua nói chung và
thời gian thực hiện luận án nói riêng.
Tơi cũng xin được gửi cảm ơn tới các Thầy Cô, các anh chị và bạn bè đồng
nghiệp trong Seminar "Toán ứng dụng" do GS. Đặng Quang Á chủ trì, đặc biệt là cá
nhân TS. Nguyễn Cơng Điều, vì những ý kiến sâu sắc, có chất lượng cao về mặt học
thuật trong các buổi trao đổi chuyên mơn. Những điều đó đã giúp tơi hồn thiện tốt
hơn các cơng trình nghiên cứu của mình.
Tơi cũng xin chân thành cảm ơn các các anh, chị và đồng nghiệp ở Bộ mơn
Tốn học, trường ĐH FPT, vì những giúp đỡ và động viên trong suốt quá trình thực
hiện luận án. Điều đó đã tạo cho tơi nhiều cảm hứng trong 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 tơi trong suốt thời gian học Đại học
và Cao học. Những bài giảng của thầy về mơn học Giải tích số và Tốn ứng dụng từ
thời Đại học đã có ảnh hưởng to lớn tới những lựa chọn sau này của tôi trên con đường
iii


nghiên cứu khoa học. Đặc biệt, Thầy cũng có rất nhiều góp ý sâu sắc và quan trọng

giúp cho luận án này được hồn thiện tốt hơn.
Tơi 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 ngồi 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.
Tơi xin chân thành cảm nhiều Giáo sư, Thầy Cô cùng nhiều bạn bè đồng nghiệp
khác vì đã dành nhiều thời gian đọc và cho tơi nhiều ý kiến giá trị về hình thức trình
bày của luận án.
Tơi 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.
Tơi 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 trong nghiên cứu khoa
học.
Cuối cùng, luận án này sẽ khơng thể được hồn thành nếu như khơng có sự
giúp đỡ, động viên và khích lệ về mọi mặt của gia đình. Tơi khơng thể diễn đạt được
hết bằng lời sự biết ơn của mình đối với gia đình. Với tất cả lịng 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 lịng vị tha đã khích lệ và
động viên tơi theo đuổi con đường nghiên cứu khoa học trong suốt những năm qua.
Hà Nội, tháng 01 năm 2021
Nghiên cứu sinh

Hoàng Mạnh Tuấn

iv


Acknowledgments

Firstly, I would like to thank my two supervisors Prof. Dr. Habil. Vũ Hoàng Linh
and especially Prof. Dr. Đặng Quang Á for the continuous support of my PhD study
and related research; for their patience, motivation and immense knowledge. Without
their help I could not have overcome the difficulties in research and study.
The wonderful research environment of the Graduate University of Sciences
and Technology, Vietnam Academy of Science and Technology, and the excellence
of its staff have helped me to complete this work within the schedule. I would like to
thank all the staff at the Graduate University of Sciences and Technology for their help
and support during the years of my PhD studies.
I would like to thank my big family for their endless love and unconditional
support.
Last but not least, I would like to thank my colleagues and many other people
beside me for their love, motivation and constant guidance.
Thanks all for your encouragement!
The author

Hoàng Mạnh Tuấn

v


List of notations and abbreviations
N

The set of natural numbers

N+

The set of non-negative nature numbers

R

The set of real numbers

R+

The set of non-negative real numbers

Rn

Real coordinate space of n-dimension

Rn+

The set of all the n-tuples with non-negative real numbers

σ(A)

The set of the eigenvalues of the matrix A

|z|

The modulus of the complex number z

x

The norm of the vector x

y(t),
˙

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

Delay differential equation

EEFD

Explicit exact finite difference

EFD

Exact finite difference

ENRK

Explicit nonstandard Runge-Kutta

ESRK

Explicit standard Runge-Kutta

FD

Finite difference

FDE

Fractional differential equation

GAS

Global asymptotic stability/Globally asymptotically stable

IEFD

Implicit exact finite difference

IVP

Initial value problem

HBV

Hepatitis B virus

LAS

Local asymptotic stability/Locally asymptotically stable

NSFD

Nonstandard finite difference

ODE

Ordinary differential equation

PDE

Partial differential equation

RK2

The second order Runge-Kutta method

RK4

The classical four stage Runge-Kutta method

SFD

Standard finite difference

w.r.t

with respect to

T r(J)

The trace of the matrix J
vi


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

vii


2.19 Numerical solutions obtained by the Euler scheme, RK4 scheme and
NFSD scheme (2.2.31) in Example 2.7 (t ∈ [0, 2100]). . . . . . . . . . . 76
2.20 Phase portrait obtained by the scheme (2.3.7) for h = 0.01 in Example 2.8. 84
2.21 Phase portrait obtained by the scheme (2.3.7) for h = 0.01 in Example 2.9. 85
2.22 Numerical solutions (Lk , Bk , Sk ) obtained by the RK4 scheme for
h = 2000/812 in Example 2.10. . . . . . . . . . . . . . . . . . . . . . . 86
2.23 Numerical solutions (Lk , Bk , Sk ) obtained by the Euler scheme for

h = 1.75 in Example 2.10. . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.24 Numerical solutions (Lk , Bk , Sk ) obtained by the NSFD schemes for
h = 2.5 in Example 2.10. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.25 The x-component obtained by the explicit Euler scheme for (x0 , y0 ) =
(100, 160), h = 1.111 after 180 iterations. . . . . . . . . . . . . . . . . . 95
2.26 The phase portrait obtained by the explicit Euler scheme for (x0 , y0 ) =
(100, 160), h = 1.111 after 180 iterations. . . . . . . . . . . . . . . . . . 96
2.27 The x-component obtained by the RK4 scheme for (x0 , y0 ) = (100, 160),
h = 1.429 after 140 iterations. . . . . . . . . . . . . . . . . . . . . . . . 96
2.28 The phase potrait obtained by the RK4 scheme for (x0 , y0 ) = (100, 160),
h = 1.429 after 140 iterations. . . . . . . . . . . . . . . . . . . . . . . . 97
2.29 The phase portrait obtained by the scheme (2.4.3)-(i) for h = 2.5,
t ∈ [0, 2000]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.30 The phase portrait obtained by the scheme (2.4.3)-(ii) for h = 2.5,
t ∈ [0, 2000] and P2∗ = (0, 0.4406). . . . . . . . . . . . . . . . . . . . . . 99
2.31 The phase portrait obtained by the scheme (2.4.3)-(iii) for h = 2.5,
t ∈ [0, 2000] and P2∗ = (0, 0.4406). . . . . . . . . . . . . . . . . . . . . . 99
2.32 The phase portrait obtained by the scheme (2.4.3)-(iv) for h = 2.5,
t ∈ [0, 2000] and P3∗ = (0.7575, 0.4422). . . . . . . . . . . . . . . . . . . 100
2.33 The phase portrait obtained by the scheme (2.4.3)-(v) for h = 2.5,
t ∈ [0, 2000] and P3∗ = (39.996, 0.0143). . . . . . . . . . . . . . . . . . . 100
2.34 The phase portrait obtained by the scheme (2.4.3)-(vi) for h = 2.5,
t ∈ [0, 2000] and P1∗ = (0.8696, 0). . . . . . . . . . . . . . . . . . . . . . 101

viii


2.35 The numerical solutions obtained by the numerical schemes in Example 2.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.36 Computational time of the numerical schemes in seconds with h = 0.8
in Example 2.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.37 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. . . 105
2.38 The required step sizes for the ode45. . . . . . . . . . . . . . . . . . . . 106
2.39 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). . . . . . . . . . . . . . . . . . . . . 117
2.40 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). . . . . . . . . . . . . . . . . . . . . 118
2.41 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). . . . . . . . . . . . . . . . . . . . . 119
2.42 The numerical solutions obtained by the RK4 scheme with t ∈ [0, 1450]
and h = 1.45 in Case (i). . . . . . . . . . . . . . . . . . . . . . . . . . . 119
2.43 Phase portrait obtained by the scheme (2.5.3) with t ∈ [0, 50] in Case (ii). 120
2.44 Phase portrait obtained by the scheme (2.5.3) with t ∈ [0, 50] in Case (iii). 121
2.45 Phase portrait obtained by the scheme (2.5.3) with t ∈ [0, 50] in Case (iv). 121
2.46 Phase portrait obtained by the scheme (2.5.3) with t ∈ [0, 500] in Case (v). 122
3.1

The exact solution and the solution generated by the EFD scheme. . . . . 133

3.2

Exact solutions and Exact difference scheme . . . . . . . . . . . . . . . . 134

ix


3.3

Graphs of the functions ϕi (h) in two cases of the paramerters. In
4

the upper figure: ϕ∗ = 1, ϕ1 = 1 − e−h , ϕ2 = he−0.12h , ϕ3 =
3

3

(1 − e−h )ϕ1 + e−h ϕ2 . In the lower figure: ϕ∗ = 1/1.2, ϕ1 = (1 −
5

4

4

e−1.2h )/1.2, ϕ2 = he−0.2h , ϕ3 = (1 − e−h )ϕ1 + e−h ϕ2 . . . . . . . . . . 149
3.4

Phase planes for the model (3.3.1) with some different inital data
obtained by ENRK54 method with ϕ3 (h) and h = 4. . . . . . . . . . . . 154

3.5

Phase portrait for the vaccination model with some different initial

6

4

data obtained by ENRK54 method for ϕ3 (h) = e−h he−0.5h + (1 −
6

e−h )(1 − e−1.6h )/1.6 and h = 2. . . . . . . . . . . . . . . . . . . . . . . 156

x


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 condition (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

i
The values τopt
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

xi


Contents

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

xii


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 metapopulation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.2.1. Complete global stability of the Amarasekare and Possingham’s metapopulation 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 DYNAMICAL 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

xiv


INTRODUCTION
1. Overview of research situation
Many essential phenomena and processes arising in real-world situations are
mathematically modeled by ODEs of the form:
dy(t)
= f y(t) ,
dt

y(t0 ) = y0 ∈ Rn ,

(0.0.1)
T

where y(t) denotes the vector-function y1 (t), y2 (t), . . . , yn (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 theory and
practice. By using appropriate definitions for the right-hand side function f , we
can 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 of
the problem (0.0.1) have become one of the prominent and most important topics in
both theoretical and applied mathematics over the past several decades. This topic has
attracted the attention of mathematicians and engineers in various aspects, such as,
the existence and uniqueness of solutions, qualitative study 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 been developed thoroughly in many books that have
become classical today.
Theoretically, it is not difficult to prove the existence, uniqueness and continuous dependence on initial data of the solutions of the problem (0.0.1) thanks to
the 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 fundamental 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, that approximates the solution
y(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:
2


• Dh (yk ) =

yk+1 − yk
, where ϕ(h) = h + O(h2 ) is a non-negative function;

ϕ(h)

• 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 be
defined similarly to Definitions 0.1 and 0.2. For many years, NSFD methods have been
strongly developed to compensate for shortcomings of the SFD methods and become
one of the most effective and powerful methods for solving differential equations
nowaday. This fact is proved convincingly in several monographs [21–23] and a great
number of publications in prestigious journals (see the works related to NSFD schemes
in References). All of these works 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 given differential
equation depending on its properties and its right-hand side function. Similarly, there
are many denominator functions satisfying ϕ(h) = h + O(h2 ) for a nonstandard
scheme, typically ϕ(h) = (1 − e−τ h )/τ , where τ > 0 (see [21–24]). Note that this
function is bounded from above by τ −1 . The derivation of the above function ϕ(h) in
particular and the nonstandard denominator functions in general were first introduced
and explained by Mickens (see [21–24]. More generally, the first derivative can be
discretized by (see [21–24]
dy
yk+1 − ψ(h)yk
→
,
dt
ϕ(h)

where ψ(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 ϕ(h) = h and the local approximation
Fh are 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
3


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 the asymptotic
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 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 dynamical

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 y = 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
4


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 of
EFD schemes was first introduced by Mickens [21–24]. More clearly, a scheme is
said to be exact if its solution coincides with the exact solution of the corresponding
differential equations at all grid nodes. Obviously, EFD schemes are the best schemes
for a differential equation. Theoretically, Mickens provided a method for constructing
exact 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 differential
equations 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 EFD
scheme should be an NSFD one.
Over the past four decades, the research direction on NSFD schemes has attracted the attention of many researchers in many different aspects and gained a great

number of interesting and significant results. All of the works confirmed the usefulness
and advantages of NSFD schemes. In major surveys [24, 38, 39] as well as several
monographs [21–23], the authors have systematically presented results on NSFD methods 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 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

5


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) = −λy α were formulated in 2011 [34]. NSFD methods having second-order
accuracy for ODEs with polynomial right-hand sides were designed in 2006 [43].
In 2004, nonstandard discrete approximations preserving stability properties of continuous mathematical models of the form (0.0.1) were studied by Solis and ChenCharpentier [44]. After that, in 1998, Mickens demonstrated that by using nonstandard
schemes, the appearance of spurious solutions when using Runge–Kutta schemes for

first-order ODEs can be eliminated, and that qualitatively correct 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 difference 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 θ-method and the RK2 method. It should
be emphasized that the above-mentioned NSFD schemes only preserve the LAS of

6


hyperbolic equilibria, and hence, equilibria must be assumed to be hyperbolic. In
2015, Wood and Kojouharov [27] designed a class of NSFD schemes preserving the
positivity of solutions and the local behavior of dynamical systems near equilibria.
These schemes are formulated by novel non-local approximations in combination with
suitable nonstandard denominator functions. Recently, Cresson and Pierre obtained
NSFD schemes preserving the positivity and LAS of a general class of two dimensional ODEs including several models in population dynamics using 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, a large
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-Volterra systems
[50–55]. In 2006 and 2008, Dimitrov and Kojouharov created positive and elementary
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 for oscillating problems
were also studied and developed, typically results of Mickens and 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,

7