Development of nonstandard finite difference methods for some classes of differential equations

Trưóc het, tôi xin bày tỏ lòng biet ơn chân thành và sâu sac tói các cánbhưóngdȁn,GS.TS.ĐngQuangÁvàGS.TSKH.VũHoàngLinh.Lunánnàysẽkhôngtheđưochoànthànhneukhôngcósụhưóngdȁnvàgiúpđõtntìnhcủ

BG I Á O DỤCVÀĐÀOTẠO VIN HÀN LÂM KHOA

HOCVÀCÔNGNGHV I TNAM

BG I Á O DỤCVÀĐÀOTẠO VIN HÀN LÂM KHOA

HOCVÀCÔNGNGHV I TNAM

LỜi camđoan

Lu n án này đưoc hoàn thành tại Hoc vi n Khoa hoc và Công ngh , Vi nHànlâmKhoahocvàcôngnghVitNamdưóisụhưóngdȁnkhoahoccủaGS.TS.ĐngQuangÁvàPGS.TSKH.VũHoàngLinh.Nhữngketquảnghiêncứuđưoctrìnhbàytronglunánlàmói,trungthụcvàchưatừngđưocaicôngbotrongbatkỳcôngtrìnhnàokhác.Cácketquảđưoccôngbochungđãđưoccánbh ư ó n g dȁnchophépsửdụngtronglu nán

HàNi,tháng01năm2021Nghiêncứusinh

HoàngMạnhTuan

ThisthesishasbeencompletedatGraduateUniversityofScienceandTechnology(GUST),Vietnam Academy of Science and Technology (VAST) under the supervisionof Prof Dr Đ ng Quang Á and Assoc.Prof Dr Habil Vũ Hoàng Linh I herebydeclare that all the results presented in thisthesis are new, original and have never beenpublishedfullyorpartiallyinanyotherwork

Theauthor

HoàngMạnhTuan

Trưóc het, tôi xin bày tỏ lòng biet ơn chân thành và sâu sac tói các cánbhưóngdȁn,GS.TS.ĐngQuangÁvàGS.TSKH.VũHoàngLinh.LunánnàysẽkhôngtheđưochoànthànhneukhôngcósụhưóngdȁnvàgiúpđõtntìnhcủacácThay.Tôivôcùng biet ơn những giúp đõ màcác Thay đã dành cho tôi không chỉ trong thòigianthụchinlunánmàcòncảtrongsuotthòigianhocĐạihocvàCaohoc.Sụquantâmvà giúp đõ củacác Thay trong cả công vi c lȁn cu c song đã giúp tôi vưot quađưocnhữngnhữngkhókhănvàthatvongđehoànthincáccôngtrìnhnghiêncứuvàhoànthànhlunán

Tôi xin gửi lòi cảm ơn tói Hoc vi n Khoa hoc và Công ngh , Vi n HànlâmKhoa hoc và Công nghVi t Nam, nơi tôi hoc t p, nghiên cứu và hoàn thành lunán.Lunánnàyđãđưochoànthànhmtcáchthunloivàđúngthòihạnlànhòvàocôngtácquảnlýđàotạochuyênnghip,môitrưònghoctpvànghiêncứukhoahoclýtưỏngcùngvóisụgiúpđõnhittìnhcủacáccánbHocvi n

Tôi xin chân thành cảm ơn Lãnh đạo cùng các đong nghi p ỏ Vi n CôngnghThông tin, Vi n Hàn lâm Khoa hoc và Công nghVi t Nam, nơi tôi đang côngtác,

vìđãdàngmoiđieukinthunloinhatchotôitrongsuotnhieunămquanóichungvàthòigianthụchinlunánnóiriêng

Tôi cũng xin đưoc gửi cảm ơn tói các Thay Cô, các anh chị và bạn bèđongnghi p trong Seminar "Toán ứng dụng" do GS Đ ng Quang Á chủ trì, đ c bi t

là cánhân TS Nguyen Công Đieu, vì những ý kien sâu sac, có chat lưong cao ve m

t hocthu t trong các buoi trao đoi chuyên môn Những đieu đó đã giúp tôi hoàn thi

n tothơncáccôngtrìnhnghiêncứucủamình

Tôi cũng xin chân thành cảm ơn các các anh, chị và đong nghi p ỏBmônToán hoc, trưòng ĐH FPT, vì những giúp đõ và đ ng viên trong suot quátrình thụchi n lu n án Đieu đó đã tạo cho tôi nhieu cảm hứng trong nghiên cứukhoa hoc vàthụchin lunán

Đ c bi t, Tôi cũng xin gửi lòi biet ơn sâu sac tói GS TSKH Phạm KỳAnh,ngưòi Thay đã giảng dạy và hưóng dȁn t n tình tôi trong suot thòi gian hoc Đại

hocvà Cao hoc Những bài giảng của thay ve môn hoc Giải tích so và Toán ứngdụng từthòiĐạihocđãcóảnhhưỏngtolóntóinhữnglụachonsaunàycủatôitrênconđưòng

nghiên cứu khoa hoc Đ c bi t, Thay cũng có rat nhieu góp ý sâu sac và quantronggiúpcho lun ánnày đưoc hoànthintot hơn.

Tôi cũng xin gửi lòi cảm ơn chân thành tói các GS R E Mickens (ClarkAtlantaUniversity), GS M Ehrhardt (Bergische Universitat Wuppertal), GS A J.Arenas(UniversidaddeCórdoba),GS.J.Cresson(UniversitédePauetdesPaysdel’Adour)cùngnhieu đong nghi p nưóc ngoài khác vì đã dành nhieu thòi gian đoc và cho tôinhieu ý kien giá trịve cả nidung lȁnhình thứctrình bày của lun án

Tôi xin chân thành cảm nhieu Giáo sư, Thay Cô cùng nhieu bạn bè đong nghipkhác vì đã dành nhieu thòi gian đoc và cho tôi nhieu ý kien giá trị ve hình thứctrìnhbàycủalun án

Tôi xin gửi lòi cảm ơn chân thành tói Ths Đ ng Quang Long (Vi n CNTT)vìnhữnggóp ý giá trị và quan trong choni dung và hình thứctrình bày của lun án

Tôi xin gửi lòi cảm ơn tói tat cả bạn bè và đong nghi p, những ngưòi đãdànhcho tôi nhieu sụ quan tâm và đ ng viên trong cu c song lȁn trong nghiên cứukhoahoc

Cuoi cùng, lu n án này sẽ không the đưoc hoàn thành neu như không cósụgiúp đõ, đ ng viên và khíchlve moi m t của gia đình Tôi không the dien đạtđưochetbanglòisụbietơncủamìnhđoivóigiađình.Vóitatcảlòngbietơnsâusac,lu n án này nói riêng cùng tat cả những đieu tot đep mà tôi đã và đang co gang thụchi n là đe gửi tói Bo Me, vo con,các anh, chị, em và những ngưòi thân tronggiađình,nhữngngưòivóisụyêuthương,đứckiênnhȁnvàlòngvịthađãkhíchlv àđngviêntôitheođuoiconđưòngnghiêncứukhoahoctrongsuotnhữngnămqua

HàNi,tháng01năm2021Nghiêncứusinh

HoàngMạnhTuan

Firstly,IwouldliketothankmytwosupervisorsProf.Dr.Habil.VũHoàngLinhand especiallyProf Dr Đ ng Quang Á for the continuous support of my PhDstudyandrelatedresearch;fortheirpatience,motivationandimmenseknowledge.WithouttheirhelpIcouldnothaveovercomethedifficultiesinresearchandstudy

The wonderful research environment of the Graduate University ofSciencesand Technology, Vietnam Academy of Science and Technology, and theexcellenceofitsstaffhavehelpedmetocompletethisworkwithintheschedule.Iwouldliketothank all the staff at the Graduate University of Sciences and Technology for theirhelpandsupportduringtheyearsofmyPhDstudies

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

Last but not least, I would like to thank my colleagues and many otherpeoplebesidemefortheirlove,motivationandconstantguidance

Thanksallforyourencouragement!

Theauthor

HoàngMạnhTuan

LAS Localasymptoticstability/LocallyasymptoticallystableNSFD Nonstandardfinitedifference

2.1 The RK4schemewithh=6.5,x(0)=0.1,y(0)=0.8 46

2.2 TheRK2schemewithh=5,x(0)=0.1,y(0)=0.4 46

2.3 TheexplicitEulerschemewithh=5,x(0)=0.4,y(0)=0.2 47

2.4 The solutionsgeneratedbythescheme(2.1.5)-(i)forh=10 48

2.5 Thesolutionscomputedbythescheme(2.1.5)-(ii)forh=10 49

2.6 Thesolutionscomputedbythescheme(2.1.5)-(iii)forh=10 49

2.7 Thenumericalsolutionscomputedbythescheme(2.1.5)-(i)forh=10 50

2.8 Thenumericalsolutionscomputedbythescheme(2.1.5)-(ii)forh=10 51

2.9 ThenumericalsolutionscomputedbytheexplicitnonstandardEuler’s scheme(2.1.5)-(iii)forh=10 51

2.10 SolutionsobtainedbytheRK4methodwith(I(0),S(0),L(0),R(0))= (0.25,0.1,0.2,0.45)andh= 2 64

2.11 SolutionsobtainedbytheEulermethodwith(I(0),S(0),L(0),R(0))= (0.25,0 1,0 2,0 45)andh=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)andh=5inExample2.3. 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)andh=5inExample2.3. 66 2.14 Graphsofthefunctionsλ i (t) 67

2.15 Numericalsolutionsobtainedbythescheme(2.2.11)withh=1and ϕ=(1−e −1.1h )/1.1inExample2.4 68

2.16 Numericalsolutionsobtainedbythescheme(2.2.11)withh=5and ϕ=(1−e −2.5h )/2.5inExample2.4 69

2.17 Numerical solutions obtained by the

scheme(2.2.31)withh= 0 1inExample2.5.

74

2.18 Numerical solutions obtained by the

scheme(2.2.31)withh= 0 1inExample2.6.

75

2.19 NumericalsolutionsobtainedbytheEulerscheme,RK4schemeand

NFSDscheme(2.2.31)inExample2.7(t∈ [0,2100]) 76 2.20 Phase portraitobtainedbythe scheme(2.3.7)forh=0.01inExample2.8.84

2.21 Phase portraitobtainedbythe scheme(2.3.7)forh=0.01inExample2.9.85

2.22 Numericalso l u ti o n s ( L k ,B k ,S k)o b t a i n e d b yt h e RK4 sc h e m e f o r

h=2000/812inExample2.10 86 2.23 Numericalsolutions(L k ,B k ,S k)obt ained bytheEulerschemefor

h=1.75inExample2.10 86 2.24 Numericalsolutions ( L k ,B k ,S k)o b t a i n e d bythe N S FD s c he me s for

h=2.5inExample2.10 87 2.25 Thex-componentobtainedbytheexplicitEulerschemefor(x0,y0)=

(100,160),h=1.111after 180iterations .95 2.26 ThephaseportraitobtainedbytheexplicitEulerschemefor(x0,y0)=

(100,160),h=1.111after 180iterations .96 2.27 Thex-componentobtainedbytheRK4schemefor(x0,y0)=(100,160),

h=1.429after140iterations 96 2.28 ThephasepotraitobtainedbytheRK4schemefor(x0,y0)=(100,160),

h=1.429after140iterations 97 2.29 Thephaseportraitobtainedbythescheme(2.4.3)-(i)forh= 2.5,

t∈ [0,2000] 98 2.30 Thephaseportraitobtainedbythescheme(2.4.3)-(ii)forh=2.5,

t∈[0,2000]andP2 ∗=(0,0.4406) 99 2.31 Thephaseportraitobtainedbythescheme(2.4.3)-(iii)forh=2.5,

t∈[0,2000]andP2 ∗=(0,0.4406) 99 2.32 Thephaseportraitobtainedbythescheme(2.4.3)-(iv)forh=2.5,

t∈ [0,2000]andP3 ∗

=( 0 7575,0.4422) 100 2.33 Thephaseportraitobtainedbythescheme(2.4.3)-(v)forh=2.5,

t∈ [0,2000]andP3 ∗

=( 3 9 996,0.0143) 100 2.34 Thephaseportraitobtainedbythescheme(2.4.3)-(vi)forh=2.5,

t∈[0,2000]andP1 ∗=(0.8696,0) 101

2.35

ThenumericalsolutionsobtainedbythenumericalschemesinExam-ple2.12 102

2.36 Computationaltimeofthenumericalschemesinsecondswithh=0.8

inExample2.12 1042.37 Numerical solutions obtained the ode45 and NSFD scheme The

withh min =0.0230andh max =0.0550andthecomputationaltimeis0.07

53seconds NSFDscheme3-(i)use

ϕ(h)=handh=1,thecomputationaltimeis1.0330e−04seconds.1 0 5

2.38 Therequiredstepsizesfortheode45 106

2.39 The numerical solutions (S-components) obtained by the RK4

schemewitht∈ [0,145]a n d h= 1 45, the Euler scheme

witht∈ [0,147]andh = 1 0 5a n d N SFD s c he me (2 5.3)wi t h t∈ [ 0

,150],ϕ (h)=

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

schemewitht∈ [0,145]a n d h= 1 45, the Euler scheme

witht∈ [0,147]andh = 1 0 5a n d N SFD s c he me (2 5.3)wi t h t∈ [ 0

,150],ϕ (h)=

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

schemewitht∈ [0,145]a n d h= 1 45, the Euler scheme

witht∈ [0,147]andh = 1 0 5a n d N SFD s c he me (2 5.3)wi t h t∈ [ 0

,150],ϕ (h)=

1−e −2h / 2, andh=2in Case(i) 119 2.42 ThenumericalsolutionsobtainedbytheRK4schemewitht∈ [0,1450]

andh=1.45inCase(i) 119 2.43 Phaseportraitobtainedbythescheme(2.5.3)witht∈ [0,50]in Case(ii) 120 2.44 Phaseportraitobtainedbythescheme(2.5.3)witht∈ [0,50]i n Case(iii).1 2 1 2.45 Phaseportraitobtainedbythescheme(2.5.3)witht∈ [0,50]in Case(iv).1 2 1 2.46 Phaseportraitobtainedbythescheme(2.5.3)w i t h t∈ [0, 500]in Case(v).1 2 2

3.1 TheexactsolutionandthesolutiongeneratedbytheEFDscheme 1333.2 ExactsolutionsandExactdifferencescheme 134

ListofTables

1.1 ThecoefficientsofanERKmethod 30

1.2 SomepopularERKmethods 31

1.3 Numberoforderconditions 31

1.4 SomeexamplesofimplicitR-Kmethods 32

1.5 EFDschemesandSFDschemesforsomeODEs 38

2.1 Thepreservedpropertiesofthedifferenceschemes 45

2.2 Thesufficientconditionsfordynamicconsistency 94

2.3 Theerrorsofthenumericalschemes 102

2.4 Thetimeoftheschemesinseconds 103

2.5 ThedynamicalpropertiesoftheNSFDscheme(2.5.3)underthecon-dition(2.5.4) 115

2.6 Parametersinnumericalsimulations 116

3.1 Errorofthemethods 137

3.2 Errorofthemethods 139

3.3 Thevaluesτ i andthedenominatorfunctionsϕ i (h)(i=1,2,3)of theENRKmethods 151

3.4 TheerrorsandratesofENRK1methods 151

3.5 TheerrorsandratesofENRK2methods 152

3.6 TheerrorsandratesofENRK43methods 152

3.7 TheerrorsandratesofENRK54methods 152

3.8 TheerrorsandratesofENRK4methods 152

3.9 TheerrorsandratesoftheWoodandKojouharovmethods 155

3.10 PositivityandelementarystabilitythresholdsforENRK 156

LỜicamđoan i

Declaration ii

LỜicamơn iii

Acknowledgments v

Listofnotationsandabbreviations vi

ListofFigures vi

ListofTables xi

INTRODUCTION 1

Chapter1.PRELIMINARIES 18

1.1 Continuous-timedynamicalsystems 18

1.1.1 Initialvalueproblems 18

1.1.2 Stabilitytheoryofcontinuous-timedynamicalsystems 20

1.2 Discrete-timedynamicalsystems 24

1.2.1 Differenceequations 24

1.2.2 Stabilitytheoryofdiscrete-timedynamicalsystems 25

1.3 Runge-KuttamethodsforsolvingODEs 29

1.3.1 ExplicitRunge-Kuttamethods 29

1.3.2 ImplicitRunge-Kuttamethods 31

1.3.3 PositivityofRunge-Kuttamethods 33

1.4 Nonstandardfinitedifferencemethods 36

1.4.1 Exactfinitedifferenceschemes 36

1.4.2 Nonstandardfinitedifferenceschemes 38

Chapter2.NONSTANDARDFINITEDIFFERENCESCHEMESFORSOM ECLASSESOFORDINARYDIFFERENTIALEQUATIONS 40

2.1 DynamicallyconsistentNSFDschemesforametapopulationmodel 40

2.1.1 Dynamicalpropertiesofthemetapopulationmodel 41

2.1.2 TheconstructionofNSFDschemes 42

2.1.3 Numericalexperiments 46

2.2 AnovelapproachforstudyingstabilityofNSFDschemesfortwometapopula-tionmodels

52 2.2.1 CompleteglobalstabilityoftheAmarasekareandPossingham’smetapo p-ulationmodel

53 2.2.2 Semi-implicitNSFDschemesformetapopulationmodel(2.2.1) 56

2.2.3 ExplicitNSFDschemesformetapopulationmodel(2.2.1) 69

2.2.4 Ani m p r o v e m e n t t o t h e s t a b i l i t y a n a l y s i s o f N S F D s c h e m e s f o r t h e metapopulationmodel(2.2.1) 76

2.3 NumericaldynamicsofNSFDschemesforacomputerviruspropagationmo del79 2.3.1 Dynamicsofacomputervirusmodelwithgradedcurerates 79

2.3.2 Nonstandardfinitedifferenceschemesforthefullmodel 81

2.3.3 Numericalsimulation 84

2.4 NSFDschemesforageneralpredator-preymodel 87

2.4.1 Continuousmodelanditsproperties 88

2.4.2 ConstructionofNSFDscheme 90

2.4.3 Stabilityanalysis 91

2.4.4 DynamicallyconsistentNSFDschemes 93

2.4.5 Numericalsimulation 94

2.5 A novelapproach forstudying globalstabilityof NSFDschemes fora mixingpropagationmodelofcomputerviruses 106

2.5.1 Mathematicalmodelanditsdynamics 107

2.5.2 PositiveNSFDschemesforModel(2.5.1) 109

2.5.3 GASa n a l y s i s f o r N S F D s c h e m e s a n d d y n a m i c a l l y c o n s i s t e n t N S F D schemes 110

2.5.4 Numericalsimulations 115

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

model1222.6.C o n c l u s i o n s 123

Chapter3.HIGH ORDER NONSTANDARD FINITE DIFFERENCE SCHEMESFOR SOME CLASSES OF GENERAL AUTONOMOUS DYNAMI-CALSYSTEMS

125 3.1 EDFs c h e m e s f o r t h r e e -d i m e n s i o n a l l i n e a r s y s t e m s w i t h c o n s t a n t c o e f f i c i e n t s 125 3.1.1 Constructionofexactfinitedifferenceschemes 126

3.1.2 ImplicitEFDschemes 129

3.1.3 ExplicitEFDschemes 131

3.1.4 Perturbationanalysis 132

3.1.5 Numericalsimulations 132

3.2 NonstandardRunge-Kuttamethodsforaclassofautonomousdynamicalsystems139 3.2.1 ElementarystableENRKmethods 142

3.2.2 PositiveENRKmethods 145

3.2.3 Thechoiceofthedenominatorfunction 146

3.3 SomeapplicationsoftheENRKmethods 149

3.3.1 ENRKmethodsforapredator-preysystem 149

3.3.2 ENRKm e thod s f or ava c c ina ti on m o d e l wi th mu ltip le e n de mi c s t a t e s 155

3.4 Conclusions 157

GENERALCONCLUSIONS 158

THEL I S T O F T H E W O R K S O F T H E A U T H O R R E L A T E D T O THETHESIS 160

BIBLIOGRAPHY 162

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

con-tothestandardmethodsofmathematicalanalysis.However,itisverychallenging,evenimpossible,

to solve the problem(0.0.1)exactly In common real-world situations,theproblemoffindingapproximatesolutionsisalmostinevitable.Consequently ,thestudyo f n u m e r i c a l m e t h o d s f o r s o l v i n g O D E s h a s b e c o m e o n e o f t h e f u n

d a m e n - tala n d p r a c t i c a l l y i m p o r t a n t r e s e a r c h c h a l l e n g e s [ 3 , 17–

2 0 ] , a n d m a n y n u m e r i c a l

methods for the problem(0.0.1), typically the finite difference methods havebeenconstructedandstronglydeveloped.Nowadays,thefinitedifferencemethodsarestillimplemented widely to numerically solve ODEs[3,17–20].The general theory of thefinite difference methods

inmanymonographs.Thesemethodswillbecalledthestandardfinitedifference(SFD)methods to distinguish them from the NSFD schemes that will be presented in theremainingparts.NotethattheRunge-

KuttaandTaylormethodscanbeconsideredasthemosttypicalandgeneralstandardone-stepdifferencemethods

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 ofessential properties However, in many problems, the SFDschemes revealed a seriousdrawback called"numerical instabilities". To describethis,Mickens,thecreatoroftheconceptofNSFDmethods,wrote:"numericalinstabilitiesare anindication that the discrete models are not able to model the correctmathematicalpropertiesofthesolutionstothedifferentialequationsofinterest"[21–

24].Inalargenumber of works, Mickens discovered and analyzed numerous examples regarding thenumerical instabilitiesoccurring when using the SFD methods for differential equations(see,forinstance,[21–24]).In1980,MickensproposedtheconceptofNSFDschemesto overcome the numericalinstabilities and to compensate for shortcoming of theSFD schemes According to the Mickens’methodology, NSFD schemes are thoseconstructed following a set of basic rulesderived from the analysis of the numericalinstabilities that occur when using SFD

thebasicrules,someauthorsproposeddefinitionsofNSFDschemesasfollows

Consideraone-stepnumericalschemewithastepsizeh,thatapproximatesthesolution y(t k)oftheproblem(0.0.1)intheform:

D h (y k )= F h (f;y k ), (0.0.2)whereD h (y k )≈dy/dt,F h (f;y k )≈f(y),andt k =t0+kh.

Definition0.1.( s e e [25,Definition1],[26,Definition3.3],[27,Definition3])The

→

one-stepfinite-differencescheme(0.0.2)forsolvingSystem(0.0.1)isanNSFDmethodifatleastoneo fthefollowingconditionsissatisfied:

).ThisformulaappearedinNSFDschemesforsystemsoflinearODEsandsomeoscillatingproblems(see[21–24])

If the traditional denominator functionϕ(h)=h and the local approximationF h are used simultaneously for the numerical scheme(0.0.1), we

classicalexplicitEulerscheme.Generally,theuseofthetraditionaldenominatorfunctionand

+ +

local approximations can generate the classical Runge-Kutta and Taylor schemes.Itshould be emphasized that the main advantage of NSFD schemes over the SFD onesisthat they are able to correctly preserve essential properties of corresponding

differentialmodelsfor all finite step sizesh>0 These properties appear in most of

importantmathematical models arising in the real world, typically positivity,boundedness,monotonicity, periodicity and asymptotic stability To make it easier tofollow, we nowrecallsomeimportantconceptsregardingpropertiesofNSFDschemes

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

propertyP.The numerical scheme(0.0.2)is called (qualitatively) stable withrespect

to propertyP(orP-stable), if for every value ofh>0the set of solutions of(0.0.2)satisfiespropertyP.

Inpractice,propertiesParediverse,typicallythepositivityandtheasymptoticstability RegardingNSFD 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 sizeh, the linear stabilityof each equilibriumy∗of System(0.0.1)is the same as the stability ofy∗as

a fixedpointofthediscretemethod(0.0.1).

Definition0 5 ( [27, Definition 1] ) Thefinite di ff eren ce method ( 0 0 2 ) is cal le d

positive,if,foranyvalueofthestepsizeh,andy0∈Rn

itssolutionremainspositive, i.e.,y k ∈R n fo r allk∈N.

In general, if the corresponding difference equations possess the samedynamicalbehavior as the continuous equations, such as local stability, bifurcations,and/or

chaos,thentheyaresaidtobedynamicallyconsistent[30].Morespecifically,Mickens[23]defin

eddynamicconsistencyasthefollowing:

Definition 0.6.Consider the differential equationyJ

=f(y) Let a finite differencescheme for the equation bey k+1 =F(y k ,h) Let the differential equation and/or

itssolutionshavepropertyP.Thediscretemodeequationisdynamicallyconsistentwiththedifferenti

alequationifitand/oritssolutionsalsohavepropertyP

Its h o u l d b e e m p h a s i z e d t h a t D e f i n i t i o n s 0 3

-0 6 w e r e s t a t e d f o r a l l f i n i t e step sizes, i.e., properties of NSFD schemes are independent of selected stepsizes.Meanwhile, the SFD schemes can only preserving essential properties ofdifferentialequations if selected step sizes are small enough, i.e., properties of the SFDschemesdependonstepsizes.However,whenstudyingdynamicalsystemsoveralongperiod,the use of smallstep sizes will lead to a very large volume of computations, and hence,the SFD schemes are not efficient in this case

SFDmethodsfailtopreservepropertiesofdifferentialequationsforanyfinitestepsize,forinstance,forproblemshavingperiodicorinvariantproperties(see[21–24])

A special case of NSFD schemes is EFD schemes The original definitionofEFD schemes was first introduced by Mickens[21–24].More clearly, a schemeissaid to be exact if its solution coincides with the exact solution of thecorrespondingdifferentialequationsatallgridnodes.Obviously,EFDschemesarethebestschemesforadifferentialequation.Theoretically,Mickensprovidedamethodforconstructingexactschemesforagivendifferentialequationbasedonitsgeneralsolution[21–24].Until now, there have been many results on

differentialequationsincludinglineardifferentialequationsandsomescalarnonlinearequations(see[21–24,31–37]).In general, an NSFD scheme is not an EFD one but anEFDschemeshouldbeanNSFDone

Over the past four decades, the research direction on NSFD schemes has tractedtheattentionofmanyresearchersinmanydifferentaspectsandgainedagreatnumber ofinteresting and significant results All of the works confirmed the usefulnessand advantages of NSFD schemes In majorsurveys[24,38,39]as well as severalmonographs[21–23],the authors havesystematically presented results on NSFD meth-ods in recent decades as well asdirections of the development in the future Nowadays,NSFD methods have been andwill continue to be widely used as a powerful andeffective approach to solve ODEs,PDEs, DDEs and FDEs For convenience, we reviewsomeimportanttopicsasfollows

at-Topic1.NSFDschemesforordinarydifferentialequations

To the best of our knowledge, this is the most exciting topic with mostpublishedworksamongthetopicsonNSFDschemes.Here,theessentialpropertiesoftheODEs

Fors c a l a r d i f f e r e n t i a l e q u a t i o n s , i n 2 0 0 3 , A n g u e l o v a n d L u b u m a p

r o p o s e d a method for constructing NSFD schemes by non-local approximations[25].Thismethod allows us toconstruct NSFD schemes preserving the monotonic properties andthe LAS ofhyperbolic equilibrium points of ODEs Then, in 2009, Roeger extendedthe result to construct

con-

Kuttaschemesforfirst-orderODEscanbeeliminated,andthatqualitativelycorrectnumericalsolutionsareobtainedforallvaluesofthestepsize[45]

For systems of ODEs, in 1994, Mickens and Ramadhani constructed aclassof finite-difference schemes for two coupled first-order ODEs such that thediffer-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 ofelementarynumericalinstabilities.In2005,DimitrovandKojouharovproposedelementarystableNSFD methods based on the explicit and implicit Euler methods, and the RK2 methodforgeneraltwo-dimensionalautonomousdynamicalsystems[28].Later,in2007,theresultwasextendedforthege

neraln-

dimensionaldynamicalsystems[29].Here,theconstructedNSFDschemesarebasedontheθ-mentionedNSFDschemesonlypreservetheLASof

methodandtheRK2method.Itshouldbeemphasizedthattheabove-hyperbolic equilibria, and hence, equilibria must be assumed to be methodandtheRK2method.Itshouldbeemphasizedthattheabove-hyperbolic.In2015, Wood and Kojouharov[27]designed a class of NSFD schemes preservingthepositivity of solutions and the local behavior of dynamical systems near equilibria.These schemes are formulated bynovel non-local approximations in combination withsuitable nonstandard denominatorfunctions Recently, Cresson and Pierre obtainedNSFD schemes preserving thepositivity and LAS of a general class of two dimen-sional ODEs including severalmodels in population dynamics using the Mickens’smethodology[26].Besides,NSFD schemes for some classes of second-order ODEswerealsoconsidered[47–49].

Along with the general differential equation models mentioned above, alargenumber of important mathematical models in the real world were transformed todynamically consistentdiscrete models It is possible to mention typical results inthis topic of Mickens andRoeger on NSFD schemes for the Lotka-Volterra systems[50–55].In2006and2008,DimitrovandKojouharovcreatedpositiveandelementarystablenonstandard numerical methods for predator-prey models[56,57].Many otherresults on NSFD forimportant mathematical models in biology, epidemiology andpharmacology are also noteworthy[58–65].NSFD results for oscillating problemswere also studied and developed,

inJournalofSoundandVibration[31,32,66–68]

In 2015, Wood’s doctoral thesis studied NSFD schemes for some classesofODEs including productive-destructive systems and autonomous dynamicalsystemswith positive solutions[69].The constructed NSFD schemes preserve twoessentialpropertiesofODEs,whicharethepositivityandLAS.Recently,Egbelowo’sdoctoralthesissuccessfullyappliedNFSDmethodsforpharmacokineticmodelsdescribedbysystems of ODEsincluding both linear and nonlinear cases[70].These resultsindicatethatNSFDschemesarebothcomputationallyefficientandeasytoimplementandcanbeusedtosolveabroadrangeofproblemsinscienceandtechnology

The improvement of the accuracy for NSFD schemes is also asignificantproblemandwasinvestigatedbysomeauthors[43,71–73].Itiswell-

knownthatmostof the constructed NSFD schemes for ODEs have only the first order ofaccuracy.ThiscanbeconsideredasacommondrawbackofNSFDschemes.Inrecentyears,

some authors have proposed some different approaches, such as, the combinationofEFD schemes and NSFD schemes[43],the Richardson’s extrapolation technique[71],extrapolationtechniques in combination with NSFD schemes[72,73],etc in ordertobuildhighlyaccurateanddynamicallyconsistentNSFDschemesforODEs.

On the other hand, EFD schemes for systems of ODEs have also attractedtheattention of some authors, especially for linear ODEs with constant coefficients.Somenotableworksinthistopiccanbefoundin[21–24,31–37]

Topic2.NSFDmethodsforpartialdifferentialequations

In recent years, the study of NSFD schemes for PDEs is also of interesttomany researchers(see[21–24,38,39]).Theclassesof equationsunder considerationarise inmany areas of science and technology and satisfy several importantphysicalproperties.SometypicalresultsinthissubjectcanbelistedasNSFDschemesforthediffusionlessBurgersequationwithlogisticreaction[74],NSFDschemesforaFisherPDE having nonlineardiffusion[75],NSFD schemes for a PDE modeling combustionwith nonlinear advection anddiffusion[76],NSFD schemes for a nonlinear PDEhaving diffusive shock wavesolutions[77].Some other results, for instance, positivity-preserving NSFD schemes forcross-diffusion equations in biosciences[78],NSFDschemesforanonlinearBlack-Scholesequation[79],NSFDschemesforconvection-diffusion equations having constant coefficients[80]and

direction with very few published works However, the existing works allconfirmedthatNSFDschemeswerealsoeffectiveforDDEs.

2 Thenecessityoftheresearch

Although the research direction on NSFD schemes for differentialequationshaveachievedalotofresultsshownbybothquantityandqualityofexistingresearchworks,real-worldsituationsalwaysposenewproblemshavingcomplexpropertiesinboth qualitative studyand numerical simulation On the other hand, there are manydifferential models that have been establishedcompletely in the qualitative aspectbut their corresponding dynamically consistentdiscrete models have not yet beenstudied Therefore, the construction of discretemodels that correctly preserve essentialproperties of differential models is trulynecessary, has scientific significance andneedstobestudied

On the other hand, the construction of NSFD schemes for ODE modelsstillfaces many difficulties and has not been completely resolved, especially formodelswithatleastoneofthefollowingcharacteristics:

hyper-FromtheabovereasonsaswellastheonesmentionedinSection1,webelievethat the followingresearch subjects are timely, have great scientific and practicalsignificance, and therefore, need to be studied

developingNSFDschemesforimportantmathematicalmodelsdescribedbysystems ofODEs,

Subject1.NSFDschemesforsomeclassesofODEsarisinginappliedfields

So far, ODEs have continued to play an especially important role in boththeoryandpractice.Becauseofthisreason,thestudyofNSFDschemesforODEsisstillofspecialinterest to mathematicians and engineers The ODE models under considerationoften possess a number of characteristicproperties, typically the positivity and theasymptotic stability (the LAS and GAS) It

is important to note that the existingNSFD schemes mainly focus on thepreservation of the LAS of continuous models.Here, the main approach is theLyapunov 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 manymodelshavingnon-hyperbolicequilibriumpoints

(ii) Only conclude the LAS of discrete models, whereas, there are manymodelsbeingGAS

(iii) Lyapunov indirect method requires the computation of the eigenvalues oftheJacobian matrix of discrete models at equilibria and the determination ofwhethereacheigenvalueisinsideoroutsidetheunitball.Theoretically,itispossibleto use the Jury stability criterion to perform the second requirement.However,thisconditionrequiresthedeterminationofthecoefficientsofthecharacteristicpolynomial,andhence,itisdifficulttousethecriterionforsystemswithhigherdimension Especially, forsystems with higher dimensions, the approach is nolonger effective because we may not have theexplicit expressions of equilibriaas well as Jacobian matrices In addition, it is not easy to apply thiscriterionbecausethecoefficientsofthecharacteristicpolynomialcanbeverycomplex

Subject2:EFDschemesforsomeODEsandtheirapplications

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, afterconstructingEFDschemesfortwo-dimensionallinearsystemswithconstantcoeffi-

3 Objectivesandcontentsofthethesis

The aim of the thesis is to study NSFD methods for solving somedifferentialequationsarisinginfieldsofscienceandtechnology.Incasewhenqualitativeproper-tiesofthedifferentialmodelshavenotbeenestablishedcompletely,wewillperformqualitativeresearchbeforeproposingNSFDschemesandinvestigatingtheirproperties

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

• Toestablishthequalitativeaspectsofthedifferentialmodels,qualitativetheoryofODEsandstabilitytheoryofcontinuous-timedynamicalsystemswillbeused

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

• The experimental method should be used to examine and demonstratetheoreticalresults,especiallyinthecasetheoreticalproofsarenotcompleted

5 Thenewcontributionsofthethesis

Inthisthesis,wehavesuccessfullydevelopedtheMickens’methodologytoconstructnonstandardfinitedifference(NSFD)methodsforsolvingsomeimportantclassesofdifferential equations arising in fields of scienceand technology The new contributionsareasfollows:

1 Proposing and analyzing NSFD schemes for some important classes of ential equations, which are mathematical models of processes andphenomenaarising in science and technology The proposed NSFD schemes

onlydynamicallyconsistentwiththedifferentialequationmodels,butalsoeasytobeimplemented; furthermore, they can be used to solve a large class of mathematicalproblemsinboththeoryandpractice

2 Proposing novel efficient approaches and techniques to study asymptoticstabilityoftheconstructedNSFDschemes

3 Constructing high-order NSFD methods for some classes of generaldynamicalsystems;consequently,thecontradictionbetweendynamicconsistencyandhighorderofaccuracyofNSFDmethodshasbeenresolved

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

equa-5 Performing many numerical experiments to confirm the theoretical resultsandto demonstrate the advantages and superiority of the proposed NSFDschemesoverthestandardnumericalschemes

Result1.NSFDschemesforsomeclassesofordinarydifferentialequations

FortheproblemsmentionedinContent1,Section3,thethesishasfocusedonODEmodelswithatleastoneofthefollowingcharacteristics:

In[A1],wehaveformulatedNSFDschemespreservingalldynamicalpropertiesofametapopulatio

n model proposed by Keymer et al in 2000 These properties includemonotone convergence, boundedness,stability and non-periodicity of solutions Itshould be emphasized that thecontinuous model has equilibria which are not onlyLAS but also GAS By usingstandard methods of mathematical analysis, we provethat the GAS of the continuousmodel is preserved by the proposed NSFD schemes forallfinitestepsizes

In [A4], we have constructed NSFD schemes preserving essential qualitativepropertiesof a computer virus propagation model These properties of the model includepositivityand boundedness of solutions, equilibria and their stability properties.Importantly, theGASofthemodelisinvestigatedbyusinganappropriateLyapunovfunction

In [A5], we have transformed a continuous-time predator-prey system withgeneralfunctional response and recruitment for both species into a discrete-timemodel byNSFD scheme We prove theoretically and confirm by numericalsimulations

thattheconstructedNSFDschemespreservetheessentialqualitativepropertiesincludingpositivityandstabilityofthecontinuousmodelforanyfinitestepsize.Wealsoshowthat some typical SFD schemes such

RK4schemecannotpreservethepropertiesofthecontinuousmodelforlargestepsizes

hyperbolicequilibriumpointoftheconstructedNSFDschemesisprovedbytheLyapunovstabilityt

solutionsofthecontinuousmodel.Especially,theGASofanon-heorem.Secondly, we have successfully designed NSFD schemes for differentialmodelshavinghigherdimensionsandcontainingmanyparameters.Toovercomedifficultieswhenconsideringthemodelsofthistype,wehaveproposedtwonovelapproachestoinvestigatethestabilityoftheproposedNSFDschemes.ThefirstapproachisbasedonextensionsoftheclassicalLyapunovstabilitytheorem.In[A2,A3],thisapproachhasbeenusedtoestablishtheasymptoticstabilityofNSFDschemsfortwometapopulationmodels.Theobtainedresultsareanimportantimprovementfortheresults constructedin[A1].Thesecondapproachisbasedontheclas

knowtheoremontheGASofCascadenonlinearsystems.In[A6],wehaveusedthisapproachtostudytheGASofNSFDschemesforamixingpropagationmodelofcom pu te r viruses [A6 ] Thanks to twoapp ro ac hes, we on ly needtoconsiderthereducedmodelswithsmallerdimensionsratherthantheoriginalmodels,and co nseq ue nt ly, c om pl ex tr ans fo rm at ions an d ca lcu la ti ons wer el im it ed significantly.Especially,in[A7],thesecondapproachhasbeenalsoutilizedtoestablishthecompleteGASofacontinuous-timepredator-preymodel Theresultsindicate thatourapproachesareeffectiveforbothdiscreteandc

sicalLyapunovstabilitytheoremanditsextensionandawell-ontinuousmodelsandcanbeappliedtoalargeclassofothermodels

Result2.EFDschemesforlinearsystemsofODEsandtheirapplications

To handle the problems stated in Content 2, Section 3, we have proposed anewapproach based on the classical Runge-Kutta methods to construct EFD schemesforsystems of three-dimensional linear systems of ODEs with constant coefficients In[A8], we have successfullyresolved Roeger’s open question on the construction of EFDschemes for 3-D linear

Importantly,theresultnotonlyresolvesRoeger’sopenequationsbutalsocanbeextendedtocon

-structEFDschemesforgeneraln-dimensionalsystems.Someimportantapplicationsshowing the

advantages of the constructed EFD schemes were also presented andanalyzedindetails

Result3.HighorderNSFDschemesgeneralautonomousdynamicalsystemsand

ToresolvetheproblemsmentionedinContent3,Section3,wehaveintroduced

a new approach which is different from some existing approaches to construct highorder NSFD schemes for a class ofautonomous dynamical systems [A9] The approachisbasedontheclassicalRunge-Kuttamethodincombinationwithnovelnonstandarddenominatorfunctions.TheproposedNSFDschemesnotonlypreservethepositivityandLASofcontinuousmodelbutalsoresolvethecontradictionbetweenthedynamicconsistency and high order of accuracy of NSFD schemes As an important conse-quenceofthis,highorderNSFDschemesforsomeimportantbiologicalsystemshavebeenformulated

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

finitedifference schemes for a computer virus propagation model, InternationalJournalofDynamicsandControl8(2020)772-778,(SCIE)

5 QuangADang,ManhTuanHoang,Nonstandardfinitedifferenceschemesfora general

predator-prey system, Journal of Computational Science 36(2019),101015,(SCIE)

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

preservingNSFDschemesforamixingpropagationmodelofcomputerviruses,Journalof

7 QuangADang,ManhTuanHoang,Exactfinitedifferenceschemesforthree-dimensional

linear systems with constant coefficients, Vietnam Journal of Math-ematics46(2018)471-492,(ESCI)

8 Quang A Dang,Manh Tuan Hoang, Positive and elementary stable

explicitnonstandard Runge-Kutta methods for a class of autonomous dynamicalsystems,InternationalJournalofComputerMathematics97(2020)2036-2054,

(SCIE)

9 Manh Tuan Hoang, On the global asymptotic stability of a predator-prey

modelwith Crowley-Martin function and stage structure for prey, Journal ofAppliedMathematicsandComputing64(2020)765-780,(SCIE)

andalsopresentedat

1 The9th,10th,11th,12thNationalConferencesonFundamentalandAppliedInformationTechnology

2 SeminarforAppliedMathematics,InstituteofInformationTechnology,VietnamAcademyofScienceTechnology

• Chapter 1: In this chapter, we recall some preliminaries and important

conceptsrelated to continuous-time and discrete-time continuous dynamicalsystems,numericalmethodsforsolvingODEs,EFDandNSFDschemesfordifferentialequations, positivity of Runge-Kutta methods These results will be used inChapters2and3

• Chapter2:Inthischapter,weproposeandanalyzeNSFDschemesforsolvingsome classes of

ODEs that describe various phenomena and processes in therealworld.TheODEmodelsunderconsiderationincludemetapopulationmodels,a general predator-prey model and computer virus propagation models Alongwith the construction ofNSFD schemes, numerical simulations are performed tosupport the theoreticalresults as well as to show the advantages of theconstructedNSFDschemesoverthestandardones

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

autonomousdynamical systems described by systems of linear and nonlinearODEs Wepropose general approaches to construct EFD schemes for three-dimensionallinear systems with constant coefficients and high order NSFDschemes for aclass of general dynamical systems Additionally, numericalsimulations areperformed to confirm the validity of the obtained theoreticalresults as well astheeffectivenessoftheproposedmethods