Conformable Dynamic Equations on Time Scales

347 Pages·2020·2.277 MB·English

by  Anderson, Douglas R., Georgiev, Svetlin G

Conformable Dynamic
Equations on Time Scales

Conformable Dynamic
Equations on Time Scales
Douglas R. Anderson
ConcordiaCollege
Svetlin G. Georgiev
Sorbonne University
Firsteditionpublished2020
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Contents
Preface ix
CHAPTER1(cid:4)Conformable Dynamic Calculus on Time Scales 1
1.1 INTRODUCTION 1
1.2 CONFORMABLE DIFFERENTIATION 2
1.3 CONFORMABLE REGRESSIVE FUNCTIONS 17
1.4 THE CONFORMABLE EXPONENTIAL FUNCTION 23
1.5 CONFORMABLE HYPERBOLIC AND TRIGONOMETRIC
FUNCTIONS 26
1.6 THE CONFORMABLE LOGARITHM FUNCTION 31
1.7 CONFORMABLE INTEGRATION 38
1.8 TAYLOR’S FORMULA 45
1.9 CALCULUS FOR THE NABLA CONFORMABLE DERIVATIVE 48
1.10 CONFORMABLE PARTIAL DERIVATIVES 51
1.11 ADVANCED PRACTICAL PROBLEMS 58
1.12 NOTES AND REFERENCES 59
CHAPTER2(cid:4)First-Order Linear Dynamic Equations 61
2.1 LINEAR FIRST-ORDER DYNAMIC EQUATIONS 61
2.2 CONFORMABLE BERNOULLI EQUATIONS 76
2.3 CONFORMABLE RICCATI EQUATIONS 82
2.4 CONFORMABLE LOGISTIC EQUATIONS 87
2.5 ADVANCED PRACTICAL PROBLEMS 90
CHAPTER3(cid:4)Conformable Dynamic Systems on Time Scales 93
3.1 STRUCTURE OF CONFORMABLE DYNAMIC SYSTEMS ON
TIME SCALES 93
v
vi (cid:4) Contents
3.2 CONSTANT COEFFICIENTS 124
3.3 ADVANCED PRACTICAL PROBLEMS 137
CHAPTER4(cid:4)Linear Conformable Inequalities 139
4.1 CONFORMABLE GRONWALL INEQUALITY 139
4.2 CONFORMABLE VOLTERRA-TYPE INTEGRAL INEQUALITIES 148
4.3 CONFORMABLE INEQUALITIES OF GAMIDOV AND
RODRIGUES 153
4.4 SIMULTANEOUS CONFORMABLE INTEGRAL INEQUALITIES 156
4.5 CONFORMABLE PACHPATTE’S INEQUALITIES 157
4.6 A CONFORMABLE INTEGRO-DYNAMIC INEQUALITY 162
CHAPTER5(cid:4)Cauchy-Type Problem for a Class of Nonlinear
Conformable Dynamic Equations 165
5.1 EXISTENCE AND UNIQUENESS OF SOLUTIONS 165
5.2 THE DEPENDENCY OF THE SOLUTION UPON THE
INITIAL DATA 169
5.3 LYAPUNOV FUNCTIONS 170
5.4 BOUNDEDNESS OF SOLUTIONS 172
5.5 EXPONENTIAL STABILITY 178
5.6 ADVANCED PRACTICAL PROBLEMS 183
CHAPTER6(cid:4)Higher-Order Linear Conformable Dynamic
Equations with Constant Coefficients 185
6.1 HOMOGENEOUS HIGHER-ORDER LINEAR CONFORMABLE
DYNAMIC EQUATIONS WITH CONSTANT COEFFICIENTS 185
6.2 NONHOMOGENEOUSHIGHER-ORDERLINEARCONFORMABLE
DYNAMIC EQUATIONS WITH CONSTANT COEFFICIENTS 190
6.3 ADVANCED PRACTICAL PROBLEMS 194
CHAPTER7(cid:4)Second-Order Conformable Dynamic Equations 197
7.1 HOMOGENEOUS SECOND-ORDER LINEAR CONFORMABLE
DYNAMIC EQUATIONS 197
7.2 REDUCTION OF ORDER 212
Contents (cid:4) vii
7.3 METHOD OF FACTORING 218
7.4 NONCONSTANT COEFFICIENTS 225
7.5 CONFORMABLE EULER-CAUCHY EQUATIONS 231
7.6 VARIATION OF PARAMETERS 241
7.7 ADVANCED PRACTICAL PROBLEMS 245
CHAPTER8(cid:4)Second-Order Self-Adjoint Conformable Dynamic
Equations 249
8.1 SELF-ADJOINT DYNAMIC EQUATIONS 249
8.2 REDUCTION-OF-ORDER THEOREMS 260
8.3 DOMINANT AND RECESSIVE SOLUTIONS 262
8.4 RICCATI EQUATION 273
8.5 CAUCHY FUNCTION AND VARIATION OF CONSTANTS
FORMULA 276
8.6 BOUNDARY VALUE PROBLEMS AND GREEN FUNCTIONS 278
8.6.1 ConjugateProblemandDisconjugacy 282
8.6.2 RightFocalProblem 286
8.6.3 PeriodicProblem 288
CHAPTER9(cid:4)The Conformable Laplace Transform 291
9.1 DEFINITION AND PROPERTIES 291
9.2 DECAY OF THE EXPONENTIAL FUNCTION 299
9.3 CONVERGENCE OF THE CONFORMABLE LAPLACE
TRANSFORM 304
9.4 APPLICATIONS TO IVPS 310
9.5 ADVANCED PRACTICAL PROBLEMS 313
APPENDIXA(cid:4)Derivatives on Banach Spaces 315
A.1 REMAINDERS 315
A.2 DEFINITION AND UNIQUENESS OF THE FRE´CHET
DERIVATIVE 317
A.3 THE GAˆTEAUX DERIVATIVE 324
viii (cid:4) Contents
APPENDIXB(cid:4)A Chain Rule 327
B.1 MEASURE CHAINS 327
B.2 PO¨TZSCHE’S CHAIN RULE 329
Bibliography 333
Index 335
Preface
The concept of derivatives of non-integer order, known as fractional derivatives, first
appeared in the letter between L’Hoˆpital and Leibniz in which the question of a half-
order derivative was posed. Since then, many formulations of fractional derivatives have
appeared. Recently, a new definition of fractional derivative, named “fractional con-
formable derivative”, has been introduced. This new fractional derivative is compatible
with the classical derivative and it has attracted attention in domains such as mechanics,
electronicsandanomalousdiffusion.
This book is devoted to the qualitative theory of conformable dynamic equations on
timescales.Itsummarizesthemostrecentcontributionsinthisarea,andvastlyexpandson
themtocreateacomprehensivetheorydevelopedexclusivelyforthisbook.Exceptforafew
sectionsinChapter1,theresultsherearepresentedforthefirsttime.Asaresult,thebookis
intendedforresearcherswhoworkondynamiccalculusontimescalesanditsapplications.
Thereareninechaptersinthisbook,allwithnewcontentnotfoundinjournals.
In Chapter 1 we introduce the concept of conformable dynamic differentiation and
integration, and deduce some of the properties of these operations. Also in this chapter,
wedefinebasicelementaryfunctions,suchasconformabledynamicexponential,trigono-
metric, hyperbolic, and logarithmic functions, and we obtain a Taylor formula. Chapter 2
isdevotedtofirst-orderlineardynamicequationsontimescales.Inparticular,wediscuss
conformable dynamic versions of Bernoulli, Riccati, and logistic equations. In Chapter
3, we investigate the structure of conformable dynamic systems with variable and con-
stant coefficients. Chapter 4 deals with linear conformable inequalities such as Gronwall,
Gamidov, Pachpatte, and Volterra-type inequalities. Chapter 5 is devoted to a Cauchy-
typeproblemforsomeclassesofnonlinearconformabledynamicequations.Forthisclass,
we prove the existence and uniqueness of solutions, and investigate the dependency of
solutions on initial data. Lyapunov functions are also discussed, including some criteria
for the boundedness and exponential stability of solutions. Higher-order linear dynamic
equations with constant coefficients are investigated in Chapter 6. Chapter 7 deals with
second-order conformable dynamic equations, and second-order self-adjoint conformable
dynamicequationsareinvestigatedinChapter8.Chapter9isdevotedtotheLaplacetrans-
form,includingitspropertiesandapplications.
Thisbookisaddressedtoawideaudienceofspecialistssuchasmathematicians,physi-
cists, engineers, and biologists. It can be used as a textbook at the graduate level and as a
referencebookforseveraldisciplines.
Theauthorswelcomeanysuggestionsforimprovementofthetext.
DouglasR.Anderson,SvetlinG.Georgiev
Moorhead,Paris,August2019
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