Conformable Dynamic Equations on Time Scales

Cover -- Half Title -- Title Page -- Copyright Page -- Contents -- Preface -- Chapter 1: Conformable Dynamic Calculus on Time Scales -- 1.1 INTRODUCTION -- 1.2 CONFORMABLE DIFFERENTIATION -- 1.3 CONFORMABLE REGRESSIVE FUNCTIONS -- 1.4 THE CONFORMABLE EXPONENTIAL FUNCTION -- 1.5 CONFORMABLE HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS -- 1.6 THE CONFORMABLE LOGARITHM FUNCTION -- 1.7 CONFORMABLE INTEGRATION -- 1.8 TAYLOR'S FORMULA -- 1.9 CALCULUS FOR THE NABLA CONFORMABLE DERIVATIVE -- 1.10 CONFORMABLE PARTIAL DERIVATIVES -- 1.11 ADVANCED PRACTICAL PROBLEMS -- 1.12 NOTES AND REFERENCES Cover......Page 1 Half Title......Page 2 Title Page......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 10 1.1 INTRODUCTION......Page 12 1.2 CONFORMABLE DIFFERENTIATION......Page 13 1.3 CONFORMABLE REGRESSIVE FUNCTIONS......Page 28 1.4 THE CONFORMABLE EXPONENTIAL FUNCTION......Page 34 1.5 CONFORMABLE HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS......Page 37 1.6 THE CONFORMABLE LOGARITHM FUNCTION......Page 42 1.7 CONFORMABLE INTEGRATION......Page 49 1.8 TAYLOR’S FORMULA......Page 56 1.9 CALCULUS FOR THE NABLA CONFORMABLE DERIVATIVE......Page 59 1.10 CONFORMABLE PARTIAL DERIVATIVES......Page 62 1.11 ADVANCED PRACTICAL PROBLEMS......Page 69 1.12 NOTES AND REFERENCES......Page 70 2.1 LINEAR FIRST-ORDER DYNAMIC EQUATIONS......Page 72 2.2 CONFORMABLE BERNOULLI EQUATIONS......Page 87 2.3 CONFORMABLE RICCATI EQUATIONS......Page 93 2.4 CONFORMABLE LOGISTIC EQUATIONS......Page 98 2.5 ADVANCED PRACTICAL PROBLEMS......Page 101 3.1 STRUCTURE OF CONFORMABLE DYNAMIC SYSTEMS ON TIME SCALES......Page 104 3.2 CONSTANT COEFFICIENTS......Page 135 3.3 ADVANCED PRACTICAL PROBLEMS......Page 148 4.1 CONFORMABLE GRONWALL INEQUALITY......Page 150 4.2 CONFORMABLE VOLTERRA-TYPE INTEGRAL INEQUALITIES......Page 159 4.3 CONFORMABLE INEQUALITIES OF GAMIDOV AND RODRIGUES......Page 164 4.4 SIMULTANEOUS CONFORMABLE INTEGRAL INEQUALITIES......Page 167 4.5 CONFORMABLE PACHPATTE’S INEQUALITIES......Page 168 4.6 A CONFORMABLE INTEGRO-DYNAMIC INEQUALITY......Page 173 5.1 EXISTENCE AND UNIQUENESS OF SOLUTIONS......Page 176 5.2 THE DEPENDENCY OF THE SOLUTION UPON THE INITIAL DATA......Page 180 5.3 LYAPUNOV FUNCTIONS......Page 181 5.4 BOUNDEDNESS OF SOLUTIONS......Page 183 5.5 EXPONENTIAL STABILITY......Page 189 5.6 ADVANCED PRACTICAL PROBLEMS......Page 194 6.1 HOMOGENEOUS HIGHER-ORDER LINEAR CONFORMABLE DYNAMIC EQUATIONS WITH CONSTANT COEFFICIENTS......Page 196 6.2 NONHOMOGENEOUS HIGHER-ORDER LINEAR CONFORMABLE DYNAMIC EQUATIONS WITH CONSTANT COEFFICIENTS......Page 201 6.3 ADVANCED PRACTICAL PROBLEMS......Page 205 7.1 HOMOGENEOUS SECOND-ORDER LINEAR CONFORMABLE DYNAMIC EQUATIONS......Page 208 7.2 REDUCTION OF ORDER......Page 223 7.3 METHOD OF FACTORING......Page 229 7.4 NONCONSTANT COEFFICIENTS......Page 236 7.5 CONFORMABLE EULER-CAUCHY EQUATIONS......Page 242 7.6 VARIATION OF PARAMETERS......Page 252 7.7 ADVANCED PRACTICAL PROBLEMS......Page 256 8.1 SELF-ADJOINT DYNAMIC EQUATIONS......Page 260 8.2 REDUCTION-OF-ORDER THEOREMS......Page 271 8.3 DOMINANT AND RECESSIVE SOLUTIONS......Page 273 8.4 RICCATI EQUATION......Page 284 8.5 CAUCHY FUNCTION AND VARIATION OF CONSTANTS FORMULA......Page 287 8.6 BOUNDARY VALUE PROBLEMS AND GREEN FUNCTIONS......Page 289 8.6.1 Conjugate Problem and Disconjugacy......Page 293 8.6.2 Right Focal Problem......Page 297 8.6.3 Periodic Problem......Page 299 9.1 DEFINITION AND PROPERTIES......Page 302 9.2 DECAY OF THE EXPONENTIAL FUNCTION......Page 310 9.3 CONVERGENCE OF THE CONFORMABLE LAPLACE TRANSFORM......Page 315 9.4 APPLICATIONS TO IVPS......Page 321 9.5 ADVANCED PRACTICAL PROBLEMS......Page 324 A.1 REMAINDERS......Page 326 A.2 DEFINITION AND UNIQUENESS OF THE FRE ́CHET DERIVATIVE......Page 328 A.3 THE GATEAUX DERIVATIVE......Page 335 B.1 MEASURE CHAINS......Page 338 B.2 POTZSCHE’S CHAIN RULE......Page 340 Bibliography......Page 344 Index......Page 346 The concept of derivatives of non-integer order, known as fractional derivatives, first appeared in the letter between L'Hopital 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, called the "fractional conformable derivative," has been introduced. This new fractional derivative is compatible with the classical derivative and it has attracted attention in areas as diverse as mechanics, electronics, and anomalous diffusion. Conformable Dynamic Equations on Time Scales is devoted to the qualitative theory of conformable dynamic equations on time scales. This book summarizes the most recent contributions in this area, and vastly expands on them to conceive of a comprehensive theory developed exclusively for this book. Except for a few sections in Chapter 1, the results here are presented for the first time. As a result, the book is intended for researchers who work on dynamic calculus on time scales and its applications. Features Can be used as a textbook at the graduate level as well as a reference book for several disciplines Suitable for an audience of specialists such as mathematicians, physicists, engineers, and biologists Contains a new definition of fractional derivative About the Authors Douglas R. Anderson is professor and chair of the mathematics department at Concordia College, Moorhead. His research areas of interest include dynamic equations on time scales and Ulam-type stability of difference and dynamic equations. He is also active in investigating the existence of solutions for boundary value problems. Svetlin G. Georgiev is currently professor at Sorbonne University, Paris, France and works in various areas of mathematics. He currently focuses on harmonic analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, dynamic calculus on time scales, and integral equations. "The concept of derivatives of non-integer order, known as fractional derivatives, first appeared in the letter between L'Hopital 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 conformable derivative", is introduced. This new fractional derivative is compatible with the classical derivative and it has attracted attention in areas as diverse as mechanics, electronics and anomalous diffusion. Conformable Dynamic Equations on Time Scales is devoted to the qualitative theory of conformable dynamic equations on time scales. This book summarizes the most recent contributions in this area, and vastly expands on them to conceive of a comprehensive theory developed exclusively for this book. Except for a few sections in Chapter 1, the results here are presented for this first time. As a result, the book is intended for researchers who work on dynamic calculus on time scales and its applications. Features Can be used as a textbook at the graduate level as well as a reference book for several disciplines Suitable for an audience of specialists, such as mathematicians, physicists, engineers and biologists Contains a new definition of fractional derivative"-- Provided by publisher This book is devoted to the qualitative theory of conformable dynamic eqs. on time scales and summarizes the most recent contributions in this area, and vastly expands on them to conceive of a comprehensive theory developed exclusively for this book.