Special functions and analysis of differential equations

Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Table of Contents......Page 6
Preface......Page 8
Editors......Page 12
Contributors......Page 14
 1.1 Introduction......Page 18
 1.2 Preliminaries and Problem Statement......Page 20
  1.2.1 Fractional Calculus......Page 21
  1.2.2 Polynomial Interpolation......Page 22
  1.2.3 Problem Statement......Page 24
  1.3.1 Chebyshev Operational Matrix of Fractional Operators......Page 25
  1.3.2 Spatial Discretization of SFFPEs......Page 28
 1.4 Convergence Analysis......Page 29
 1.5 Numerical Examples......Page 31
 References......Page 34
2 Special Functions and Their Link with Nonlinear Rod Theory......Page 38
 2.1 Introduction......Page 39
  2.1.1 A Short Sketch of the Elastica History......Page 41
  2.2.2 The Elastica Nonlinear ODE......Page 42
  2.2.3 Phase Portrait Analysis......Page 44
  2.2.4 Integration......Page 45
  2.3.2 A 3D Rod Model......Page 49
  2.3.4 Rod Local Rotations Parametrized through the Arc: ɸ = ɸ(s)......Page 51
   2.3.4.2 The Free End Rotation ɸ0......Page 52
  2.3.5 Elastica Coordinates x(s), y(s) Parametrized through Its Arc......Page 54
  2.4.1 The Main Assumptions......Page 56
  2.4.3 The Heavy Cantilever......Page 58
  2.4.4 A Tip Sheared Horizontal Cantilever......Page 59
   2.4.4.2 A Treatment by Elliptic Integrals......Page 60
   2.4.4.3 Consistency with the Literature......Page 62
   2.4.4.4 How to Compute the Tip Position after the Strain......Page 63
  2.4.5 The Cantilever Loaded by Sinusoidal Bending Moment......Page 64
  2.4.6 The Cantilever Inflected by Hydrostatic Pressure......Page 65
  2.5.2 The Heavy Flagpole under a Transverse Wind......Page 66
  2.5.3 The Analytical Solution of the Third Order ODE......Page 68
  2.6.2 Hypergeometric Tools......Page 71
   2.6.3.1 First Subsystem: The Heavy Cantilever......Page 72
   2.6.3.3 Consistence and Detection of the Statically Redundant Unknown......Page 74
  2.6.4 Conclusions about the Statically Indeterminate Unknowns......Page 75
 References......Page 77
 3.1 Introduction......Page 80
 3.2 Definitions and Mathematical Preliminaries......Page 84
  3.3.1 Wavelets and the SKCWs......Page 85
  3.3.3 Convergence and Error Analysis......Page 86
  3.3.4 The Operational Matrix of Variable-Order Fractional Derivative (OMV-FD)......Page 89
 3.4 The Proposed Method......Page 90
 3.5 Illustrative Examples......Page 92
 3.6 Conclusion......Page 96
 References......Page 97
 4.1 Introduction......Page 104
 4.2 Preliminaries Results......Page 105
 4.3 Hyers–Ulam–Rassias Stability in a Finite Interval......Page 109
 4.4 Hyers–Ulam Stability in a Finite Interval......Page 112
 4.5 Hyers–Ulam–Rassias Stability in an Infinite Interval......Page 114
 4.6 Conclusions......Page 116
 References......Page 117
 5.1 Introduction......Page 120
 5.2 Mathematical Modeling......Page 122
  5.3.1 Solution of Energy Equation......Page 124
  5.3.2 Solution of Momentum Equation......Page 125
 5.4 Parametric Studies......Page 126
 Appendix 5.A......Page 131
 References......Page 132
6 The Hyperbolic Maximum Principle Approach to the Construction of Generalized Convolutions......Page 136
 6.1 Introduction......Page 137
  6.2.1 Solutions of the Sturm–Liouville Equation......Page 140
  6.2.2 Sturm–Liouville Type Transforms......Page 142
  6.2.3 Diffusion Processes......Page 145
 6.3 The Hyperbolic Equation ℓ[sub(x)]f = ℓ[sub(y)]f......Page 146
  6.3.1 Existence and Uniqueness of Solution......Page 147
  6.3.2 Maximum Principle and Positivity of Solution......Page 150
  6.4.1 Definition and First Properties......Page 152
  6.4.2 Sturm–Liouville Transform of Measures......Page 154
 6.5 The Product Formula......Page 156
 6.6 Harmonic Analysis on L[sub(p)] Spaces......Page 158
  6.7.1 Infinite Divisibility of Measures and the Lévy–Khintchine Representation......Page 161
  6.7.2 Convolution Semigroups and Their Contraction Properties......Page 162
  6.7.3 Additive and Lévy Processes......Page 164
 6.8 Examples......Page 169
 References......Page 173
 7.1 Introduction......Page 178
  7.2.1 Definition......Page 179
  7.2.2 Integral Representation of F(Z) and Twisted Cohomology......Page 180
  7.2.3 Twisted Homology and Twisted Cycles......Page 181
  7.2.4 Differential Equations of F(Z)......Page 182
  7.2.5 Nonprojected Formulation......Page 184
 7.3 Generalized Hypergeometric Functions on Gr(2, n + 1)......Page 185
  7.4.1 Basics of Gauss’ Hypergeometric Function......Page 186
  7.4.2 Reduction to Gauss’ Hypergeometric Function 1: From Defining Equations......Page 187
  7.4.3 Reduction to Gauss’ Hypergeometric Function 2: Use of Twisted Cohomology......Page 188
  7.4.4 Reduction to Gauss’ Hypergeometric Function 3: Permutation Invariance......Page 191
  7.4.5 Summary......Page 193
 References......Page 194
 8.1 Prerequisites......Page 196
  8.2.1 Laurent Expansion to q-Expansion......Page 198
  8.2.2 RHP → D......Page 200
  8.2.3 Fourier Series as an Intrinsic Property of the Monolog......Page 203
  8.2.4 Boundary Functions of Certain Lambert Series......Page 205
  8.2.5 Control Theory in the Unit Disc......Page 207
  8.3.1 GNP......Page 208
  8.3.2 Robust Stabilizer......Page 211
 References......Page 212
 9.1 Introduction......Page 214
 9.2 Transforms Definition......Page 215
  9.3.2 Transforms of Power Functions......Page 216
  9.3.3 Convolution Property......Page 217
  9.3.6 Mellin Transform......Page 218
  9.3.7 Fractional Differentiation and Integration......Page 219
  9.3.8 Limit Behavior......Page 221
   9.3.9.2 Exponential, Trigonometric, and Hyperbolic Functions......Page 222
  9.4.1 Integral Representation of the Wright Function......Page 223
  9.4.3 Fractional Differential Equations......Page 224
   9.4.3.2 Equations with Caputo and Weyl Derivatives......Page 225
   9.4.3.3 Fundamental Solution for a Higher-Order Parabolic Equation......Page 226
 References......Page 227
 10.1 Introduction......Page 230
 10.2 Maxwell Equations as Wave Equations......Page 234
 10.3 A Titbit about Differential Forms......Page 236
 10.4 Algebraic Introduction to Cohomology......Page 237
 10.5 Vectorial Stokes Theorem......Page 238
 References......Page 243
 11.1 Introduction......Page 244
 11.2 Concepts of Fractional Integral Operators......Page 245
 11.3 Integral Inequalities via Fractional Integral Operators......Page 249
 11.4 New Results via Generalized Fractional Integral Operators......Page 254
 References......Page 258
 12.1 Introduction......Page 260
 12.2 Braaksma Revisited......Page 261
 12.3 Expansion in the Neighborhood of the Singular Point......Page 265
 References......Page 270
13 Categories and Zeta & Möbius Functions: Applications to Universal Fractional Operators......Page 272
 13.1 Riemann Zeta Function and Heuristic Approach of Riemann Hypothesis......Page 273
  13.1.1 Introduction to Zeta Riemann Function Based on Universal Transfer Function: N-Measure in Prime Ξ(s)-Space......Page 274
  13.2.1 An Outlook about Category Theory......Page 275
  13.2.2 Lattices and Exponentiation: First Step to Applications in Physics......Page 277
  13.2.3 Monads: Categorical Foundations of Coarse Graining......Page 278
  13.2.4 Kan Extension and Functorial Division......Page 280
  13.2.5 Adjunction, Order Structures: Emergence of a Pair of Scaling Parameters......Page 281
 13.3 Physical Application: Fractional Differentiation, α-Exponential, and Arrow of Time......Page 283
   13.3.1.1 Set Derivative and Additive Systems......Page 287
   13.3.1.2 Toward Non-Integer Derivatives......Page 289
  13.3.2 Countable versus Real Representation: Hamel Basis, Cauchy Additive Functional, and Extended Autosimilarity......Page 291
   13.3.3.1 Kan Extension and Completion of the Universal Dynamic Models through the Construction of a Topos......Page 294
 13.4 Conclusions and Outlook......Page 297
 References......Page 302
 14.1 Introduction......Page 308
 14.2 General Properties of MEFM......Page 309
 14.3 Implementation of the Method......Page 310
 14.4 Conclusions......Page 318
 References......Page 320
15 Statistical Approach of Mixed Convective Flow of Third-Grade Fluid towards an Exponentially Stretching Surface with Convective Boundary Condition......Page 324
 15.2 Mathematical Formulation......Page 325
  15.3.1 Zero[sup(th)]-Order Deformation Problem......Page 327
  15.3.2 m[sup(th)]-Order Deformation Problems......Page 328
 15.5 Results and Discussion......Page 329
  15.5.1 Analysis of Velocity Profile......Page 331
 15.6 Statistical Paradigm......Page 333
  15.7.1 Statistical Proclamation......Page 334
 15.8 Concluding Remarks......Page 335
 References......Page 336
 16.1 Introduction and Formulation of the Problem......Page 338
 16.2 Representation of Solution of the Equation......Page 340
 16.3 The Main Results......Page 343
 16.4 Conclusion......Page 348
 References......Page 349
 17.1 Introduction......Page 352
 17.2 Mathematical Preliminaries......Page 353
 17.3 Design of the Slave System......Page 354
 17.4 Numerical Method for Fractional Conformable Derivative in the RL Sense......Page 355
  17.5.1 Moore–Spiegel system......Page 356
  17.5.2 Arneodo’s System......Page 358
  17.5.3 Van der Pol Oscillator (VPO)......Page 359
  17.5.4 Chua’s Circuit Sine Function Approach......Page 362
 17.6 Conclusions......Page 364
 Conflicts of Interest......Page 365
 References......Page 366
Index......Page 370