Functional fractional calculus:
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Berlin ; Heidelberg
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2011
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| Edition: | 2. ed. |
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| Item Description: | Literaturangaben |
| Physical Description: | XXVIII, 612 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9783642205446 3642205445 |
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| 100 | 1 | |a Das, Shantanu |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Functional fractional calculus |c Shantanu Das |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c 2011 | |
| 300 | |a XXVIII, 612 S. |b Ill., graph. Darst. |c 24 cm | ||
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IMAGE 1
CONTENTS
I I
BIRTH OF FRACTIONAL CALCULUS 2
FRACTIONAL CALCULUS A GENERALIZATION OF INTEGER ORDER CALCULUS 3
HISTORICAL DEVELOPMENT OF FRACTIONAL CALCULUS 5
1.4.1 THE POPULAR DEFINITIONS OF FRACTIONAL DERIVATIVES/INTEGRALS IN
FRACTIONAL CALCULUS 10
1 INTRODUCTION TO FRACTIONAL CALCULUS., 1.1 INTRODUCTION 1.2 1.3 1.4
1.5
.4. .1 RIEMANN-LIOUVILLE 10
.2 GRUNWALD-LETNIKOV: (DIFFERINTEGRALS) 11
.3 M. CAPUTO (1967) 11
.4 OLDHAM AND SPANIER (1974) 11
.5 K.S. MILLERAND B.ROSS (1993) 11
.6 KOLWANKAR AND GANGAL (1994) 11
ABOUT FRACTIONAL INTEGRATION DERIVATIVES AND DIFFERINTEGRATION 12 .5.1
FRACTIONAL INTEGRATION RIEMANN-LIOUVILLE (RL) 12 FRACTIONAL INTEGRATION
WEYL'S (W) 14
NATURE OF KERNEL FOR FRACTIONAL INTEGRATION 14
FRACTIONAL DERIVATIVES RIEMANN-LIOUVILLE (RL) LEFT HAND DEFINITION (LHD)
15
FRACTIONAL DERIVATIVES CAPUTO RIGHT HAND DEFINITION (RHD) 17 FRACTIONAL
DERIVATIVES OF SAME ORDER BUT DIFFERENT TYPES RL-CAPUTO 19
FRACTIONAL DIFFERINTEGRALS GRUNWALD LETNIKOV (GL) 20 FRACTIONAL
DERIVATIVE WEYL'S 21
SCALE INVARIANCE AND POWER LAW 22
FOURIER TRANSFORM OF FRACTIONAL DERIVATIVE 25
1.6
1.7 1.8
1.9
.5.2 .5.3 .5.4
.5.5 .5.6
.5.7 .5.8 .5.9 .5.10 .5.11 COMPOSITION AND PROPERTY 26
.5.12 FRACTIONAL DERIVATIVE FOR SOME STANDARD FUNCTION 28 SOLUTION OF
FRACTIONAL DIFFERENTIAL EQUATIONS 30
.6.1 ABEL'S FRACTIONAL INTEGRAL EQUATION OF TAUTOCHRONE 31 .6.2
FRACTIONAL DAMPED MOTION 34
.6.3 FORMAL DEFINITION OF FRACTIONAL DIFFERENTIAL AND FRACTIONAL
INTEGRAL EQUATION 35
FRACTIONAL CALCULUS AND LAW OF IRREVERSIBILITY NON-LOCALITY 37 STABLE
RANDOM VARIABLES AND GENERALIZATION OF NORMAL PROBABILITY DENSITY
FUNCTION 38
CONSERVATION OF PROBABILITY 40
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1010602349
DIGITALISIERT DURCH
IMAGE 2
XVIII CONTENTS
1.10 HALF ORDER FRACTIONAL DIFFERENTIATION EMBEDDED IN STANDARD FICK'S
LAW AND ITS EXTENSION TO DESCRIBE ANAMOLOUS DIFFUSION 44 1.11 FRACTIONAL
BROWNIAN MOTION 46
1.12 A THOUGHT EXPERIMENT 48
1.13 QUOTABLE QUOTES ABOUT FRACTIONAL CALCULUS 50
1.14 CONCLUDING COMMENTS 50
2 FUNCTIONS USED IN FRACTIONAL CALCULUS 51
2.1 INTRODUCTION 51
2.2 FUNCTIONS FOR THE FRACTIONAL CALCULUS 51
2.2.1 GAMMA FUNCTION 51
2.2.1.1 REPRESENTATION OF GAMMA FUNCTION 52
2.2.1.2 BASIC PROPERTIES OF GAMMA FUNCTION 52
2.2.2 HYPERGEOMETRIC FUNCTIONS 60
2.2.3 MITTAG-LEFFLER FUNCTION 61
2.2.3.1 ONE-PARAMETER MITTAG-LEFFLER FUNCTION 62
2.2.3.2 TWO PARAMETER MITTAG-LEFFLER FUNCTIONS 63 2.2.3.3 VARIANTS OF
MITTAG-LEFFLER FUNCTION 65
2.2.3.4 LAPLACE TRANSFORMS OF MITTAG-LEFFLER FUNCTION 66 2.2.4 AGARWAL
FUNCTION 67
2.2.5 ERDELYI'S FUNCTION 67
2.2.6 ROBOTNOV-HARTLEY FUNCTION 68
2.2.7 MILLER ROSS FUNCTION 68
2.2.8 GENERALIZED COSINE AND SINE FUNCTION 71
2.2.9 GENERALIZED R FUNCTION AND G FUNCTION 73
2.2.9 A RELATION TO ELEMENTARY FUNCTIONS 74
2.2.9.2 RELATIONSHIP OF/? FUNCTION TO OTHER GENERALIZED FUNCTION 74
2.2.9.3 FURTHER GENERALIZED FUNCTION (G FUNCTION) 75 2.2.10 BESSEL
FUNCTION 75
2.3 LIST OF LAPLACE AND INVERSE LAPLACE TRANSFORMS RELATED TO FRACTIONAL
CALCULUS 77
2.4 PARADOXIAL CONDITIONS FOR USING GENERALIZED DIFFERENTIATION AND
INTEGRATION EXPRESSIONS AND CAUTIONS 81
2.5 NON-EXPONENTIAL RELAXATION POWER LAW AND MEMORY INTEGRALS 83 2.6
BOLTZMANN'S SUPERPOSITION PRINCIPLE 86
2.7 MOTIVATION TO USE HIGHER TRANSCENDENTAL FUNCTIONS TO SOLVE
FRACTIONAL DIFFERENTIAL EQUATIONS 87
2.8 FRACTIONAL DERIVATIVES AND INTEGRALS OF IMPORTANT FUNCTIONS WITH USE
OF HIGHER TRANSCENDENTAL FUNCTIONS 89
2.9 IRREGULAR FUNCTIONS AND MEASURE OF IRREGULARITY (ROUGHNESS) WITH BOX
DIMMENSION, HOLDER AND HURST'S EXPONENTS 92
2.9.1 MEASURE OF ROUGHNESS OF GRAPH 93
2.9.2 GENERATION OF IRREGULAR GRAPH 94
2.9.3 DETERMINATION OF BOX-DIMENSION OF AN IRREGULAR GRAPH 95
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CONTENTS XIX
2.9.4 DIFFERENCE IN PERSISTENT ANTI PERSISTENT NOISE AND MOTION FROM
POWER LAW OF POWER SPECTRAL DENSITY 97
2.10 CONCLUDING COMMENTS 98
3 OBSERVATION OF FRACTIONAL CALCULUS IN PHYSICAL SYSTEM DESCRIPTION 101
3.1 INTRODUCTION 101
3.2 TEMPERATURE HEAT FLUX RELATIONSHIP FOR HEAT FLOWING IN SEMI-INFINITE
CONDUCTOR 102
3.3 SINGLE THERMOCOUPLE JUNCTION TEMPERATURE IN MEASUREMENT OF HEAT FLUX
104
3.4 HEAT TRANSFER 107
3.5 DRIVING POINT IMPEDANCE OF SEMI-INFINITE LOSSY TRANSMISSION LINE 110
3.5.1 PRACTICAL APPLICATION OF THE SEMI-INFINITE LINE IN CIRCUITS 116
3.5.1.1 SEMI-INTEGRATOR CIRCUIT 116
3.5.1.2 SEMI-DIFFERENTIATOR CIRCUIT 118
3.5.2 APPLICATION OF FRACTIONAL INTEGRAL AND FRACTIONAL DIFFERENTIATOR
CIRCUIT IN CONTROL SYSTEM 120
3.5.3 BODE'S INTEGRALS 122
3.6 SEMI INFINITE LOSSLESS TRANSMISSION LINE 124
3.7 PARTIAL DIFFERENTIAL EQUATIONS AND OPERATIONAL CALCULUS 130 3.8
FICK'S DIFFUSION DISCUSSION 132
3.9 CATTANEO DIFFUSION 137
3.10 ANOMALOUS DIFFUSION 139
3.11 TRUNCATION OF SEMI-INFINITE SYSTEM TO A FINITE SYSTEM 140 3.12
APPROXIMATING THE HALF ORDER BY SELF SIMILAR STRUCTURE AND ITS RELATION
TO CONTINUED FRACTION EXPANSION 143
3.13 DYNAMICS OF CHAIN NETWORK 147
3.14 DYNAMICS OF CHARGED CHAIN NETWORK IN ELECTRIC FIELD 153 3.15
CONCLUDING COMMENTS 156
4 CONCEPT OF FRACTIONAL DIVERGENCE AND FRACTIONAL CURL 157 4.1
INTRODUCTION 157
4.2 CONCEPT OF FRACTIONAL DIVERGENCE FOR PARTICLE FLUX 157
4.3 FRACTIONAL KINETIC EQUATION 159
4.4 DISCRETE DIFFERENCE AND CONTINUM LIMIT AND DIFFERENTIAL OPERATOR IN
RANDOM WALK CONTEXT 162
4.4.1 INTEGER ORDER DISCRETE DIFFERENCE AND CONTINUUM LIMIT AND
DIFFERENTIAL OPERATOR 162
4.4.2 FRACTIONAL ORDER DISCRETE DIFFERENCE AND CONTINUUM LIMIT AND
FRACTIONAL DIFFERENTIAL OPERATOR 163
4.4.3 FOURIER REPRESENTATION OF FRACTIONAL DIFFERENCE AND DERIVATIVE 165
4.4.4 STOCHASTIC FRACTIONAL DIFFERENCE EQUATIONS 166
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XX CONTENTS
4.4.5 RANDOM WALKER WITH MEMORY CONCEPT OF PERSISTENCE AND
ANTI-PERSISTENCE WALK WITH LONG MEMORY AND SHORT TERM MEMORY 170
4.5 NUCLEAR REACTOR NEUTRON FLUX DESCRIPTION 172
4.6 CLASSICAL CONSTITUTIVE NEUTRON DIFFUSION EQUATION 173
4.6.1 DISCUSSION ON CLASSICAL CONSTITUTIVE EQUATIONS 174 4.6.2 GRAPHICAL
EXPLANATION 175
4.6.3 ABOUT SURFACE FLUX CURVATURE 176
4.6.4 STATISTICAL AND GEOMETRICAL EXPLANATION FOR NON-LOCAL DIVERGENCE
177
4.6.5 POINT KINETIC EQUATION IN HETEROGENEOUS BACKGROUND 178 4.6.6
REVISITING THE REALM OF BROWNIAN MOTION 181
4.6.7 THE CONTINUOUS TIME RANDOM WALK (CTRW) MODEL 182 4.7 DIFFUSION
WITH LONG RESTS 184
4.8 DIFFUSION WITH LONG JUMPS 186
4.9 FRACTIONAL DIVERGENCE IN NEUTRON DIFFUSION EQUATIONS 190 4.9.1
SOLUTION OF CLASSICAL CONSTITUTIVE NEUTRON DIFFUSION EQUATION (INTEGER
ORDER) 192
4.9.2 SOLUTION OF FRACTIONAL DIVERGENCE BASED NEUTRON DIFFUSION EQUATION
(FRACTIONAL ORDER) 193
4.9.3 FRACTIONAL GEOMETRICAL BUCKLING AND NON-POINT REACTOR KINETICS 194
4.9.4 FRACTIONAL REACTOR KINETIC EQUATION 195
4.9.5 GROWTH OF NEUTRON FLUX WITH TIME FOR DIFFERENT VALUES OF
FRACTIONAL ORDERS AND FRACTIONAL CRITICALITY 199
4.10 CONCEPT OF FRACTIONAL CURL IN ELECTROMAGNETICS 200
4.10.1 CONCEPT OF CHIRALITY 200
4.10.2 DUALITY OF SOLUTIONS 200
4.10.3 FRACTIONAL CURL OPERATOR 201
4.10.4 WAVE PROPAGATION IN UNBOUNDED CHIRAL MEDIUM 201 4.10.5 REFLECTION
IN CHIRAL MEDIUM 203
4.10.6 TRANSVERSE WAVE IMPEDANCE 205
4.10.7 PROPAGATION OF ELECTROMAGNETIC WAVES IN BI-ISOTROPIC MEDIUM 207
4.10.8 FRACTIONAL NON-SYMMETRIC TRANSMISSION LINE 208 4.10.9 INPUT
IMPEDANCE OF TERMINATED FRACTIONAL NON-SYMMETRIC LINE 209
4.11 CONCLUDING COMMENTS 210
5 FRACTIONAL DIFFERINTEGRATIONS INSIGHT CONCEPTS 213
5.1 INTRODUCTION 213
5.2 CALCULATING FRACTIONAL INTEGRAL 213
5.2.1 EXISTENCE OF FRACTIONAL DIFFEREINTEGRATION 214
5.2.2 USEFUL PROCEDURE FOR CALCULATING FRACTIONAL INTEGRAL 215 5.2.3
CALCULATING FRACTIONAL INTEGRAL WITH NON-ZERO LOWER LIMIT 217 5.2.4
FRACTIONAL INTEGRAL FOR ANALYTICAL FUNCTION 217
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CONTENTS XXI
5.3 FRACTIONAL DIFFERINTEGRATION OF PRODUCT OF TWO FUNCTIONS 218 5.4
SYMBOL STANDARDIZATION AND DESCRIPTION FOR DIFFERINTEGRATION 221 5.5
RIEMANN-LIOUVILLE FRACTIONAL DIFFERINTEGRAL 222
5.5.1 SCALE TRANSFORMATION 222
5.5.2 CHANGING SHAPE OF CURVE WHILE OBTAINING FRACTIONAL INTEGRATION AND
DIFFERENTIATION 225
5.5.3 HOMOGENEOUS AND HETEROGENEOUS SCALES IN FRACTIONAL
INTEGRATION/DIFFERENTIATION 226
5.5.4 CONVOLUTION EXAMPLE 227
5.5.5 PRACTICAL EXAMPLE OF RL DIFFERITEGRATION IN ELECTRICAL CIRCUIT
ELEMENT DESCRIPTION 231
5.6 GRUNWALD-LETNIKOV FRACTIONAL DIFFERINTERATION 234
5.7 UNIFICATION OF DIFFERINTEGRATION THROUGH BINOMIAL COEFFICIENTS 237
5.8 SHORT MEMORY PRINCIPLE- A MOVING START POINT APPROXIMATION AND ITS
ERROR 240
5.9 MATRIX APPROACH TO DISCRETIZE FRACTIONAL DIFFERINTEGRATION AND
WEIGHTS 242
5.10 USE OF DISCRETE FRACTIONAL ORDER DIFFERINTEGRATION IN FRACTIONAL
ORDER SIGNAL PROCESSING 244
5.11 INFINITESIMAL ELEMENT GEOMETRICAL INTERPRETATION OF FRACTIONAL
DIFFERINTEGRATIONS 247
5.11.1 INTEGRATION 247
5.11.2 DIFFERENTIATION 249
5.12 LOCAL FRACTIONAL DERIVATIVES (LFD) 250
5.12.1 KG- LFD FOR ORDER LESS THAN UNITY 251
5.12.2 KG- LFD FOR ORDER GREATER THAN UNITY 252
5.12.3 CRITICAL ORDER OF A FUNCTION AND ITS RELATION TO THE BOX
DIMENSION 252
5.12.4 INFORMATION CONTENT IN LFD 255
5.12.5 FINDING HOLDER EXPONENT FOR SINGULARITY AT A POINT 259 5.13
NUMERICAL SOLUTION OF FRACTIONAL ORDER DIFFERENTIAL EQUATION BY USE OF
GRUNWALD-LETNIKOV TECHNIQUE 260
5.13.1 THE ALGORITHM 260
5.13.2 OBTAINING THE STEP RESPONSE 261
5.13.3 FRACTIONAL ORDER SYSTEM AND INTEGER ORDER SYSTEM COMPARISION 261
5.13.3.1 ORDER OF THE FOS-N 261
5.13.3.2 SIGNIFICANCE OF PARAMETERS A AND B 262
5.13.3.3 EFFECT OF INITIAL CONDITIONS 264
5.14 LINE, SURFACE AND VOLUME INTEGRATION OF FRACTAL DISTRIBUTIONS 265
5.15 FRACTIONAL GENERALIZATION OF GAUSS'S LAW AND STROKE'S LAW 268 5.16
CONCLUDING COMMENTS 269
6 INITIALIZED DIFFERINTEGRALS AND GENERALIZED CALCULUS
. 271 6.1 INTRODUCTION 271
6.2 NOTATIONS OF DIFFERINTEGRALS 272
6.3 REQUIREMENT OF INITIALIZATION 273
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XXII CONTENTS
6.4 INITIALIZATION FRACTIONAL INTEGRATION (RIEMANN-LIOUVILLE APPROACH)
274 6.4.1 TERMINAL INITIALIZATION 275
6.4.2 SIDE-INITIALIZATION 276
6.5 INITIALIZING FRACTIONAL DERIVATIVE (RIEMANN-LIOUVELLE APPROACH) 277
6.5.1 TERMINAL INITIALIZATION 278
6.5.2 SIDE-INITIALIZATION 279
6.6 INITIALIZING FRACTIONAL DIFFERINTEGRALS (GRUNWALD-LETNIKOV APPROACH)
280
6.7 PROPERTIES AND CRITERIA FOR GENERALIZED DIFFERINTEGRALS 282 6.7.1
TERMINAL CHARGING 285
6.7.2 SIDE-CHARGING 286
6.8 INITIALIZATION WITH CAPUTO DERIVATIVE AND ITS DIFFICULTIES 286 6.8.1
RELATION BETWEEN CAPUTO AND RIEMAN-LIOUVELLI (RL) FRACTIONAL DERIVATIVE
AND ISSUES RELATING TO INITIALIZATION 287 6.8.2 UN-INITIALIZED
DERIVATIVES RL AND CAPUTO 288
6.8.3 EVALUATION OF RL AND CAPUTO DERIVATIVE FROM THE START POINT OF THE
FUNCTION 291
6.8.4 INITIALIZATION OF CAPUTO DERIVATIVE 292
6.8.5 GENERALIZATION OF RL AND CAPUTO FORMULATIONS 298
6.8.6 OBSERVATIONS REGARDING DIFFICULTIES IN CAPUTO INITIALIZATION AND
DEMANDING PHYSICAL CONDITIONS VIS-A-VIS RL INITIALIZATION CONDITIONS AND
RELATION TO PHYSICS IN SOLVING FRACTIONAL ORDER DIFFERENTIAL EQUATIONS
300
6.9 FRACTIONAL DIFFERINTEGRATIONS FOR PERIODIC SIGNALS 301
6.9.1 FRACTIONAL DERIVATIVE/INTEGRAL OF GENERALIZED PERIODIC FUNCTION
301
6.9.2 FRACTIONAL DERIVATIVE OF PERIODIC FUNCTION WITH LOWER TERMINAL NOT
AT MINUS INFINITY 303
6.10 FRACTIONAL ADVECTION DISPERSION EQUATION AND ITS SOLUTION 305 6.11
IDENTIFICATION OF RANDOM DELAYS 307
6.11.1 RANDOM DELAY A STOCHASTIC BEHAVIOR 307
6.11.2 ABOUT LEVY DISTRIBUTION 309
6.11.3 FRACTIONAL STOCHASTIC DYNAMIC MODEL 311
6.11.4 FRACTIONAL DELAY DYNAMICS 317
6.11.5 THE RANDOM DYNAMICS OF COMPUTER CONTROL SYSTEM 320 6.12
CONCLUDING COMMENTS 321
7 GENERALIZED LAPLACE TRANSFORM FOR FRACTIONAL DIFFERINTEGRALS 323 7.1
INTRODUCTION 323
7.2 RECALLING LAPLACE TRANSFORM FUNDAMENTALS 323
7.3 LAPLACE TRANSFORM OF FRACTIONAL INTEGRALS 329
7.3.1 DECOMPOSITION OF FRACTIONAL INTEGRAL IN INTEGER ORDER 330 7.3.2
DECOMPOSITION OF FRACTIONAL ORDER INTEGRAL IN FRACTIONAL ORDER 334
7.4 LAPLACE TRANSFORMATION OF FRACTIONAL DERIVATIVES 336
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CONTENTS XXII!
7.4.1 DECOMPOSITION OF FRACTIONAL ORDER DERIVATIVE IN INTEGER ORDER 338
7.4.2 DECOMPOSITION OF FRACTIONAL DERIVATIVE IN FRACTIONAL ORDER. 342
7.4.3 EFFECT OF TERMINAL CHARGING ON LAPLACE TRANSFORMS 343 7.5 START
POINT SHIFT EFFECT 344
7.5.1 FRACTIONAL INTEGRAL 344
7.5.2 FRACTIONAL DERIVATIVE 345
7.6 LAPLACE TRANSFORM OF INITIALIZATION FUNCTION 345
7.6.1 FRACTIONAL INTEGRAL 345
7.6.2 FRACTIONAL DERIVATIVE 346
7.7 EXAMPLES OF INITIALIZATION IN FRACTIONAL DIFFERENTIAL EQUATIONS 346
7.8 THE FUNDAMENTAL FRACTIONAL ORDER DIFFERENTIAL EQUATION 350 7.8.1 THE
GENERALIZED IMPULSE RESPONSE FUNCTION 351
7.9 PROBLEM OF SCALAR INITIALIZATION 355
7.10 PROBLEM OF VECTOR INITIALIZATION 357
7.11 LAPLACE TRANSFORM S -» W PLANE FOR FRACTIONAL CONTROLS
STABILITY. 360 7.12 RATIONAL APPROXIMATIONS OF FRACTIONAL LAPLACE
OPERATOR 362 7.12.1 FINDING ARBITRARY ROOT OF POLYNOMIAL APPROXIMATION
FOR FRACTIONAL LAPLACE OPERATOR 363
7.12.2 FRACTIONAL POWER POLE AND FRACTIONAL POWER ZERO TO APPROXIMATE
FRACTIONAL LAPLACE OPERATOR 364 7.12.2.1 SINGULARITY STRUCTURE FOR A
SINGLE FRACTIONAL POWER POLE (FPP) 365
7.12.2.2 GEOMETRICAL DERIVATION OF RECURRING RELATIONSHIP OF FRACTIONAL
POWER POLE FOR FRACTIONAL INTEGRATION 366
7.12.2.3 RECURSIVE ALGORITHM FOR FRACTIONAL POWER POLE. 368 7.12.2.4
SINGULARITY STRUCTURE FOR A SINGLE FRACTIONAL POWER ZERO (FPZ) 370
7.13 REALIZATION OF CONSTANT PHASE ELEMENT 371
7.13.1 ASYMPTOTIC BODE PHASE PLOT 372
7.13.2 POLE ZERO CALCULATION FOR CONSTANT PHASE 373
7.13.3 CALCULATION FOR POLE-ZERO POSITION OF FRACTIONAL ORDER IMPEDANCE
376
7.13.4 ALGORITHM 376
7.13.5 DESIGN AND PERFORMANCE OF FRACTIONAL ORDER IMPEDANCE 378 7.14
LAPLACE TRANSFORM AND CHARATERIZATION OF TYPE OF FRACTIONAL DERIVATIVE
380
7.15 GENERALIZED STATIONARY CONDITIONS 385
7.16 CONCLUDING COMMENTS 386
8 APPLICATION OF GENERALIZED FRACTIONAL CALCULUS IN ELECTRICAL CIRCUIT
ANALYSIS AND ELECTROMAGNETICS 387
8.1 INTRODUCTION 387
8.2 ELECTRONICS OPERATIONAL AMPLIFIER CIRCUITS 387
IMAGE 8
XXIV CONTENTS
8.2.1 OPERATIONAL AMPLIFIER CIRCUIT WITH LUMPED COMPONENTS 387 8.2.2
OPERATIONAL AMPLIFIER INTEGRATOR WITH LUMPED ELEMENT 389 8.2.3
OPERATIONAL AMPLIFIER INTEGRATOR WITH DISTRIBUTED ELEMENT 390
8.2.4 OPERATIONAL AMPLIFIER DIFFERENTIAL CIRCUIT WITH LUMPED ELEMENTS
392
8.2.5 OPERATIONAL AMPLIFIER DIFFERENTIATOR WITH DISTRIBUTED ELEMENT 393
8.2.6 OPERATIONAL AMPLIFIER AS ZERO ORDER GAIN WITH LUMPED COMPONENTS
394
8.2.7 OPERATIONAL AMPLIFIER AS ZERO ORDER GAIN WITH DISTRIBUTED ELEMENTS
395
8.2.8 OPERATIONAL AMPLIFIER CIRCUIT FOR SEMI-DIFFERINTEGRATION BY
SEMI-INFINITE LOSSY LINE 396
8.2.9 OPERATIONAL AMPLIFIER CIRCUIT FOR SEMI-INTEGRATOR 397 8.2.10
OPERATIONAL AMPLIFIER CIRCUIT FOR SEMI-DIFFERENTIATOR 398 8.2.11
CASCADED SEMI-INTEGRATORS 399
8.2.12 SEMI-INTEGRATOR SERIES WITH SEMI-DIFFERENTIATOR CIRCUIT 400 8.3
BATTERY DYNAMICS 400
8.3.1 BATTERY AS FRACTIONAL ORDER SYSTEM 400
8.3.2 BATTERY CHARGING PHASE 401
8.3.3 BATTERY DISCHARGE PHASE 405
8.4 TRACKING FILTER 407
8.5 FRACTIONAL ORDER STATE VECTOR REPRESENTATION IN CIRCUIT THEORY 410
8.6 REALIZATION OF FRACTIONAL ORDER TRANSFER FUNCTION F O R ^ "^ 415
8.6.1 FRACTIONAL ORDER PID CONTROLLER APPROXIMATION BY FPP AND FPZ 415
8.6.2 FRACTIONAL ORDER INTEGRATOR 415
8.6.2.1 RATIONAL APPROXIMATION 415
8.6.3 FRACTIONAL ORDER DIFFERENTIATOR 418
8.6.3.1 RATIONAL APPROXIMATION 418
8.6.4 FRACTIONAL P F T/ CONTROLLER 419
8.6.4.1 RATIONAL APPROXIMATION 419
8.6.5 REALIZATION OF FRACTIONAL ORDER ELEMENT BY CIRCUIT NETWORK .
420 8.6.5.1 IMPEDANCE FUNCTIONS OF A SINGLE PORT NETWORK 420 8.6.5.2
IMPEDANCE FUNCTIONS OF A TWO PORT NETWORK 421 8.6.5.3 IMPROVED TWO PORT
NETWORK 421
8.7 ADVANCE DIGITAL ALGORITHMS REALIZATION FOR FRACTIONAL CONTROLS 424
8.7.1 CONCEPT OF GENERATING FUNCTION 425
8.7.2 DIGITAL FILTER REALIZATION BY RATIONAL FUNCTION APPROXIMATION FOR
FRACTIONAL OPERATOR 426
8.7.3 FILTER STABILITY CONSIDERATION 428
8.8 CHARGE CONSERVATION FOR FRACTAL DISTRIBUTION 429
8.9 ELECTRIC FIELD OF FRACTAL DISTRIBUTION 430
8.9.1 ELECTRIC FIELD AND COULOMB'S LAW FOR FRACTAL DISTRIBUTION 430
8.9.2 GAUSS'S LAW FOR FRACTAL DISTRIBUTION 430
IMAGE 9
CONTENTS XXV
8.10 MAGNETIC FIELD OF FRACTAL DISTRIBUTION 431
8.10.1 BIOT-SAVART LAW FOR FRACTAL DISTRIBUTION 431
8.10.2 AMPERE'S LAW FOR FRACTAL DISTRIBUTION 432
8.11 MAXWELL EQUATION FOR FRACTAL DISTRIBUTION 432
8.12 ELECTRIC DIPOLE MOMENTS FOR FRACTAL DISTRIBUTION 434
8.13 CONCLUDING COMMENTS 436
9 APPLICATION OF GENERALIZED FRACTIONAL CALCULUS IN OTHER SCIENCE AND
ENGINEERING FIELDS 437
9.1 INTRODUCTION 437
9.2 DIFFUSION MODEL IN ELECTROCHEMISTRY 437
9.3 ELECTRODE-ELECTROLYTE INTERFACE IMPEDANCE 438
9.3.1 NORMAL DIFFUSION IN A FINITE BOUNDARY SYSTEM 439
9.3.2 ANOMALOUS DIFFUSION IN FINITE BOUNDARY SYSTEM 441 9.3.2.1
DIFFUSION WITH FRACTIONAL CONTINUITY EQUATION 441 9.3.2.2 DIFFUSION WITH
FRACTIONAL DIFFERENTIAL CONSTITUTIVE EQUATION 442
9.3.2.3 DIFFUSION WITH FRACTIONAL INTEGRAL CONSTITUTIVE EQUATION 443
9.4 CAPACITOR THEORY 444
9.5 FRACTANCE CIRCUIT 446
9.6 FEEDBACK CONTROL SYSTEM 448
9.6.1 CONCEPT OF ISO-DAMPING 457
9.6.2 FREQUENCY DOMAIN DESIGN FOR FRACTIONAL ORDER PLANT AND FRACTIONAL
ORDER CONTROLLER TUNING 459
9.6.3 FAMILY OF FRACTIONAL ORDER CONTROLLERS 462
9.6.4 FRACTIONAL VECTOR FEEDBACK CONTROLLER 462
9.6.5 OBSERVER IN FRACTIONAL VECTOR SYSTEM 463
9.6.6 MODERN ASPECTS OF FRACTIONAL CONTROL 465
9.7 FRACTIONAL COMPENSATOR 467
9.7.1 GENERALIZED COMPENSATOR 467
9.7.2 FREQUENCY CHARACTERISTICS OF THE LEAD COMPENSATOR 467 9.7.3
COMPENSATION USING A FRACTIONAL LEAD COMPENSATOR 469 9.8 PHASE SHAPING
WITH FRACTIONAL ORDER DIFFER-INTEGRATOR 473 9.8.1 APPLICATION OF BODE'S
PHASE INTEGRAL 473
9.8.2 PLANT WITH TUNED WITH INTEGER ORDER PID MADE ISO-DAMPED WITH
ADDITIONAL FRACTIONAL DIFFER-INTEGRATOR 476 9.9 VISCOELASTICITY
(STRESS-STRAIN) 482
9.10 VIBRATION DAMPING SYSTEM 487
9.11 THE NON-NEWTONIAN FLUID ANAMOLOUS BEHAVIOR WITH MEMORY 488 9.12
CONCLUDING COMMENTS 492
10 SYSTEM ORDER IDENTIFICATION AND CONTROL 493
10.1 INTRODUCTION 493
10.2 FRACTIONAL ORDER SYSTEMS 493
10.3 CONTINUOUS ORDER DISTRIBUTION 495
IMAGE 10
XXVI CONTENTS
10.4 DETERMINATION OF ORDER DISTRIBUTION FROM FREQUENCY DOMAIN
EXPERIMENTAL DATA 499
10.5 ANALYSIS OF CONTINUOUS ORDER DISTRIBUTION 501
10.6 VARIABLE ORDER SYSTEM 513
10.6.1 RL DEFINITION FOR VARIABLE ORDER 513
10.6.2 LAPLACE TRANSFORMS AND TRANSFER FUNCTION OF VARIABLE ORDER SYSTEM
515
10.6.3 GL DEFINITION FOR VARIABLE ORDER 517
10.7 GENERALIZED PID-CONTROLS 518
10.8 CONTINUUM ORDER FEED BACK CONTROL SYSTEM 520
10.9 TIME DOMAIN RESPONSE OF SINUSOIDAL INPUTS FOR FRACTIONAL ORDER
OPERATOR 522
10.10 FREQUENCY DOMAIN RESPONSE OF SINUSOIDAL INPUTS FOR FRACTIONAL
ORDER OPERATOR 523
10.11 ULTRA-DAMPED SYSTEM RESPONSE 524
10.12 HYPER-DAMPED SYSTEM RESPONSE 525
10.13 COMPLEX ORDER DIFFERINTEGRATIONS 526
10.14 ORDERING THE DISORDER OF SYSTEM 531
10.14.1 DISORDERED RELAXATION WITH MULTIPLE STATES AND RELAXATION
CONSTANTS 531
10.14.2 APPEARANCE OF FRACTIONAL DERIVATIVE IN DISORDERED RELAXATION 532
10.14.3 GENERALIZATION OF DISORDERED RELAXATION 533 10.14.3.1
INTERMITTENCY DISORDER 534
10.14.3.2 STRONG INTENSE RELAXATION 536
10.14.3.3 WEAK INTERMITTENT RELAXATION 537
10.14.3.4 OSCILLATING RELAXATION 537
10.14.3.5 GENERALIZED DYNAMIC CRITICAL INDEX OF RELAXATION WITH
INTERMITTENCY 538
10.14.3.6 SPATIAL DISORDER 540
10.14.3.7 HYBRID DISORDER WITH INTERMITTENCY AND SPATIAL HETEROGENEITY
541
10.15 IDENTIFICATION OF FRACTIONAL STOCHASTIC PROCESSES 543
10.15.1 FITTING STOCHASTIC DATA INTO PARAMETERS OF LEVY STABLE
DISTRIBUTION 543
10.15.2 ESTIMATION OF HURST INDEX BY RESCALED RANGE (R/S METHOD) FOR
STOCHASTIC DATA 545
10.16 THE CONCEPT OF SYSTEM ORDER AND DISADVANTAGE OF FRACTIONAL ORDER
SYSTEM 546
10.17 CONCLUDING COMMENTS 548
11 SOLUTION OF GENERALIZED DIFFERENTIAL EQUATION SYSTEMS 549 11.1
INTRODUCTION 549
11.2 GENERALIZED DYNAMIC SYSTEM AND EVOLUTION OF IT'S SOLUTION BY
PRINCIPLE OF ACTION REACTION 550
IMAGE 11
CONTENTS XXVII
11.3 PHYSICAL REASONING TO SOLVE FIRST ORDER SYSTEM AND ITS MODE
DECOMPOSITION 551
11.4 PHYSICAL REASONING TO SOLVE SECOND ORDER SYSTEM AND ITS
MODE-DECOMPOSITION 555
11.5 ADOMIAN DECOMPOSITION FUNDAMENTALS AND ADOMIAN POLYNOMIALS 558
11.6 GENERALIZATION OF PHYSICAL LAW OF NATURE VIS-A-VIS ADM 564 11.7 ADM
APPLIED TO FIRST ORDER LINEAR DIFFERENTIAL EQUATION AND
MODE-DECOMPOSITION SOLUTION 565
11.8 ADM APPLIED TO SECOND ORDER LINEAR DIFFERENTIAL EQUATION SYSTEM AND
MODE-DECOMPOSITION 567
11.9 ADM FOR FIRST ORDER LINEAR DIFFERENTIAL EQUATION SYSTEM WITH HALF
ORDER ELEMENT AND MODE-DECOMPOSITION 569 11.10 ADM FOR SECOND ORDER
SYSTEM, WITH HALF ORDER ELEMENT AND IT'S PHYSICS 570
11.10.1 FORCING FUNCTION AS DELTA FUNCTION 570
11.10.2 FORCING FUNCTION AS STEP FUNCTION 572
11.10.3 EXPLANATION PHYSICAL ACTION REACTION PROCESS VIS-A-VISADM 573
11.11 APPLICATION OF DECOMPOSITION METHOD IN RL-FORMULATED PARTIAL
FRACTIONAL DIFFERENTIAL EQUATIONS LINEAR DIFFUSION WAVE EQUATION AND
SOLUTION TO IMPULSE FORCING FUNCTION 575 11.12 GENERALIZATION OF
FRACTIONAL ORDER LEADING TERMS IN
DIFFERENTIAL EQUATIONS FORMULATED WITH RIEMANN-LIOUVELLI AND CAPUTO
DEFINITIONS-AND USE OF INTEGER ORDER INITIAL/BOUNDARY CONDITIONS-WITH
DECOMPOSITION METHOD 578
11.12.1 DECOMPOSITION OF CAPUTO DERIVATIVE IN FRACTIONAL DIFFERENTIAL
EQUATIONS 578
11.12.2 RIEMANN-LIOUVELLI (RL) DERIVATIVE AND ITS DECOMPOSITION FOR
SOLVING FRACTIONAL DIFFERENTIAL EQUATION-WITH INTEGER ORDER INITIAL
CONDITION 579 11.13 APPLICATION OF DECOMPOSITION METHOD IN RL FORMULATED
FRACTIONAL DIFFERENTIAL EQUATIONS (NON-LINEAR) AND ITS SOLUTION 581
11.14 APPLICATION OF DECOMPOSITION METHOD IN RL-FORMULATED PARTIAL
FRACTIONAL DIFFERENTIAL EQUATIONS NON-LINEAR
DIFFUSION-WAVE EQUATION AND SOLUTION 583
11.15 DECOMPOSITION METHOD FOR GENERALIZED EQUATION OF MOTION 584 11.16
DECOMPOSITION METHOD FOR DELAY DIFFERENTIAL EQUATION SYSTEM. 586
11.17 PROPOSITION 587
11.17.1 FRACTIONAL INITIAL STATES-CLASSICAL SOLUTION TO FDE 587 11.17.2
BASIC FRACTIONAL ORDER DIFFERENTIAL EQUATION SYSTEM AND ITS CLASSICAL
SOLUTION 590
11.17.3 CLASSICAL SOLUTION TO FRACTIONAL FOKKER-PLANK KOLMOGOROV
EQUATION (FFPK) BY FOURIER-LAPLACE TECHNIQUE 591
IMAGE 12
XXVIII CONTENTS
11.17.4 DECOMPOSITION OF FRACTIONAL DIFFERENTIAL EQUATION PRINCIPLE-AND
EQUIVALENCE OF RL AND CAPUTO DEFINITIONS TO SOLVE FDE WITH INTEGER ORDER
INITIAL STATES 592
11.17.5 APPLICATION TO FRACTIONAL DIFFUSION-WAVE EQUATION WITH INPUT
SINE EXCITATION WITH RL-FORMULATION 596 11.18 OBSERVATIONS 597
11.19 CONCLUDING COMMENTS 598
REFERENCES 599 |
| any_adam_object | 1 |
| author | Das, Shantanu |
| author_facet | Das, Shantanu |
| author_role | aut |
| author_sort | Das, Shantanu |
| author_variant | s d sd |
| building | Verbundindex |
| bvnumber | BV039130376 |
| ctrlnum | (OCoLC)725022668 (DE-599)DNB1010602349 |
| dewey-full | 515.83 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.83 |
| dewey-search | 515.83 |
| dewey-sort | 3515.83 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV039130376 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:14:17Z |
| institution | BVB |
| isbn | 9783642205446 3642205445 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024148733 |
| oclc_num | 725022668 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM |
| owner_facet | DE-19 DE-BY-UBM |
| physical | XXVIII, 612 S. Ill., graph. Darst. 24 cm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| spelling | Das, Shantanu Verfasser aut Functional fractional calculus Shantanu Das 2. ed. Berlin ; Heidelberg Springer 2011 XXVIII, 612 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Literaturangaben Gebrochene Analysis (DE-588)4722475-7 gnd rswk-swf Gebrochene Analysis (DE-588)4722475-7 s DE-604 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3702280&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024148733&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Das, Shantanu Functional fractional calculus Gebrochene Analysis (DE-588)4722475-7 gnd |
| subject_GND | (DE-588)4722475-7 |
| title | Functional fractional calculus |
| title_auth | Functional fractional calculus |
| title_exact_search | Functional fractional calculus |
| title_full | Functional fractional calculus Shantanu Das |
| title_fullStr | Functional fractional calculus Shantanu Das |
| title_full_unstemmed | Functional fractional calculus Shantanu Das |
| title_short | Functional fractional calculus |
| title_sort | functional fractional calculus |
| topic | Gebrochene Analysis (DE-588)4722475-7 gnd |
| topic_facet | Gebrochene Analysis |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3702280&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024148733&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT dasshantanu functionalfractionalcalculus |