Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
San Diego [u.a.]
Acad. Press
1999
|
| Series: | Mathematics in science and engineering
198 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 313 - 335 |
| Physical Description: | XXIV, 340 S. graph. Darst. |
| ISBN: | 0125588402 |
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| 100 | 1 | |a Podlubny, Igor |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Fractional differential equations |b an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications |c by Igor Podlubny |
| 264 | 1 | |a San Diego [u.a.] |b Acad. Press |c 1999 | |
| 300 | |a XXIV, 340 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics in science and engineering |v 198 | |
| 500 | |a Literaturverz. S. 313 - 335 | ||
| 650 | 4 | |a Ableitung gebrochener Ordnung | |
| 650 | 7 | |a Differentiaalvergelijkingen. |2 gtt | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Differential equations |x Numerical solutions | |
| 650 | 4 | |a Fractional calculus | |
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| 655 | 7 | |0 (DE-588)4151278-9 |a Einführung |2 gnd-content | |
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| _version_ | 1804136086820093952 |
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| adam_text | FRACTIONAL DIFFERENTIAL EQUATIONS AN INTRODUCTION TO FRACTIONAL
DERIVATIVES, FRACTIONAL DIFFERENTIAL EQUATIONS, TO METHODS OF THEIR
SOLUTION AND SOME OF THEIR APPLICATIONS BY IGOR PODLUBNY TECHNICAL
UNIVERSITY OF KOSICE, SLOVAK REPUBLIC ACADEMIC PRESS SAN DIEGO * BOSTON
* NEW YORK * LONDON * SYDNEY * TOKYO * TORONTO CONTENTS PREFACE XVII
ACKNOWLEDGEMENTS XXIII 1 SPECIAL FUNCTIONS OF THE FRACTIONAL CALCULUS 1
1.1 GAMMA FUNCTION 1 1.1.1 DEFINITION OF THE GAMMA FUNCTION 1 1.1.2 SOME
PROPERTIES OF THE GAMMA FUNCTION . . . . 2 1.1.3 LIMIT REPRESENTATION OF
THE GAMMA FUNCTION . 4 1.1.4 BETA FUNCTION 6 1.1.5 CONTOUR INTEGRAL
REPRESENTATION 10 1.1.6 CONTOUR INTEGRAL REPRESENTATION OF 1/F(Z) ... 12
1.2 MITTAG-LEFFLER FUNCTION 16 1.2.1 DEFINITION AND RELATION TO SOME
OTHER FUNCTIONS 17 1.2.2 THE LAPLACE TRANSFORM OF THE MITTAG-LEFFLER
FUNCTION IN TWO PARAMETERS 20 1.2.3 DERIVATIVES OF THE MITTAG-LEFFLER
FUNCTION . ... 21 1.2.4 DIFFERENTIAL EQUATIONS FOR THE MITTAG-LEFFLER
FUNCTION 23 1.2.5 SUMMATION FORMULAS 23 1.2.6 INTEGRATION OF THE
MITTAG-LEFFLER FUNCTION . . . . 24 1.2.7 ASYMPTOTIC EXPANSIONS 29 1.3
WRIGHT FUNCTION 37 1.3.1 DEFINITION 37 1.3.2 INTEGRAL REPRESENTATION 37
1.3.3 RELATION TO OTHER FUNCTIONS 38 2 FRACTIONAL DERIVATIVES AND
INTEGRALS 41 2.1 THE NAME OF THE GAME 41 2.2 GRUENWALD-LETNIKOV
FRACTIONAL DERIVATIVES 43 VII I CONTENTS 2.2.1 UNIFICATION OF
INTEGER-ORDER DERIVATIVES AND INTEGRALS 43 2.2.2 INTEGRALS OF ARBITRARY
ORDER 48 2.2.3 DERIVATIVES OF ARBITRARY ORDER 52 2.2.4 FRACTIONAL
DERIVATIVE OF (T - A) SS 55 2.2.5 COMPOSITION WITH INTEGER-ORDER
DERIVATIVES ... 57 2.2.6 COMPOSITION WITH FRACTIONAL DERIVATIVES . . . .
59 2.3 RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES 62 2.3.1 UNIFICATION OF
INTEGER-ORDER DERIVATIVES AND INTEGRALS 63 2.3.2 INTEGRALS OF ARBITRARY
ORDER 65 2.3.3 DERIVATIVES OF ARBITRARY ORDER 68 2.3.4 FRACTIONAL
DERIVATIVE OF (T - A) SS 72 2.3.5 COMPOSITION WITH INTEGER-ORDER
DERIVATIVES ... 73 2.3.6 COMPOSITION WITH FRACTIONAL DERIVATIVES . . . .
74 2.3.7 LINK TO THE GRUENWALD-LETNIKOV APPROACH . ... 75 2.4 SONIC OTHER
APPROACHES 77 2.4.1 CAPUTO S FRACTIONAL DERIVATIVE 78 2.4.2 GENERALIZED
FUNCTIONS APPROACH 81 2.5 SEQUENTIAL FRACTIONAL DERIVATIVES 86 2.6 LEFT
AND RIGHT FRACTIONAL DERIVATIVES 88 2.7 PROPERTIES OF FRACTIONAL
DERIVATIVES 90 2.7.1 LINEARITY 90 2.7.2 THE LEIBNIZ RULE FOR FRACTIONAL
DERIVATIVES ... 91 2.7.3 FRACTIONAL DERIVATIVE OF A COMPOSITE FUNCTION .
97 2.7.4 RIEMANN- LIOUVILLE FRACTIONAL DIFFERENTIATION OF AN INTEGRAL
DEPENDING ON A PARAMETER . . . . 98 2.7.5 BEHAVIOUR NEAR THE LOWER
TERMINAL 99 2.7.6 BEHAVIOUR FAR FROM THE LOWER TERMINAL 101 2.8 LAPLACE
TRANSFORMS OF FRACTIONAL DERIVATIVES 103 2.8.1 BASIC FACTS ON THE
LAPLACE TRANSFORM 103 2.8.2 LAPLACE TRANSFORM OF THE RIEMANN-LIOUVILLE
FRACTIONAL DERIVATIVE 104 2.8.3 LAPLACE TRANSFORM OF THE CAPUTO
DERIVATIVE . . 106 2.8.4 LAPLACE TRANSFORM OF THE GRUENWALD-LETNIKOV
FRACTIONAL DERIVATIVE 106 2.8.5 LAPLACE TRANSFORM OF THE MILLER-ROSS
SEQUENTIAL FRACTIONAL DERIVATIVE 108 2.9 FOURIER TRANSFORMS OF
FRACTIONAL DERIVATIVES . 109 2.9.1 BASIC FACTS ON THE FOURIER TRANSFORM
109 CONTENTS IX 2.UE.2 FOURIER TRANSFORM OF FRACTIONAL INTEGRALS . . . .
110 2.9.3 FOURIER TRANSFORM OF FRACTIONAL DERIVATIVES . . . 111 2.10
MOLLIN TI ANSFORMS OF FRACTIONAL DERIVATIVES 112 2.10.1 BASIC FACTS ON
THE MOLLIN TRANSFORM 112 2.10.2 MEILIN TRANSFORM OF THE
RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL 115 2.10.3 MELLIN TRANSFORM OF THE
RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE 115 2.10.4 MELLIN TRANSFORM OF
THE CAPUTO FRACTIONAL DERIVATIVE 116 2.10.5 MELLIN TRANSFORM OF THE
MILLER-ROSS FRACTIONAL DERIVATIVE 117 3 EXISTENCE AND UNIQUENESS
THEOREMS 121 3.1 LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 122 3.2
FRACTIONAL DIFFERENTIAL EQUATION OF A GENERAL FORM . . . 126 3.3
EXISTENCE AND UNIQUENESS THEOREM AS A METHOD OF SOLUTION 131 3.4
DEPENDENCC OF A SOLUTION ON INITIAL CONDITIONS 133 4 THE LAPLACE
TRANSFORM METHOD 137 4.1 STANDARD FRACTIONAL DIFFERENTIAL EQUATIONS 138
4.1.1 ORDINARY LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 138 4.1.2
PARTIAL LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 140 4.2 SEQUENTIAL
FRACTIONAL DIFFERENTIAL EQUATIONS 144 4.2.1 ORDINARY LINEAR FRACTIONAL
DIFFERENTIAL EQUATIONS 144 4.2.2 PARTIAL LINEAR FRACTIONAL DIFFERENTIAL
EQUATIONS 146 5 FRACTIONAL GREEN S FUNCTION 149 5.1 DEFINITION AND SOME
PROPERTIES 150 5.1.1 DEFINITION 150 5.1.2 PROPERTIES 150 5.2 ONE-TERM
EQUATION 153 5.3 TWO-TERM EQUATION 154 5.4 THREE-TERM EQUATION 155 5.5
FOUR-TERM EQUATION 156 X CONTENTS 5.6 GENERAL CASE: N-TERM EQUATION 15 7
6 OTHER METHODS FOR THE SOLUTION OF FRACTIONAL-ORDER EQUATIONS 159 6.1
THE MEILIN TRANSFORM METHOD 159 6.2 POWER SERIES METHOD 161 6.2.1
ONE-TERM EQUATION 162 6.2.2 EQUATION WITH NON-CONSTANT COEFFICIENTS . .
. . 166 6.2.3 TWO-TERM NON-LINEAR EQUATION 167 6.3 BABENKO S SYMBOLIC
CALCULUS METHOD 168 6.3.1 THE IDEA OF THE METHOD 169 6.3.2 APPLICATION
IN HEAT AND MASS TRANSFER 170 6.3.3 LINK TO THE LAPLACE TRANSFORM METHOD
172 6.4 METHOD OF ORTHOGONAL POLYNOMIALS 173 6.4.1 THE IDEA OF THE
METHOD 174 6.4.2 GENERAL SCHEME OF THE METHOD 179 6.4.3 RIESZ FRACTIONAL
POTENTIAL 181 6.4.4 LEFT RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND
DERIVATIVES 186 6.4.5 OTHER SPECTRAL RELATIONSHIPS FOR THE LEFT
RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS 188 6.4.6 SPECTRAL RELATIONSHIPS
FOR THE RIGHT RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS 189 6.4.7 SOLUTION
OF ARUTYUNYAN S EQUATION IN CREEP THEORY 191 6.4.8 SOLUTION OF ABEL S
EQUATION . * 192 6.4.9 FINITE-PART INTEGRALS 192 6.4.10 JACOBI
POLYNOMIALS ORTHOGONAL WITH NON-INTEGRABLE WEIGHT FUNCTION 195 7
NUMERICAL EVALUATION OF FRACTIONAL DERIVATIVES 199 7.1 RIEMANN-LIOUVILLE
AND GRUENWALD-LETNIKOV DEFINITIONS OF THE FRACTIONAL-ORDER DERIVATIVE 199
7.2 APPROXIMATION OF FRACTIONAL DERIVATIVES 200 7.2.1 FRACTIONAL
DIFFERENCE APPROACH 200 7.2.2 THE USE OF QUADRATURE FORMULAS 200 7.3 THE
SHORT-MEMORY PRINCIPLE 203 7.4 ORDER OF APPROXIMATION 204 7.5
COMPUTATION OF COEFFICIENTS 208 7.6 HIGHER-ORDER APPROXIMATIONS 209
CONTENTS XI 7.7 CALCULATION OF HEAT LOAD INTENSITY CHANGE IN BLAST
FURNACE WALLS 210 7.7.1 INTRODUCTION TO THE PROBLEM 211 7.7.2
FRACTIONAL-ORDER DIFFERENTIATION AND INTEGRATION 211 7.7.3 CALCULATION
OF THE HEAT FLUEX BY FRACTIONAL ORDER DERIVATIVES - METHOD A 212 7.7.4
CALCULATION OF THE HEAT FLUX BASED ON THE SIMULATION OF THE THERMAL
FIELD OF THE FURNACE WALL - METHOD B 215 7.7.5 COMPARISON OF THE METHODS
218 7.8 FINITE-PART INTEGRALS AND FRACTIONAL DERIVATIVES 219 7.8.1
EVALUATION OF FINITE-PART INTEGRALS USING FRACTIONAL DERIVATIVES 220
7.8.2 EVALUATION OF FRACTIONAL DERIVATIVES USING FINITE-PART INTEGRALS
220 NUMERICAL SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS 223 8.1
INITIAL CONDITIONS: WHICH PROBLEM TO SOLVE? 223 8.2 NUMERICAL SOLUTION
224 8.3 EXAMPLES OF NUMERICAL SOLUTIONS 224 8.3.1 RELAXATION-OSCILLATION
EQUATION 224 8.3.2 EQUATION WITH CONSTANT COEFFICIENTS: MOTION OF AN
IMMERSED PLATE 225 8.3.3 EQUATION WITH NON-CONSTANT COEFFICIENTS:
SOLUTION OF A GAS IN A FLUID 231 8.3.4 NON-LINEAR PROBLEM: COOLING OF A
SEMI-INFINITE BODY BY RADIATION . . 235 8.4 THE SHORT-MEMORY PRINCIPLE
IN INITIAL VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS 242
FRACTIONAL-ORDER SYSTEMS AND CONTROLLERS 243 9.1 FRACTIONAL-ORDER
SYSTEMS AND FRACTIONAL-ORDER CONTROLLERS 244 9.1.1 FRACTIONAL-ORDER
CONTROL SYSTEM 244 9.1.2 FRACTIONAL-ORDER TRANSFER FUNCTIONS 245 9.1.3
NEW FUNCTION OF THE MITTAG-LEFFLER TYPE . . . . 246 9.1.4 GENERAL
FORMULA 247 XLL CONTENTS 9.1.5 THE UNIT-IMPULSE AND UNIT-STEP RESPONSE .
. . 248 9.1.6 SOME SPECIAL CASES 248 9.1.7 PJ A _D -CONTROLLER 249 9.1.8
OPEN-LOOP SYSTEM RESPONSE 250 9.1.9 CLOSED-LOOP SYSTEM RESPONSE 250 9.2
EXAMPLE 251 9.2.1 FRACTIONAL-ORDER CONTROLLED SYSTEM 252 9.2.2
INTEGER-ORDER APPROXIMATION 252 9.2.3 INTEGER-ORDER PZ)-CONTROLLER 253
9.2.4 FRACTIONAL-ORDER CONTROLLER 256 9.3 ON FRACTIONAL-ORDER SYSTEM
IDENTIFICATION 257 9.4 CONCLUSION 259 10 SURVEY OF APPLICATIONS OF THE
FRACTIONAL CALCULUS 261 10.1 ABEL S INTEGRAL EQUATION 261 10.1.1 GENERAL
REMARKS 262 10.1.2 SOME EQUATIONS REDUCIBLE TO ABEL S EQUATION . 263
10.2 VISCOELASTICITY 268 10.2.1 INTEGER-ORDER MODELS 268 10.2.2
FRACTIONAL-ORDER MODELS 271 10.2.3 APPROACHES RELATED TO THE FRACTIONAL
CALCULUS . 275 10.3 BODE S ANALYSIS OF FEEDBACK AMPLIFIERS 277 10.4
FRACTIONAL CAPACITOR THEORY 278 10.5 ELECTRICAL CIRCUITS 279 10.5.1 TREE
FRACTANCE 279 10.5.2 CHAIN FRACTANCE 280 10.5.3 ELECTRICAL ANALOGUE
MODEL OF A POROUS DYKE . . 282 10.5.4 WESTERLUND S GENERALIZED VOLTAGE
DIVIDER . . . . 282 10.5.5 FRACTIONAL-ORDER CHUA-HARTLEY SYSTEM 286 10.6
ELECTROANALYTICAL CHEMISTRY 290 10.7 ELECTRODE-ELECTROLYTE INTERFACE 291
10.8 FRACTIONAL MULTIPOLES 293 10.9 BIOLOGY 294 10.9.1 ELECTRIC
CONDUCTANCE OF BIOLOGICAL SYSTEMS . . . 294 10.9.2 FRACTIONAL-ORDER
MODEL OF NEURONS 295 10.10 FRACTIONAL DIFFUSION EQUATIONS 296 10.11
CONTROL THEORY 298 10.12 FITTING OF EXPERIMENTAL DATA 299 10.12.1
DISADVANTAGES OF CLASSICAL REGRESSION MODELS . . 299 10.12.2 FRACTIONAL
DERIVATIVE APPROACH 300 CONTENTS XIII 10.12.3 EXAMPLE: WIRES AT NIZNA
SLANA MINES 301 10.13 FRACTIONAL-ORDER PHYSICS? 305 APPENDIX: TABLES
OF FRACTIONAL DERIVATIVES 309 BIBLIOGRAPHY 313 INDEX 337
|
| adam_txt |
FRACTIONAL DIFFERENTIAL EQUATIONS AN INTRODUCTION TO FRACTIONAL
DERIVATIVES, FRACTIONAL DIFFERENTIAL EQUATIONS, TO METHODS OF THEIR
SOLUTION AND SOME OF THEIR APPLICATIONS BY IGOR PODLUBNY TECHNICAL
UNIVERSITY OF KOSICE, SLOVAK REPUBLIC ACADEMIC PRESS SAN DIEGO * BOSTON
* NEW YORK * LONDON * SYDNEY * TOKYO * TORONTO CONTENTS PREFACE XVII
ACKNOWLEDGEMENTS XXIII 1 SPECIAL FUNCTIONS OF THE FRACTIONAL CALCULUS 1
1.1 GAMMA FUNCTION 1 1.1.1 DEFINITION OF THE GAMMA FUNCTION 1 1.1.2 SOME
PROPERTIES OF THE GAMMA FUNCTION . . . . 2 1.1.3 LIMIT REPRESENTATION OF
THE GAMMA FUNCTION . 4 1.1.4 BETA FUNCTION 6 1.1.5 CONTOUR INTEGRAL
REPRESENTATION 10 1.1.6 CONTOUR INTEGRAL REPRESENTATION OF 1/F(Z) . 12
1.2 MITTAG-LEFFLER FUNCTION 16 1.2.1 DEFINITION AND RELATION TO SOME
OTHER FUNCTIONS 17 1.2.2 THE LAPLACE TRANSFORM OF THE MITTAG-LEFFLER
FUNCTION IN TWO PARAMETERS 20 1.2.3 DERIVATIVES OF THE MITTAG-LEFFLER
FUNCTION . . 21 1.2.4 DIFFERENTIAL EQUATIONS FOR THE MITTAG-LEFFLER
FUNCTION 23 1.2.5 SUMMATION FORMULAS 23 1.2.6 INTEGRATION OF THE
MITTAG-LEFFLER FUNCTION . . . . 24 1.2.7 ASYMPTOTIC EXPANSIONS 29 1.3
WRIGHT FUNCTION 37 1.3.1 DEFINITION 37 1.3.2 INTEGRAL REPRESENTATION 37
1.3.3 RELATION TO OTHER FUNCTIONS 38 2 FRACTIONAL DERIVATIVES AND
INTEGRALS 41 2.1 THE NAME OF THE GAME 41 2.2 GRUENWALD-LETNIKOV
FRACTIONAL DERIVATIVES 43 VII I CONTENTS 2.2.1 UNIFICATION OF
INTEGER-ORDER DERIVATIVES AND INTEGRALS 43 2.2.2 INTEGRALS OF ARBITRARY
ORDER 48 2.2.3 DERIVATIVES OF ARBITRARY ORDER 52 2.2.4 FRACTIONAL
DERIVATIVE OF (T - A) SS 55 2.2.5 COMPOSITION WITH INTEGER-ORDER
DERIVATIVES . 57 2.2.6 COMPOSITION WITH FRACTIONAL DERIVATIVES . . . .
59 2.3 RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES 62 2.3.1 UNIFICATION OF
INTEGER-ORDER DERIVATIVES AND INTEGRALS 63 2.3.2 INTEGRALS OF ARBITRARY
ORDER 65 2.3.3 DERIVATIVES OF ARBITRARY ORDER 68 2.3.4 FRACTIONAL
DERIVATIVE OF (T - A) SS 72 2.3.5 COMPOSITION WITH INTEGER-ORDER
DERIVATIVES . 73 2.3.6 COMPOSITION WITH FRACTIONAL DERIVATIVES . . . .
74 2.3.7 LINK TO THE GRUENWALD-LETNIKOV APPROACH . . 75 2.4 SONIC OTHER
APPROACHES 77 2.4.1 CAPUTO'S FRACTIONAL DERIVATIVE 78 2.4.2 GENERALIZED
FUNCTIONS APPROACH 81 2.5 SEQUENTIAL FRACTIONAL DERIVATIVES 86 2.6 LEFT
AND RIGHT FRACTIONAL DERIVATIVES 88 2.7 PROPERTIES OF FRACTIONAL
DERIVATIVES 90 2.7.1 LINEARITY 90 2.7.2 THE LEIBNIZ RULE FOR FRACTIONAL
DERIVATIVES . 91 2.7.3 FRACTIONAL DERIVATIVE OF A COMPOSITE FUNCTION .
97 2.7.4 RIEMANN- LIOUVILLE FRACTIONAL DIFFERENTIATION OF AN INTEGRAL
DEPENDING ON A PARAMETER . . . . 98 2.7.5 BEHAVIOUR NEAR THE LOWER
TERMINAL 99 2.7.6 BEHAVIOUR FAR FROM THE LOWER TERMINAL 101 2.8 LAPLACE
TRANSFORMS OF FRACTIONAL DERIVATIVES 103 2.8.1 BASIC FACTS ON THE
LAPLACE TRANSFORM 103 2.8.2 LAPLACE TRANSFORM OF THE RIEMANN-LIOUVILLE
FRACTIONAL DERIVATIVE 104 2.8.3 LAPLACE TRANSFORM OF THE CAPUTO
DERIVATIVE . . 106 2.8.4 LAPLACE TRANSFORM OF THE GRUENWALD-LETNIKOV
FRACTIONAL DERIVATIVE 106 2.8.5 LAPLACE TRANSFORM OF THE MILLER-ROSS
SEQUENTIAL FRACTIONAL DERIVATIVE 108 2.9 FOURIER TRANSFORMS OF
FRACTIONAL DERIVATIVES '. 109 2.9.1 BASIC FACTS ON THE FOURIER TRANSFORM
109 CONTENTS IX 2.UE.2 FOURIER TRANSFORM OF FRACTIONAL INTEGRALS . . . .
110 2.9.3 FOURIER TRANSFORM OF FRACTIONAL DERIVATIVES . . . 111 2.10
MOLLIN TI'ANSFORMS OF FRACTIONAL DERIVATIVES 112 2.10.1 BASIC FACTS ON
THE MOLLIN TRANSFORM 112 2.10.2 MEILIN TRANSFORM OF THE
RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL 115 2.10.3 MELLIN TRANSFORM OF THE
RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE 115 2.10.4 MELLIN TRANSFORM OF
THE CAPUTO FRACTIONAL DERIVATIVE 116 2.10.5 MELLIN TRANSFORM OF THE
MILLER-ROSS FRACTIONAL DERIVATIVE 117 3 EXISTENCE AND UNIQUENESS
THEOREMS 121 3.1 LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 122 3.2
FRACTIONAL DIFFERENTIAL EQUATION OF A GENERAL FORM . . . 126 3.3
EXISTENCE AND UNIQUENESS THEOREM AS A METHOD OF SOLUTION 131 3.4
DEPENDENCC OF A SOLUTION ON INITIAL CONDITIONS 133 4 THE LAPLACE
TRANSFORM METHOD 137 4.1 STANDARD FRACTIONAL DIFFERENTIAL EQUATIONS 138
4.1.1 ORDINARY LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 138 4.1.2
PARTIAL LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 140 4.2 SEQUENTIAL
FRACTIONAL DIFFERENTIAL EQUATIONS 144 4.2.1 ORDINARY LINEAR FRACTIONAL
DIFFERENTIAL EQUATIONS 144 4.2.2 PARTIAL LINEAR FRACTIONAL DIFFERENTIAL
EQUATIONS 146 5 FRACTIONAL GREEN'S FUNCTION 149 5.1 DEFINITION AND SOME
PROPERTIES 150 5.1.1 DEFINITION 150 5.1.2 PROPERTIES 150 5.2 ONE-TERM
EQUATION 153 5.3 TWO-TERM EQUATION 154 5.4 THREE-TERM EQUATION 155 5.5
FOUR-TERM EQUATION 156 X CONTENTS 5.6 GENERAL CASE: N-TERM EQUATION 15 7
6 OTHER METHODS FOR THE SOLUTION OF FRACTIONAL-ORDER EQUATIONS 159 6.1
THE MEILIN TRANSFORM METHOD 159 6.2 POWER SERIES METHOD 161 6.2.1
ONE-TERM EQUATION 162 6.2.2 EQUATION WITH NON-CONSTANT COEFFICIENTS . .
. . 166 6.2.3 TWO-TERM NON-LINEAR EQUATION 167 6.3 BABENKO'S SYMBOLIC
CALCULUS METHOD 168 6.3.1 THE IDEA OF THE METHOD 169 6.3.2 APPLICATION
IN HEAT AND MASS TRANSFER 170 6.3.3 LINK TO THE LAPLACE TRANSFORM METHOD
172 6.4 METHOD OF ORTHOGONAL POLYNOMIALS 173 6.4.1 THE IDEA OF THE
METHOD 174 6.4.2 GENERAL SCHEME OF THE METHOD 179 6.4.3 RIESZ FRACTIONAL
POTENTIAL 181 6.4.4 LEFT RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND
DERIVATIVES 186 6.4.5 OTHER SPECTRAL RELATIONSHIPS FOR THE LEFT
RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS 188 6.4.6 SPECTRAL RELATIONSHIPS
FOR THE RIGHT RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS 189 6.4.7 SOLUTION
OF ARUTYUNYAN'S EQUATION IN CREEP THEORY 191 6.4.8 SOLUTION OF ABEL'S
EQUATION . * 192 6.4.9 FINITE-PART INTEGRALS 192 6.4.10 JACOBI
POLYNOMIALS ORTHOGONAL WITH NON-INTEGRABLE WEIGHT FUNCTION 195 7
NUMERICAL EVALUATION OF FRACTIONAL DERIVATIVES 199 7.1 RIEMANN-LIOUVILLE
AND GRUENWALD-LETNIKOV DEFINITIONS OF THE FRACTIONAL-ORDER DERIVATIVE 199
7.2 APPROXIMATION OF FRACTIONAL DERIVATIVES 200 7.2.1 FRACTIONAL
DIFFERENCE APPROACH 200 7.2.2 THE USE OF QUADRATURE FORMULAS 200 7.3 THE
"SHORT-MEMORY" PRINCIPLE 203 7.4 ORDER OF APPROXIMATION 204 7.5
COMPUTATION OF COEFFICIENTS 208 7.6 HIGHER-ORDER APPROXIMATIONS 209
CONTENTS XI 7.7 CALCULATION OF HEAT LOAD INTENSITY CHANGE IN BLAST
FURNACE WALLS 210 7.7.1 INTRODUCTION TO THE PROBLEM 211 7.7.2
FRACTIONAL-ORDER DIFFERENTIATION AND INTEGRATION 211 7.7.3 CALCULATION
OF THE HEAT FLUEX BY FRACTIONAL ORDER DERIVATIVES - METHOD A 212 7.7.4
CALCULATION OF THE HEAT FLUX BASED ON THE SIMULATION OF THE THERMAL
FIELD OF THE FURNACE WALL - METHOD B 215 7.7.5 COMPARISON OF THE METHODS
218 7.8 FINITE-PART INTEGRALS AND FRACTIONAL DERIVATIVES 219 7.8.1
EVALUATION OF FINITE-PART INTEGRALS USING FRACTIONAL DERIVATIVES 220
7.8.2 EVALUATION OF FRACTIONAL DERIVATIVES USING FINITE-PART INTEGRALS
220 NUMERICAL SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS 223 8.1
INITIAL CONDITIONS: WHICH PROBLEM TO SOLVE? 223 8.2 NUMERICAL SOLUTION
224 8.3 EXAMPLES OF NUMERICAL SOLUTIONS 224 8.3.1 RELAXATION-OSCILLATION
EQUATION 224 8.3.2 EQUATION WITH CONSTANT COEFFICIENTS: MOTION OF AN
IMMERSED PLATE 225 8.3.3 EQUATION WITH NON-CONSTANT COEFFICIENTS:
SOLUTION OF A GAS IN A FLUID 231 8.3.4 NON-LINEAR PROBLEM: COOLING OF A
SEMI-INFINITE BODY BY RADIATION . . 235 8.4 THE "SHORT-MEMORY" PRINCIPLE
IN INITIAL VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS 242
FRACTIONAL-ORDER SYSTEMS AND CONTROLLERS 243 9.1 FRACTIONAL-ORDER
SYSTEMS AND FRACTIONAL-ORDER CONTROLLERS 244 9.1.1 FRACTIONAL-ORDER
CONTROL SYSTEM 244 9.1.2 FRACTIONAL-ORDER TRANSFER FUNCTIONS 245 9.1.3
NEW FUNCTION OF THE MITTAG-LEFFLER TYPE . . . . 246 9.1.4 GENERAL
FORMULA 247 XLL CONTENTS 9.1.5 THE UNIT-IMPULSE AND UNIT-STEP RESPONSE .
. . 248 9.1.6 SOME SPECIAL CASES 248 9.1.7 PJ A _D"-CONTROLLER 249 9.1.8
OPEN-LOOP SYSTEM RESPONSE 250 9.1.9 CLOSED-LOOP SYSTEM RESPONSE 250 9.2
EXAMPLE 251 9.2.1 FRACTIONAL-ORDER CONTROLLED SYSTEM 252 9.2.2
INTEGER-ORDER APPROXIMATION 252 9.2.3 INTEGER-ORDER PZ)-CONTROLLER 253
9.2.4 FRACTIONAL-ORDER CONTROLLER 256 9.3 ON FRACTIONAL-ORDER SYSTEM
IDENTIFICATION 257 9.4 CONCLUSION 259 10 SURVEY OF APPLICATIONS OF THE
FRACTIONAL CALCULUS 261 10.1 ABEL'S INTEGRAL EQUATION 261 10.1.1 GENERAL
REMARKS 262 10.1.2 SOME EQUATIONS REDUCIBLE TO ABEL'S EQUATION . 263
10.2 VISCOELASTICITY 268 10.2.1 INTEGER-ORDER MODELS 268 10.2.2
FRACTIONAL-ORDER MODELS 271 10.2.3 APPROACHES RELATED TO THE FRACTIONAL
CALCULUS . 275 10.3 BODE'S ANALYSIS OF FEEDBACK AMPLIFIERS 277 10.4
FRACTIONAL CAPACITOR THEORY 278 10.5 ELECTRICAL CIRCUITS 279 10.5.1 TREE
FRACTANCE 279 10.5.2 CHAIN FRACTANCE 280 10.5.3 ELECTRICAL ANALOGUE
MODEL OF A POROUS DYKE . . 282 10.5.4 WESTERLUND'S GENERALIZED VOLTAGE
DIVIDER . . . . 282 10.5.5 FRACTIONAL-ORDER CHUA-HARTLEY SYSTEM 286 10.6
ELECTROANALYTICAL CHEMISTRY 290 10.7 ELECTRODE-ELECTROLYTE INTERFACE 291
10.8 FRACTIONAL MULTIPOLES 293 10.9 BIOLOGY 294 10.9.1 ELECTRIC
CONDUCTANCE OF BIOLOGICAL SYSTEMS . . . 294 10.9.2 FRACTIONAL-ORDER
MODEL OF NEURONS 295 10.10 FRACTIONAL DIFFUSION EQUATIONS 296 10.11
CONTROL THEORY 298 10.12 FITTING OF EXPERIMENTAL DATA 299 10.12.1
DISADVANTAGES OF CLASSICAL REGRESSION MODELS . . 299 10.12.2 FRACTIONAL
DERIVATIVE APPROACH 300 CONTENTS XIII 10.12.3 EXAMPLE: WIRES AT NIZNA
SLANA MINES 301 10.13 "FRACTIONAL-ORDER" PHYSICS? 305 APPENDIX: TABLES
OF FRACTIONAL DERIVATIVES 309 BIBLIOGRAPHY 313 INDEX 337 |
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| id | DE-604.BV022112405 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:15:47Z |
| indexdate | 2024-07-09T20:50:49Z |
| institution | BVB |
| isbn | 0125588402 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015327280 |
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| physical | XXIV, 340 S. graph. Darst. |
| publishDate | 1999 |
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| publisher | Acad. Press |
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| series | Mathematics in science and engineering |
| series2 | Mathematics in science and engineering |
| spelling | Podlubny, Igor Verfasser aut Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications by Igor Podlubny San Diego [u.a.] Acad. Press 1999 XXIV, 340 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics in science and engineering 198 Literaturverz. S. 313 - 335 Ableitung gebrochener Ordnung Differentiaalvergelijkingen. gtt Differential equations Differential equations Numerical solutions Fractional calculus Laplace-Transformation (DE-588)4034577-4 gnd rswk-swf Fraktal (DE-588)4123220-3 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Differentialgleichung (DE-588)4012249-9 s DE-604 Laplace-Transformation (DE-588)4034577-4 s Fraktal (DE-588)4123220-3 s Mathematics in science and engineering 198 (DE-604)BV000001196 198 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015327280&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Podlubny, Igor Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications Mathematics in science and engineering Ableitung gebrochener Ordnung Differentiaalvergelijkingen. gtt Differential equations Differential equations Numerical solutions Fractional calculus Laplace-Transformation (DE-588)4034577-4 gnd Fraktal (DE-588)4123220-3 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| subject_GND | (DE-588)4034577-4 (DE-588)4123220-3 (DE-588)4012249-9 (DE-588)4151278-9 |
| title | Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications |
| title_auth | Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications |
| title_exact_search | Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications |
| title_exact_search_txtP | Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications |
| title_full | Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications by Igor Podlubny |
| title_fullStr | Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications by Igor Podlubny |
| title_full_unstemmed | Fractional differential equations an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications by Igor Podlubny |
| title_short | Fractional differential equations |
| title_sort | fractional differential equations an introduction to fractional derivatives fractional differential equations to methods of their solution and some of their applications |
| title_sub | an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications |
| topic | Ableitung gebrochener Ordnung Differentiaalvergelijkingen. gtt Differential equations Differential equations Numerical solutions Fractional calculus Laplace-Transformation (DE-588)4034577-4 gnd Fraktal (DE-588)4123220-3 gnd Differentialgleichung (DE-588)4012249-9 gnd |
| topic_facet | Ableitung gebrochener Ordnung Differentiaalvergelijkingen. Differential equations Differential equations Numerical solutions Fractional calculus Laplace-Transformation Fraktal Differentialgleichung Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015327280&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000001196 |
| work_keys_str_mv | AT podlubnyigor fractionaldifferentialequationsanintroductiontofractionalderivativesfractionaldifferentialequationstomethodsoftheirsolutionandsomeoftheirapplications |