Theory and applications of fractional differential equations:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
2006
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| Edition: | First edition |
| Series: | North-Holland Mathematics Studies
204 |
| Subjects: | |
| Online Access: | Publisher description Table of contents Inhaltsverzeichnis |
| Physical Description: | XV, 523 Seiten |
| ISBN: | 0444518320 9780444518323 |
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MARC
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| 100 | 1 | |a Kilbas, Anatoly A. |d 1948-2010 |e Verfasser |0 (DE-588)1042268665 |4 aut | |
| 245 | 1 | 0 | |a Theory and applications of fractional differential equations |c Anatoly A. Kilbas, Belarusian State University, Minsk, Belarus, Hari M. Srivastava, University of Victoria, Victoria, British Columbia, Canada, Juan J. Trujillo, Universidad de La Laguna, La Laguna (Tenerife), Canary Islands, Spain |
| 250 | |a First edition | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2006 | |
| 300 | |a XV, 523 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a North-Holland Mathematics Studies |v 204 | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Fractional calculus | |
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| 689 | 0 | 1 | |a Gebrochene Analysis |0 (DE-588)4722475-7 |D s |
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| 700 | 1 | |a Srivastava, Hari M. |d 1940- |e Verfasser |0 (DE-588)12358874X |4 aut | |
| 700 | 1 | |a Trujillo, Juan J. |e Verfasser |4 aut | |
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| adam_text | THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS ANATOLYA.
KILBAS BELARUSIAN STATE UNIVERSITY MINSK, BELARUS HARI M. SRIVASTAVA
UNIVERSITY OF VICTORIA VICTORIA, BRITISH COLUMBIA, CANADA JUAN J.
TRUJILLO UNIVERSIDAD DE LA LAGUNA LA LAGUNA (TENERIFE) CANARY ISLANDS,
SPAIN ELSEVIER AMSTERDAM - BOSTON - HEIDELBERG - LONDON - NEW YORK -
OXFORD PARIS - SAN DIEGO - SAN FRANCISCO - SINGAPORE - SYDNEY - TOKYO
CONTENTS 1 PRELIMINARIES 1 1.1 SPACES OF INTEGRABLE, ABSOLUTELY
CONTINUOUS, AND CONTINUOUS FUNC- TIONS 1 1.2 GENERALIZED FUNCTIONS 6
1.3 FOURIER TRANSFORMS 10 1.4 LAPLACE AND MELLIN TRANSFORMS 18 1.5 THE
GAMMA FUNCTION AND RELATED SPECIAL FUNCTIONS 24 1.6 HYPERGEOMETRIC
FUNCTIONS 27 1.7 BESSEL FUNCTIONS 32 1.8 CLASSICAL MITTAG-LEFFLER
FUNCTIONS 40 1.9 GENERALIZED MITTAG-LEFFLER FUNCTIONS 45 1.10 FUNCTIONS
OF THE MITTAG-LEFFLER TYPE 49 1.11 WRIGHT FUNCTIONS 54 1.12 THE
^-FUNCTION 58 1.13 FIXED POINT THEOREMS 67 2 FRACTIONAL INTEGRALS AND
FRACTIONAL DERIVATIVES 69 2.1 RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND
FRACTIONAL DERI- VATIVES 69 2.2 LIOUVILLE FRACTIONAL INTEGRALS AND
FRACTIONAL DERIVATIVES ON THE HALF- AXIS 79 2.3 LIOUVILLE FRACTIONAL
INTEGRALS AND FRACTIONAL DERIVATIVES ON THE REAL AXIS 87 2.4 CAPUTO
FRACTIONAL DERIVATIVES 90 2.5 FRACTIONAL INTEGRALS AND FRACTIONAL
DERIVATIVES OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION 99 2.6
ERDELYI-KOBER TYPE FRACTIONAL INTEGRALS AND FRACTIONAL DERIVA- TIVES 105
2.7 HADAMARD TYPE FRACTIONAL INTEGRALS AND FRACTIONAL DERIVATIVES . .
110 2.8 GRIINWALD-LETNIKOV FRACTIONAL DERIVATIVES 121 2.9 PARTIAL AND
MIXED FRACTIONAL INTEGRALS AND FRACTIONAL DERIVATIVES 123 2.10 RIESZ
FRACTIONAL INTEGRO-DIFFERENTIATION 127 2.11 COMMENTS AND OBSERVATIONS
132 XII THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS 3
ORDINARY FRACTIONAL DIFFERENTIAL EQUATIONS. EXISTENCE AND UNIQUENESS
THEOREMS 135 3.1 INTRODUCTION AND A BRIEF OVERVIEW OF RESULTS 135 3.2
EQUATIONS WITH THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE IN THE SPACE
OF SUMMABLE FUNCTIONS 144 3.2.1 EQUIVALENCE OF THE CAUCHY TYPE PROBLEM
AND THE VOLTERRA INTEGRAL EQUATION 145 3.2.2 EXISTENCE AND UNIQUENESS OF
THE SOLUTION TO THE CAUCHY TYPE PROBLEM 148 3.2.3 THE WEIGHTED CAUCHY
TYPE PROBLEM 151 3.2.4 GENERALIZED CAUCHY TYPE PROBLEMS 153 3.2.5 CAUCHY
TYPE PROBLEMS FOR LINEAR EQUATIONS 15 7 3.2.6 MISCELLANEOUS EXAMPLES 160
3.3 EQUATIONS WITH THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE IN THE
SPACE OF CONTINUOUS FUNCTIONS. GLOBAL SOLUTION 162 3.3.1 EQUIVALENCE OF
THE CAUCHY TYPE PROBLEM AND THE VOLTERRA INTEGRAL EQUATION 163 3.3.2
EXISTENCE AND UNIQUENESS OF THE GLOBAL SOLUTION TO THE CAUCHY TYPE
PROBLEM 164 3.3.3 THE WEIGHTED CAUCHY TYPE PROBLEM 167 3.3.4 GENERALIZED
CAUCHY TYPE PROBLEMS 168 3.3.5 CAUCHY TYPE PROBLEMS FOR LINEAR EQUATIONS
170 3.3.6 MORE EXACT SPACES 171 3.3.7 FURTHER EXAMPLES 177 3.4 EQUATIONS
WITH THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE IN THE SPACE OF
CONTINUOUS FUNCTIONS. SEMI-GLOBAL AND LOCAL SOLUTIONS . 182 3.4.1 THE
CAUCHY TYPE PROBLEM WITH INITIAL CONDITIONS AT THE ENDPOINT OF THE
INTERVAL. SEMI-GLOBAL SOLUTION 183 3.4.2 THE CAUCHY PROBLEM WITH INITIAL
CONDITIONS AT THE INNER POINT OF THE INTERVAL. PRELIMINARIES 186 3.4.3
EQUIVALENCE OF THE CAUCHY PROBLEM AND THE VOLTERRA INTEGRAL EQUATION .
189 3.4.4 THE CAUCHY PROBLEM WITH INITIAL CONDITIONS AT THE INNER POINT
OF THE INTERVAL. THE UNIQUENESS OF SEMI-GLOBAL AND LOCAL SOLUTIONS 191
3.4.5 A SET OF EXAMPLES 196 3.5 EQUATIONS WITH THE CAPUTO DERIVATIVE IN
THE SPACE OF CONTINUOUSLY DIFFERENTIABLE FUNCTIONS 198 3.5.1 THE CAUCHY
PROBLEM WITH INITIAL CONDITIONS AT THE ENDPOINT OF THE INTERVAL. GLOBAL
SOLUTION 199 3.5.2 THE CAUCHY PROBLEMS WITH INITIAL CONDITIONS AT THE
END AND INNER POINTS OF THE INTERVAL. SEMI-GLOBAL AND LOCAL SOLUTIONS
205 3.5.3 ILLUSTRATIVE EXAMPLES 209 CONTENTS XIII 3.6 EQUATIONS WITH THE
HADAMARD FRACTIONAL DERIVATIVE IN THE SPACE OF CONTINUOUS FUNCTIONS 212
4 METHODS FOR EXPLICITLY SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS 221
4.1 METHOD OF REDUCTION TO VOLTERRA INTEGRAL EQUATIONS 221 4.1.1 THE
CAUCHY TYPE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH THE
RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES 222 4.1.2 THE CAUCHY PROBLEMS
FOR ORDINARY DIFFERENTIAL EQUA- TIONS 228 4.1.3 THE CAUCHY PROBLEMS FOR
DIFFERENTIAL EQUATIONS WITH THE CAPUTO FRACTIONAL DERIVATIVES 230 4.1.4
THE CAUCHY TYPE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH HADAMARD
FRACTIONAL DERIVATIVES 234 4.2 COMPOSITIONAL METHOD 238 4.2.1
PRELIMINARIES 238 4.2.2 COMPOSITIONAL RELATIONS 239 4.2.3 HOMOGENEOUS
DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH RIEMANN-LIOUVILLE
FRACTIONAL DERIVATIVES 242 4.2.4 NONHOMOGENEOUS DIFFERENTIAL EQUATIONS
OF FRACTIONAL OR- DER WITH RIEMANN-LIOUVILLE AND LIOUVILLE FRACTIONAL
DERIVA- TIVES WITH A QUASI-POLYNOMIAL FREE TERM 245 4.2.5 DIFFERENTIAL
EQUATIONS OF ORDER 1/2 248 4.2.6 CAUCHY TYPE PROBLEM FOR NONHOMOGENEOUS
DIFFERENTIAL EQUATIONS WITH RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES AND
WITH A QUASI-POLYNOMIAL FREE TERM 251 4.2.7 SOLUTIONS TO HOMOGENEOUS
FRACTIONAL DIFFERENTIAL EQUATIONS WITH LIOUVILLE FRACTIONAL DERIVATIVES
IN TERMS OF BESSEL- TYPE FUNCTIONS 257 4.3 OPERATIONAL METHOD 260
4.3.1 LIOUVILLE FRACTIONAL INTEGRATION AND DIFFERENTIATION OPERA- TORS
IN SPECIAL FUNCTION SPACES ON THE HALF-AXIS 261 4.3.2 OPERATIONAL
CALCULUS FOR THE LIOUVILLE FRACTIONAL CALCULUS OPERATORS 263 4.3.3
SOLUTIONS TO CAUCHY TYPE PROBLEMS FOR FRACTIONAL DIFFEREN- TIAL
EQUATIONS WITH LIOUVILLE FRACTIONAL DERIVATIVES 26 6 4.3.4 OTHER RESULTS
270 4.4 NUMERICAL TREATMENT 272 5 INTEGRAL TRANSFORM METHOD FOR EXPLICIT
SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS 279 5.1 INTRODUCTION AND
A BRIEF SURVEY OF RESULTS 279 5.2 LAPLACE TRANSFORM METHOD FOR SOLVING
ORDINARY DIFFERENTIAL EQUA- TIONS WITH LIOUVILLE FRACTIONAL DERIVATIVES
283 XIV THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS
5.2.1 HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS 283 5.2.2
NONHOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS . . . 295 5.2.3
EQUATIONS WITH NONCONSTANT COEFFICIENTS 303 5.2.4 CAUCHY TYPE FOR
FRACTIONAL DIFFERENTIAL EQUATIONS 309 5.3 LAPLACE TRANSFORM METHOD FOR
SOLVING ORDINARY DIFFERENTIAL EQUA- TIONS WITH CAPUTO FRACTIONAL
DERIVATIVES 312 5.3.1 HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS
312 5.3.2 NONHOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS . . . 322
5.3.3 CAUCHY PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS . . . 326
5.4 MELLIN TRANSFORM METHOD FOR SOLVING NONHOMOGENEOUS FRACTIONAL
DIFFERENTIAL EQUATIONS WITH LIOUVILLE DERIVATIVES 329 5.4.1 GENERAL
APPROACH TO THE PROBLEMS 329 5.4.2 EQUATIONS WITH LEFT-SIDED FRACTIONAL
DERIVATIVES 331 5.4.3 EQUATIONS WITH RIGHT-SIDED FRACTIONAL DERIVATIVES
336 5.5 FOURIER TRANSFORM METHOD FOR SOLVING NONHOMOGENEOUS DIFFEREN-
TIAL EQUATIONS WITH RIESZ FRACTIONAL DERIVATIVES 341 5.5.1
MULTI-DIMENSIONAL EQUATIONS 341 5.5.2 ONE-DIMENSIONAL EQUATIONS 344 6
PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS 347 6.1 OVERVIEW OF RESULTS
347 6.1.1 PARTIAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 347 6.1.2
FRACTIONAL PARTIAL DIFFERENTIAL DIFFUSION EQUATIONS 351 6.1.3 ABSTRACT
DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 359 6.2 SOLUTION OF CAUCHY
TYPE PROBLEMS FOR FRACTIONAL DIFFUSION-WAVE EQUATIONS 362 6.2.1 CAUCHY
TYPE PROBLEMS FOR TWO-DIMENSIONAL EQUATIONS . . 362 6.2.2 CAUCHY TYPE
PROBLEMS FOR MULTI-DIMENSIONAL EQUATIONS . . 366 6.3 SOLUTION OF CAUCHY
PROBLEMS FOR FRACTIONAL DIFFUSION-WAVE EQUA- TIONS 373 6.3.1 CAUCHY
PROBLEMS FOR TWO-DIMENSIONAL EQUATIONS 374 6.3.2 CAUCHY PROBLEMS FOR
MULTI-DIMENSIONAL EQUATIONS 377 6.4 SOLUTION OF CAUCHY PROBLEMS FOR
FRACTIONAL EVOLUTION EQUATIONS . 380 6.4.1 SOLUTION OF THE SIMPLEST
PROBLEM 380 6.4.2 SOLUTION TO THE GENERAL PROBLEM 384 6.4.3 SOLUTIONS OF
CAUCHY PROBLEMS VIA THE H-FUNCTIONS 388 7 SEQUENTIAL LINEAR DIFFERENTIAL
EQUATIONS OF FRACTIONAL ORDER 393 7.1 SEQUENTIAL LINEAR DIFFERENTIAL
EQUATIONS OF FRACTIONAL ORDER . . . . 394 7.2 SOLUTION OF LINEAR
DIFFERENTIAL EQUATIONS WITH CONSTANT COEF- FICIENTS 400 7.2.1 GENERAL
SOLUTION IN THE HOMOGENEOUS CASE 400 7.2.2 GENERAL SOLUTION IN THE
NON-HOMOGENEOUS CASE. FRACTIONAL GREEN FUNCTION 403 CONTENTS XV 7.3
NON-SEQUENTIAL LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT CO-
EFFICIENTS 407 7.4 SYSTEMS OF EQUATIONS ASSOCIATED WITH
RIEMANN-LIOUVILLE AND CA- PUTO DERIVATIVES 409 7.4.1 GENERAL THEORY 409
7.4.2 GENERAL SOLUTION FOR THE CASE OF CONSTANT COEFFICIENTS. FRAC-
TIONAL GREEN VECTORIAL FUNCTION 412 7.5 SOLUTION OF FRACTIONAL
DIFFERENTIAL EQUATIONS WITH VARIABLE COEF- FICIENTS. GENERALIZED METHOD
OF FROBENIUS 415 7.5.1 INTRODUCTION 41 5 7.5.2 SOLUTIONS AROUND AN
ORDINARY POINT OF A FRACTIONAL DIFFER- ENTIAL EQUATION OF ORDER A 418
7.5.3 SOLUTIONS AROUND AN ORDINARY POINT OF A FRACTIONAL DIFFER- ENTIAL
EQUATION OF ORDER 2A 421 7.5.4 SOLUTION AROUND AN A-SINGULAR POINT OF A
FRACTIONAL DIFFER- ENTIAL EQUATION OF ORDER A 424 7.5.5 SOLUTION AROUND
AN A-SINGULAR POINT OF A FRACTIONAL DIFFER- ENTIAL EQUATION OF ORDER 2A
427 7.6 SOME APPLICATIONS OF LINEAR ORDINARY FRACTIONAL DIFFERENTIAL
EQUATIONS 433 7.6.1 DYNAMICS OF A SPHERE IMMERSED IN AN INCOMPRESSIBLE
VIS- COUS FLUID. BASSET S PROBLEM 434 7.6.2 OSCILLATORY PROCESSES WITH
FRACTIONAL DAMPING 436 7.6.3 STUDY OF THE TENSION-DEFORMATION
RELATIONSHIP OF VISCOELAS- TIC MATERIALS 439 8 FURTHER APPLICATIONS OF
FRACTIONAL MODELS 449 8.1 PRELIMINARY REVIEW 449 8.1.1 HISTORICAL
OVERVIEW 450 8.1.2 COMPLEX SYSTEMS 452 8.1.3 FRACTIONAL INTEGRAL AND
FRACTIONAL DERIVATIVE OPERATORS . . 456 8.2 FRACTIONAL MODEL FOR THE
SUPER-DIFFUSION PROCESSES 458 8.3 DIRAC EQUATIONS FOR THE ORDINARY
DIFFUSION EQUATION 462 8.4 APPLICATIONS DESCRIBING CARRIER TRANSPORT IN
AMORPHOUS SEMICON- DUCTORS WITH MULTIPLE TRAPPING 463 BIBLIOGRAPHY 469
SUBJECT INDEX 521
|
| adam_txt |
THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS ANATOLYA.
KILBAS BELARUSIAN STATE UNIVERSITY MINSK, BELARUS HARI M. SRIVASTAVA
UNIVERSITY OF VICTORIA VICTORIA, BRITISH COLUMBIA, CANADA JUAN J.
TRUJILLO UNIVERSIDAD DE LA LAGUNA LA LAGUNA (TENERIFE) CANARY ISLANDS,
SPAIN ELSEVIER AMSTERDAM - BOSTON - HEIDELBERG - LONDON - NEW YORK -
OXFORD PARIS - SAN DIEGO - SAN FRANCISCO - SINGAPORE - SYDNEY - TOKYO
CONTENTS 1 PRELIMINARIES 1 1.1 SPACES OF INTEGRABLE, ABSOLUTELY
CONTINUOUS, AND CONTINUOUS FUNC- TIONS ' 1 1.2 GENERALIZED FUNCTIONS 6
1.3 FOURIER TRANSFORMS 10 1.4 LAPLACE AND MELLIN TRANSFORMS 18 1.5 THE
GAMMA FUNCTION AND RELATED SPECIAL FUNCTIONS 24 1.6 HYPERGEOMETRIC
FUNCTIONS 27 1.7 BESSEL FUNCTIONS 32 1.8 CLASSICAL MITTAG-LEFFLER
FUNCTIONS 40 1.9 GENERALIZED MITTAG-LEFFLER FUNCTIONS 45 1.10 FUNCTIONS
OF THE MITTAG-LEFFLER TYPE 49 1.11 WRIGHT FUNCTIONS 54 1.12 THE
^-FUNCTION 58 1.13 FIXED POINT THEOREMS 67 2 FRACTIONAL INTEGRALS AND
FRACTIONAL DERIVATIVES 69 2.1 RIEMANN-LIOUVILLE FRACTIONAL INTEGRALS AND
FRACTIONAL DERI- VATIVES 69 2.2 LIOUVILLE FRACTIONAL INTEGRALS AND
FRACTIONAL DERIVATIVES ON THE HALF- AXIS 79 2.3 LIOUVILLE FRACTIONAL
INTEGRALS AND FRACTIONAL DERIVATIVES ON THE REAL AXIS 87 2.4 CAPUTO
FRACTIONAL DERIVATIVES 90 2.5 FRACTIONAL INTEGRALS AND FRACTIONAL
DERIVATIVES OF A FUNCTION WITH RESPECT TO ANOTHER FUNCTION 99 2.6
ERDELYI-KOBER TYPE FRACTIONAL INTEGRALS AND FRACTIONAL DERIVA- TIVES 105
2.7 HADAMARD TYPE FRACTIONAL INTEGRALS AND FRACTIONAL DERIVATIVES . .
110 2.8 GRIINWALD-LETNIKOV FRACTIONAL DERIVATIVES 121 2.9 PARTIAL AND
MIXED FRACTIONAL INTEGRALS AND FRACTIONAL DERIVATIVES 123 2.10 RIESZ
FRACTIONAL INTEGRO-DIFFERENTIATION 127 2.11 COMMENTS AND OBSERVATIONS
132 XII THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS 3
ORDINARY FRACTIONAL DIFFERENTIAL EQUATIONS. EXISTENCE AND UNIQUENESS
THEOREMS 135 3.1 INTRODUCTION AND A BRIEF OVERVIEW OF RESULTS 135 3.2
EQUATIONS WITH THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE IN THE SPACE
OF SUMMABLE FUNCTIONS 144 3.2.1 EQUIVALENCE OF THE CAUCHY TYPE PROBLEM
AND THE VOLTERRA INTEGRAL EQUATION 145 3.2.2 EXISTENCE AND UNIQUENESS OF
THE SOLUTION TO THE CAUCHY TYPE PROBLEM 148 3.2.3 THE WEIGHTED CAUCHY
TYPE PROBLEM 151 3.2.4 GENERALIZED CAUCHY TYPE PROBLEMS 153 3.2.5 CAUCHY
TYPE PROBLEMS FOR LINEAR EQUATIONS 15 7 3.2.6 MISCELLANEOUS EXAMPLES 160
3.3 EQUATIONS WITH THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE IN THE
SPACE OF CONTINUOUS FUNCTIONS. GLOBAL SOLUTION 162 3.3.1 EQUIVALENCE OF
THE CAUCHY TYPE PROBLEM AND THE VOLTERRA INTEGRAL EQUATION 163 3.3.2
EXISTENCE AND UNIQUENESS OF THE GLOBAL SOLUTION TO THE CAUCHY TYPE
PROBLEM 164 3.3.3 THE WEIGHTED CAUCHY TYPE PROBLEM 167 3.3.4 GENERALIZED
CAUCHY TYPE PROBLEMS 168 3.3.5 CAUCHY TYPE PROBLEMS FOR LINEAR EQUATIONS
170 3.3.6 MORE EXACT SPACES 171 3.3.7 FURTHER EXAMPLES 177 3.4 EQUATIONS
WITH THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE IN THE SPACE OF
CONTINUOUS FUNCTIONS. SEMI-GLOBAL AND LOCAL SOLUTIONS . 182 3.4.1 THE
CAUCHY TYPE PROBLEM WITH INITIAL CONDITIONS AT THE ENDPOINT OF THE
INTERVAL. SEMI-GLOBAL SOLUTION 183 3.4.2 THE CAUCHY PROBLEM WITH INITIAL
CONDITIONS AT THE INNER POINT OF THE INTERVAL. PRELIMINARIES 186 3.4.3
EQUIVALENCE OF THE CAUCHY PROBLEM AND THE VOLTERRA INTEGRAL EQUATION .
189 3.4.4 THE CAUCHY PROBLEM WITH INITIAL CONDITIONS AT THE INNER POINT
OF THE INTERVAL. THE UNIQUENESS OF SEMI-GLOBAL AND LOCAL SOLUTIONS 191
3.4.5 A SET OF EXAMPLES 196 3.5 EQUATIONS WITH THE CAPUTO DERIVATIVE IN
THE SPACE OF CONTINUOUSLY DIFFERENTIABLE FUNCTIONS 198 3.5.1 THE CAUCHY
PROBLEM WITH INITIAL CONDITIONS AT THE ENDPOINT OF THE INTERVAL. GLOBAL
SOLUTION 199 3.5.2 THE CAUCHY PROBLEMS WITH INITIAL CONDITIONS AT THE
END AND INNER POINTS OF THE INTERVAL. SEMI-GLOBAL AND LOCAL SOLUTIONS
205 3.5.3 ILLUSTRATIVE EXAMPLES 209 CONTENTS XIII 3.6 EQUATIONS WITH THE
HADAMARD FRACTIONAL DERIVATIVE IN THE SPACE OF CONTINUOUS FUNCTIONS 212
4 METHODS FOR EXPLICITLY SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS 221
4.1 METHOD OF REDUCTION TO VOLTERRA INTEGRAL EQUATIONS 221 4.1.1 THE
CAUCHY TYPE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH THE
RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES 222 4.1.2 THE CAUCHY PROBLEMS
FOR ORDINARY DIFFERENTIAL EQUA- TIONS 228 4.1.3 THE CAUCHY PROBLEMS FOR
DIFFERENTIAL EQUATIONS WITH THE CAPUTO FRACTIONAL DERIVATIVES 230 4.1.4
THE CAUCHY TYPE PROBLEMS FOR DIFFERENTIAL EQUATIONS WITH HADAMARD
FRACTIONAL DERIVATIVES 234 4.2 COMPOSITIONAL METHOD 238 4.2.1
PRELIMINARIES 238 4.2.2 COMPOSITIONAL RELATIONS 239 4.2.3 HOMOGENEOUS
DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH RIEMANN-LIOUVILLE
FRACTIONAL DERIVATIVES 242 4.2.4 NONHOMOGENEOUS DIFFERENTIAL EQUATIONS
OF FRACTIONAL OR- DER WITH RIEMANN-LIOUVILLE AND LIOUVILLE FRACTIONAL
DERIVA- TIVES WITH A QUASI-POLYNOMIAL FREE TERM 245 4.2.5 DIFFERENTIAL
EQUATIONS OF ORDER 1/2 248 4.2.6 CAUCHY TYPE PROBLEM FOR NONHOMOGENEOUS
DIFFERENTIAL EQUATIONS WITH RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES AND
WITH A QUASI-POLYNOMIAL FREE TERM 251 4.2.7 SOLUTIONS TO HOMOGENEOUS
FRACTIONAL DIFFERENTIAL EQUATIONS WITH LIOUVILLE FRACTIONAL DERIVATIVES
IN TERMS OF BESSEL- TYPE FUNCTIONS \ 257 4.3 OPERATIONAL METHOD 260
4.3.1 LIOUVILLE FRACTIONAL INTEGRATION AND DIFFERENTIATION OPERA- TORS
IN SPECIAL FUNCTION SPACES ON THE HALF-AXIS 261 4.3.2 OPERATIONAL
CALCULUS FOR THE LIOUVILLE FRACTIONAL CALCULUS OPERATORS 263 4.3.3
SOLUTIONS TO CAUCHY TYPE PROBLEMS FOR FRACTIONAL DIFFEREN- TIAL
EQUATIONS WITH LIOUVILLE FRACTIONAL DERIVATIVES 26 6 4.3.4 OTHER RESULTS
270 4.4 NUMERICAL TREATMENT 272 5 INTEGRAL TRANSFORM METHOD FOR EXPLICIT
SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS 279 5.1 INTRODUCTION AND
A BRIEF SURVEY OF RESULTS 279 5.2 LAPLACE TRANSFORM METHOD FOR SOLVING
ORDINARY DIFFERENTIAL EQUA- TIONS WITH LIOUVILLE FRACTIONAL DERIVATIVES
283 XIV THEORY AND APPLICATIONS OF FRACTIONAL DIFFERENTIAL EQUATIONS
5.2.1 HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS 283 5.2.2
NONHOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS . . . 295 5.2.3
EQUATIONS WITH NONCONSTANT COEFFICIENTS 303 5.2.4 CAUCHY TYPE FOR
FRACTIONAL DIFFERENTIAL EQUATIONS 309 5.3 LAPLACE TRANSFORM METHOD FOR
SOLVING ORDINARY DIFFERENTIAL EQUA- TIONS WITH CAPUTO FRACTIONAL
DERIVATIVES 312 5.3.1 HOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS
312 5.3.2 NONHOMOGENEOUS EQUATIONS WITH CONSTANT COEFFICIENTS . . . 322
5.3.3 CAUCHY PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS . . . 326
5.4 MELLIN TRANSFORM METHOD FOR SOLVING NONHOMOGENEOUS FRACTIONAL
DIFFERENTIAL EQUATIONS WITH LIOUVILLE DERIVATIVES 329 5.4.1 GENERAL
APPROACH TO THE PROBLEMS 329 5.4.2 EQUATIONS WITH LEFT-SIDED FRACTIONAL
DERIVATIVES 331 5.4.3 EQUATIONS WITH RIGHT-SIDED FRACTIONAL DERIVATIVES
336 5.5 FOURIER TRANSFORM METHOD FOR SOLVING NONHOMOGENEOUS DIFFEREN-
TIAL EQUATIONS WITH RIESZ FRACTIONAL DERIVATIVES 341 5.5.1
MULTI-DIMENSIONAL EQUATIONS 341 5.5.2 ONE-DIMENSIONAL EQUATIONS 344 6
PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS 347 6.1 OVERVIEW OF RESULTS
347 6.1.1 PARTIAL DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 347 6.1.2
FRACTIONAL PARTIAL DIFFERENTIAL DIFFUSION EQUATIONS 351 6.1.3 ABSTRACT
DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER 359 6.2 SOLUTION OF CAUCHY
TYPE PROBLEMS FOR FRACTIONAL DIFFUSION-WAVE EQUATIONS 362 6.2.1 CAUCHY
TYPE PROBLEMS FOR TWO-DIMENSIONAL EQUATIONS . . 362 6.2.2 CAUCHY TYPE
PROBLEMS FOR MULTI-DIMENSIONAL EQUATIONS . . 366 6.3 SOLUTION OF CAUCHY
PROBLEMS FOR FRACTIONAL DIFFUSION-WAVE EQUA- TIONS 373 6.3.1 CAUCHY
PROBLEMS FOR TWO-DIMENSIONAL EQUATIONS 374 6.3.2 CAUCHY PROBLEMS FOR
MULTI-DIMENSIONAL EQUATIONS 377 6.4 SOLUTION OF CAUCHY PROBLEMS FOR
FRACTIONAL EVOLUTION EQUATIONS . 380 6.4.1 SOLUTION OF THE SIMPLEST
PROBLEM 380 6.4.2 SOLUTION TO THE GENERAL PROBLEM 384 6.4.3 SOLUTIONS OF
CAUCHY PROBLEMS VIA THE H-FUNCTIONS 388 7 SEQUENTIAL LINEAR DIFFERENTIAL
EQUATIONS OF FRACTIONAL ORDER 393 7.1 SEQUENTIAL LINEAR DIFFERENTIAL
EQUATIONS OF FRACTIONAL ORDER . . . . 394 7.2 SOLUTION OF LINEAR
DIFFERENTIAL EQUATIONS WITH CONSTANT COEF- FICIENTS 400 7.2.1 GENERAL
SOLUTION IN THE HOMOGENEOUS CASE 400 7.2.2 GENERAL SOLUTION IN THE
NON-HOMOGENEOUS CASE. FRACTIONAL GREEN FUNCTION 403 CONTENTS XV 7.3
NON-SEQUENTIAL LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT CO-
EFFICIENTS 407 7.4 SYSTEMS OF EQUATIONS ASSOCIATED WITH
RIEMANN-LIOUVILLE AND CA- PUTO DERIVATIVES 409 7.4.1 GENERAL THEORY 409
7.4.2 GENERAL SOLUTION FOR THE CASE OF CONSTANT COEFFICIENTS. FRAC-
TIONAL GREEN VECTORIAL FUNCTION 412 7.5 SOLUTION OF FRACTIONAL
DIFFERENTIAL EQUATIONS WITH VARIABLE COEF- FICIENTS. GENERALIZED METHOD
OF FROBENIUS 415 7.5.1 INTRODUCTION 41 5 7.5.2 SOLUTIONS AROUND AN
ORDINARY POINT OF A FRACTIONAL DIFFER- ENTIAL EQUATION OF ORDER A 418
7.5.3 SOLUTIONS AROUND AN ORDINARY POINT OF A FRACTIONAL DIFFER- ENTIAL
EQUATION OF ORDER 2A 421 7.5.4 SOLUTION AROUND AN A-SINGULAR POINT OF A
FRACTIONAL DIFFER- ENTIAL EQUATION OF ORDER A 424 7.5.5 SOLUTION AROUND
AN A-SINGULAR POINT OF A FRACTIONAL DIFFER- ENTIAL EQUATION OF ORDER 2A
427 7.6 SOME APPLICATIONS OF LINEAR ORDINARY FRACTIONAL DIFFERENTIAL
EQUATIONS 433 7.6.1 DYNAMICS OF A SPHERE IMMERSED IN AN INCOMPRESSIBLE
VIS- COUS FLUID. BASSET'S PROBLEM 434 7.6.2 OSCILLATORY PROCESSES WITH
FRACTIONAL DAMPING 436 7.6.3 STUDY OF THE TENSION-DEFORMATION
RELATIONSHIP OF VISCOELAS- TIC MATERIALS 439 8 FURTHER APPLICATIONS OF
FRACTIONAL MODELS 449 8.1 PRELIMINARY REVIEW 449 8.1.1 HISTORICAL
OVERVIEW 450 8.1.2 COMPLEX SYSTEMS 452 8.1.3 FRACTIONAL INTEGRAL AND
FRACTIONAL DERIVATIVE OPERATORS . . 456 8.2 FRACTIONAL MODEL FOR THE
SUPER-DIFFUSION PROCESSES 458 8.3 DIRAC EQUATIONS FOR THE ORDINARY
DIFFUSION EQUATION 462 8.4 APPLICATIONS DESCRIBING CARRIER TRANSPORT IN
AMORPHOUS SEMICON- DUCTORS WITH MULTIPLE TRAPPING 463 BIBLIOGRAPHY 469
SUBJECT INDEX 521 |
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| author | Kilbas, Anatoly A. 1948-2010 Srivastava, Hari M. 1940- Trujillo, Juan J. |
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| id | DE-604.BV021552588 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T14:32:12Z |
| indexdate | 2024-07-09T20:38:27Z |
| institution | BVB |
| isbn | 0444518320 9780444518323 |
| language | English |
| lccn | 2005044764 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014768638 |
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| physical | XV, 523 Seiten |
| publishDate | 2006 |
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| publisher | Elsevier |
| record_format | marc |
| series | North-Holland Mathematics Studies |
| series2 | North-Holland Mathematics Studies |
| spelling | Kilbas, Anatoly A. 1948-2010 Verfasser (DE-588)1042268665 aut Theory and applications of fractional differential equations Anatoly A. Kilbas, Belarusian State University, Minsk, Belarus, Hari M. Srivastava, University of Victoria, Victoria, British Columbia, Canada, Juan J. Trujillo, Universidad de La Laguna, La Laguna (Tenerife), Canary Islands, Spain First edition Amsterdam [u.a.] Elsevier 2006 XV, 523 Seiten txt rdacontent n rdamedia nc rdacarrier North-Holland Mathematics Studies 204 Differential equations Fractional calculus Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Gebrochene Analysis (DE-588)4722475-7 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Gebrochene Analysis (DE-588)4722475-7 s DE-604 Srivastava, Hari M. 1940- Verfasser (DE-588)12358874X aut Trujillo, Juan J. Verfasser aut North-Holland Mathematics Studies 204 (DE-604)BV000003247 204 http://www.loc.gov/catdir/enhancements/fy0624/2005044764-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0624/2005044764-t.html Table of contents GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014768638&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kilbas, Anatoly A. 1948-2010 Srivastava, Hari M. 1940- Trujillo, Juan J. Theory and applications of fractional differential equations North-Holland Mathematics Studies Differential equations Fractional calculus Differentialgleichung (DE-588)4012249-9 gnd Gebrochene Analysis (DE-588)4722475-7 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4722475-7 |
| title | Theory and applications of fractional differential equations |
| title_auth | Theory and applications of fractional differential equations |
| title_exact_search | Theory and applications of fractional differential equations |
| title_exact_search_txtP | Theory and applications of fractional differential equations |
| title_full | Theory and applications of fractional differential equations Anatoly A. Kilbas, Belarusian State University, Minsk, Belarus, Hari M. Srivastava, University of Victoria, Victoria, British Columbia, Canada, Juan J. Trujillo, Universidad de La Laguna, La Laguna (Tenerife), Canary Islands, Spain |
| title_fullStr | Theory and applications of fractional differential equations Anatoly A. Kilbas, Belarusian State University, Minsk, Belarus, Hari M. Srivastava, University of Victoria, Victoria, British Columbia, Canada, Juan J. Trujillo, Universidad de La Laguna, La Laguna (Tenerife), Canary Islands, Spain |
| title_full_unstemmed | Theory and applications of fractional differential equations Anatoly A. Kilbas, Belarusian State University, Minsk, Belarus, Hari M. Srivastava, University of Victoria, Victoria, British Columbia, Canada, Juan J. Trujillo, Universidad de La Laguna, La Laguna (Tenerife), Canary Islands, Spain |
| title_short | Theory and applications of fractional differential equations |
| title_sort | theory and applications of fractional differential equations |
| topic | Differential equations Fractional calculus Differentialgleichung (DE-588)4012249-9 gnd Gebrochene Analysis (DE-588)4722475-7 gnd |
| topic_facet | Differential equations Fractional calculus Differentialgleichung Gebrochene Analysis |
| url | http://www.loc.gov/catdir/enhancements/fy0624/2005044764-d.html http://www.loc.gov/catdir/enhancements/fy0624/2005044764-t.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014768638&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003247 |
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