Computational methods in the fractional calculus of variations:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
London
ICP Imperial College Press
[2015]
|
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xii, 266 Seiten Diagramme |
| ISBN: | 9781783266401 |
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| 245 | 1 | 0 | |a Computational methods in the fractional calculus of variations |c Ricardo Almeida (University of Aveiro, Portugal), Shakoor Pooseh (Technische Universitat Dresden, Germany), Delfim F.M. Torres (University of Aveiro, Portugal) |
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Record in the Search Index
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| adam_text | Computational Methods in the
ractional alculus
of ariations
This book fills a gap in the literature by introducing numerical
techniques to solve problems of fractional calculus of variations
(FCV). In most cases, finding the analytic solution to such
problems is extremely difficult or even impossible, and numerical
methods need to be used.
The authors are well-known researchers in the area of FCV and
the book contains some of their recent results, serving as a
companion volume to Introduction to the Fractional Calculus of
Variations by A B Malinowska and D F M Torres, where analytical
methods are presented to solve FCV problems. After some
preliminaries on the subject, different techniques are presented
in detail with numerous examples to help the reader to better
understand the methods. The techniques presented may be used
not only to deal with FCV problems but also in other contexts of
fractional calculus, such as fractional differential equations and
fractional optimal control. It is suitable as an advanced book for
graduate students in mathematics, physics and engineering, as
well as for researchers interested in fractional calculus.
Contents#
Preface v
1. Introduction 1
2. The calculus of variations and optimal control 9
2.1 The calculus of variations.................................. 9
2.1.1 From light beams to the Brachistochrone problem 9
2.1.2 Contemporary mathematical formulation.............. 10
2.1.3 Solution methods .................................. 13
2.2 Optimal control theory..................................... 16
2.2.1 Mathematical formulation........................... 16
2.2.2 Necessary optimality conditions.................... 18
2.2.3 Pontryagin’s minimum principle..................... 18
3. Fractional calculus 21
3.1 Special functions.......................................... 21
3.2 A historical review........................................ 22
3.3 The relation between the Riemann-Liouville and Caputo
derivatives................................................ 27
3.4 Integration by parts ...................................... 28
4. Fractional variational problems 29
4.1 Fractional calculus of variations and optimal control ... 29
4.2 A general formulation ..................................... 30
4.3 Fractional Euler-Lagrange equations........................ 32
4.4 Infinite horizon fractional variational problems .......... 33
ix
Computational Methods in the Fractional Calculus of Variations
4.4.1 The Euler-Lagrange equation ......................... 34
4.4.2 Optimal control problem....................... 38
4.4.3 Example....................................... 41
4.5 Variational problems with the Riesz-Caputo derivative . . 42
4.5.1 The Euler-Lagrange equation ......................... 42
4.5.2 The fractional isoperimetric problem........... 45
4.5.3 Optimal time problem.......................... 47
4.6 Solution methods...................................... 49
5. Numerical methods for fractional variational problems 51
6. Approximating fractional derivatives 57
6.1 Riemann-Liouville derivative.......................... 57
6.1.1 Approximation by a sum of integer-order
derivatives................................... 57
6.1.2 Approximation using moments of a function ... 59
6.1.3 Numerical evaluation of fractional derivatives ... 66
6.1.4 Fractional derivatives of tabular data......... 68
6.1.5 Applications to fractional differential equations . . 72
6.2 Hadamard derivatives ....................................... 74
6.2.1 Approximation by a sum of integer-order
derivatives.................................... 74
6.2.2 Approximation using moments of a function ... 76
6.2.3 Examples....................................... 78
6.3 Error analysis........................................ 80
7. Approximating fractional integrals 85
7.1 Riemann-Liouville fractional integral................. 85
7.1.1 Approximation by a sum of integer-order
derivatives.................................... 85
7.1.2 Approximation using moments of a function ... 86
7.1.3 Numerical evaluation of fractional integrals .... 90
7.1.4 Applications to fractional integral equations ... 96
7.2 Hadamard fractional integrals......................... 97
7.2.1 Approximation by a sum of integer-order
derivatives.................................... 97
7.2.2 Approximation using moments of a function ... 97
7.2.3 Examples...................................... 103
Contents xi
7.3 Error analysis............................................... 105
8. Direct methods 111
8.1 Finite differences for fractional derivatives................ Ill
8.2 Euler-like direct method for variational problems............ 112
8.2.1 Euler’s classic direct method..................... 112
8.2.2 Euler-like direct method.......................... 113
8.2.3 Examples.......................................... 116
8.3 A discrete-time method on the first variation................ 122
8.3.1 Basic fractional variational problems............. 125
8.3.2 An isoperimetric fractional variational problem . . 126
9. Indirect methods 129
9.1 Expansion to integer orders.................................. 132
9.2 Expansion through the moments of a function.................. 134
10. Fractional optimal control with free end-points 139
10.1 Necessary optimality conditions.............................. 139
10.1.1 Fractional necessary conditions................... 140
10.1.2 Approximated integer-order necessary optimality
conditions .......................................... 145
10.2 A generalization............................................. 146
10.3 Sufficient optimality conditions............................. 149
10.4 Numerical treatment and examples............................. 161
10.4.1 Fixed final time.................................. 161
10.4.2 Free final time................................... 166
IF An expansion formula for fractional operators of variable order 159
11.1 Introduction................................................. 169
11.2 Fractional calculus of variable order........................ 160
11.3 Expansion formulas with higher-order derivatives............. 161
11.4 Examples..................................................... 169
11.4.1 Test function..................................... 169
11.4.2 Fractional differential equations of variable order . 169
11.4.3 Fractional variational calculus of variable order . . 172
12. Discrete-time fractional calculus of variations 177
12.1 Introduction................................................. 177
xii Computational Methods in the Fractional Calculus of Variations
12.2 Discrete fractional calculus.............................. 178
12.3 Basic definitions on time scales.......................... 180
12.4 Calculus of variations on time scales.................... 186
12.5 Fractional variational problems in T = Z.................. 189
12.5.1 Introduction..................................... 189
12.5.2 Preliminaries..................................... 190
12.5.3 Fractional summation by parts..................... 194
12.5.4 Necessary optimality conditions................... 197
12.5.5 Examples...........................................206
12.5.6 Conclusion........................................ 211
12.6 Fractional Variational Problems in T = (/¿Z)a............ 212
12.6.1 Introduction...................................... 212
12.6.2 Preliminaries..................................... 212
12.6.3 Fractional /¿֊summation by parts.................. 221
12.6.4 Necessary optimality conditions....................224
12.6.5 Examples...........................................233
12.6.6 Conclusion.........................................237
13. Conclusion 239
Appendix A MATLAB® code 243
Bibliography 251
Index 265
|
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| author | Almeida, Ricardo Pooseh, Shakoor Torres, Delfim F. M. 1971- |
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| id | DE-604.BV042403501 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:20:36Z |
| institution | BVB |
| isbn | 9781783266401 |
| language | English |
| lccn | 014049076 |
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| owner | DE-703 |
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| physical | xii, 266 Seiten Diagramme |
| publishDate | 2015 |
| publishDateSearch | 2015 |
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| publisher | ICP Imperial College Press |
| record_format | marc |
| spelling | Almeida, Ricardo Verfasser aut Computational methods in the fractional calculus of variations Ricardo Almeida (University of Aveiro, Portugal), Shakoor Pooseh (Technische Universitat Dresden, Germany), Delfim F.M. Torres (University of Aveiro, Portugal) London ICP Imperial College Press [2015] xii, 266 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Fractional calculus Calculus of variations Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s DE-604 Pooseh, Shakoor Verfasser (DE-588)1075478235 aut Torres, Delfim F. M. 1971- Verfasser (DE-588)1029177880 aut Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027839142&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027839142&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Almeida, Ricardo Pooseh, Shakoor Torres, Delfim F. M. 1971- Computational methods in the fractional calculus of variations Fractional calculus Calculus of variations Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4062355-5 |
| title | Computational methods in the fractional calculus of variations |
| title_auth | Computational methods in the fractional calculus of variations |
| title_exact_search | Computational methods in the fractional calculus of variations |
| title_full | Computational methods in the fractional calculus of variations Ricardo Almeida (University of Aveiro, Portugal), Shakoor Pooseh (Technische Universitat Dresden, Germany), Delfim F.M. Torres (University of Aveiro, Portugal) |
| title_fullStr | Computational methods in the fractional calculus of variations Ricardo Almeida (University of Aveiro, Portugal), Shakoor Pooseh (Technische Universitat Dresden, Germany), Delfim F.M. Torres (University of Aveiro, Portugal) |
| title_full_unstemmed | Computational methods in the fractional calculus of variations Ricardo Almeida (University of Aveiro, Portugal), Shakoor Pooseh (Technische Universitat Dresden, Germany), Delfim F.M. Torres (University of Aveiro, Portugal) |
| title_short | Computational methods in the fractional calculus of variations |
| title_sort | computational methods in the fractional calculus of variations |
| topic | Fractional calculus Calculus of variations Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Fractional calculus Calculus of variations Variationsrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027839142&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027839142&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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