An introduction to the fractional calculus and fractional differential equations:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Wiley
1993
|
| Series: | A Wiley interscience publication
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 366 S. graph. Darst. |
| ISBN: | 0471588849 |
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MARC
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| 245 | 1 | 0 | |a An introduction to the fractional calculus and fractional differential equations |c Kenneth S. Miller ; Bertram Ross |
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
Preface xi
I. Historical Survey 1
1. The Origin of the Fractional Calculus, 1
2. The Contributions of Abel and Liouville, 3
3. A Longstanding Controversy, 6
4. Riemann s Contribution, Errors by Noted
Mathematicians, 7
5. The Mid Nineteenth Century, 9
6. The Origin of the Riemann Liouville Definition, 9
7. The Last Decade of the Nineteenth Century, 13
8. The Twentieth Century, 15
9. Bibliography, 16
II. The Modern Approach 21
1. Introduction, 21
2. The Iterated Integral Approach, 23
3. The Differential Equation Approach, 25
4. The Complex Variable Approach, 28
5. The Weyl Transform, 33
6. The Fractional Derivative, 35
7. The Definitions of Grunwald and Marchaud, 38
vii
viii CONTENTS
III. The Riemann Liouville Fractional Integral 44
1. Introduction, 44
2. Definition of the Fractional Integral, 45
3. Some Examples of Fractional Integrals, 47
4. Dirichlet s Formula, 56
5. Derivatives of the Fractional Integral and the
Fractional Integral of Derivatives, 59
6. Laplace Transform of the Fractional Integral, 67
7. Leibniz s Formula for Fractional Integrals, 73
IV. The Riemann Liouville Fractional Calculus 80
1. Introduction, 80
2. The Fractional Derivative, 82
3. A Class of Functions, 87
4. Leibniz s Formula for Fractional Derivatives, 95
5. Some Further Examples, 97
6. The Law of Exponents, 104
7. Integral Representations, 111
8. Representations of Functions, 116
9. Integral Relations, 118
10. Laplace Transform of the Fractional Derivative, 121
V. Fractional Differential Equations 126
1. Introduction, 126
2. Motivation: Direct Approach, 128
3. Motivation: Laplace Transform, 133
4. Motivation: Linearly Independent Solutions, 136
5. Solution of the Homogeneous Equation, 139
6. Explicit Representation of Solution, 145
7. Relation to the Green s Function, 153
8. Solution of the Nonhomogeneous Fractional
Differential Equation, 157
9. Convolution of Fractional Green s Functions, 165
10. Reduction of Fractional Differential Equations
to Ordinary Differential Equations, 171
11. Semidifferential Equations, 174
CONTENTS ix
VI. Further Results Associated with Fractional
Differential Equations 185
1. Introduction, 185
2. Fractional Integral Equations, 186
3. Fractional Differential Equations with Nonconstant
Coefficients, 194
4. Sequential Fractional Differential Equations, 209
5. Vector Fractional Differential Equations, 217
6. Some Comparisons with Ordinary Differential
Equations, 229
VII. The Weyl Fractional Calculus 236
1. Introduction, 236
2. Good Functions, 237
3. A Law of Exponents for Fractional Integrals, 239
4. The Weyl Fractional Derivative, 240
5. The Algebra of the Weyl Transform, 244
6. A Leibniz Formula, 245
7. Some Further Examples, 247
8. An Application to Ordinary Differential
Equations, 251
VIII. Some Historical Arguments 255
1. Introduction, 255
2. Abel s Integral Equation and the Tautochrone
Problem, 255
3. Heaviside Operational Calculus and the Fractional
Calculus, 261
4. Potential Theory and Liouville s Problem, 264
5. Fluid Flow and the Design of a Weir Notch, 269
Appendix A. Some Algebraic Results 275
1. Introduction, 275
2. Some Identities Associated with Partial Fraction
Expansions, 275
3. Zeros of Multiplicity Greater than One, 285
x CONTENTS
4. Complementary Polynomials, 290
5. Some Reduction Formulas, 292
6. Some Algebraic Identities, 294
Appendix B. Higher Transcendental
Functions 297
1. Introduction, 297
2. The Gamma Function and Related Functions, 297
3. Bessel Functions, 301
4. Hypergeometric Functions, 303
5. Legendre and Laguerre Functions, 307
Appendix C. The Incomplete Gamma Function and
Related Functions 308
1. Introduction, 308
2. The Incomplete Gamma Function, 309
3. Some Functions Related to the Incomplete Gamma
Function, 314
4. Laplace Transforms, 321
5. Numerical Results, 330
Appendix D. A Brief Table of Fractional Integrals
and Derivatives 352
References 357
Index of Symbols 361
Index 363
|
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| id | DE-604.BV008408818 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:19:14Z |
| institution | BVB |
| isbn | 0471588849 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005538590 |
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| publishDate | 1993 |
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| publisher | Wiley |
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| series2 | A Wiley interscience publication |
| spelling | Miller, Kenneth S. Verfasser (DE-588)1159045585 aut An introduction to the fractional calculus and fractional differential equations Kenneth S. Miller ; Bertram Ross New York u.a. Wiley 1993 XIII, 366 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier A Wiley interscience publication Calculus Differential equations Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Differentialoperator (DE-588)4012251-7 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Differentialrechnung (DE-588)4012252-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s DE-604 Differentialoperator (DE-588)4012251-7 s Partielle Differentialgleichung (DE-588)4044779-0 s Differentialrechnung (DE-588)4012252-9 s Ross, Bertram Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005538590&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Miller, Kenneth S. Ross, Bertram An introduction to the fractional calculus and fractional differential equations Calculus Differential equations Differentialgleichung (DE-588)4012249-9 gnd Differentialoperator (DE-588)4012251-7 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Differentialrechnung (DE-588)4012252-9 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4012251-7 (DE-588)4044779-0 (DE-588)4012252-9 |
| title | An introduction to the fractional calculus and fractional differential equations |
| title_auth | An introduction to the fractional calculus and fractional differential equations |
| title_exact_search | An introduction to the fractional calculus and fractional differential equations |
| title_full | An introduction to the fractional calculus and fractional differential equations Kenneth S. Miller ; Bertram Ross |
| title_fullStr | An introduction to the fractional calculus and fractional differential equations Kenneth S. Miller ; Bertram Ross |
| title_full_unstemmed | An introduction to the fractional calculus and fractional differential equations Kenneth S. Miller ; Bertram Ross |
| title_short | An introduction to the fractional calculus and fractional differential equations |
| title_sort | an introduction to the fractional calculus and fractional differential equations |
| topic | Calculus Differential equations Differentialgleichung (DE-588)4012249-9 gnd Differentialoperator (DE-588)4012251-7 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Differentialrechnung (DE-588)4012252-9 gnd |
| topic_facet | Calculus Differential equations Differentialgleichung Differentialoperator Partielle Differentialgleichung Differentialrechnung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005538590&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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