An Introduction to the Fractional Calculus and Fractional Differential Equations / Edition 1

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Commences with the historical development of fractional calculus, its mathematical theory--particularly the Riemann-Liouville version. Numerous examples and theoretical applications of the theory are presented. Features topics associated with fractional differential equations. Discusses Weyl fractional calculus and some of its uses. Includes selected physical problems which lead to fractional differential or integral equations.
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Product Details

  • ISBN-13: 9780471588849
  • Publisher: Wiley, John & Sons, Incorporated
  • Publication date: 10/28/1993
  • Edition number: 1
  • Pages: 384
  • Product dimensions: 6.30 (w) x 9.45 (h) x 0.98 (d)

Table of Contents

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
II The Modern Approach 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
III The Riemann-Liouville Fractional Integral 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
2 The Fractional Derivative 82
3 A Class of Functions 87
4 Leibniz's Formula for Fractional Derivatives 95
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
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
VI Further Results Associated with Fractional Differential Equations 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
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
8 An Application to Ordinary Differential Equations 251
VIII Some Historical Arguments 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
2 Some Identities Associated with Partial Fraction Expansions 275
3 Zeros of Multiplicity Greater than One 285
4 Complementary Polynomials 290
5 Some Reduction Formulas 292
6 Some Algebraic Identities 294
Appendix B. Higher Transcendental Functions 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
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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