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

Hardcover (Print)
Not Available on BN.com
 

Overview

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.
Read More Show Less

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

Preface
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
Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star

(0)

4 Star

(0)

3 Star

(0)

2 Star

(0)

1 Star

(0)

Your Rating:

Your Name: Create a Pen Name or

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation

Reminder:

  • - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
  • - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

 
Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)