Research Catalog

Title

An introduction to the fractional calculus and fractional differential equations / Kenneth S. Miller, Bertram Ross.

Author

Miller, Kenneth S.

Publication

New York : Wiley, [1993], ©1993.

Items in the Library & Off-site

Holdings

Details

Additional Authors

Ross, Bertram.

Description

xiii, 366 pages : illustrations; 25 cm

Alternative Title

Fractional calculus and fractional differential equations.

Subjects

• Differential equations
• Calculus

Note

• "A Wiley-Interscience publication."

Bibliography (note)

• Includes bibliographical references and index.

Contents

• I. Historical Survey. 1. The Origin of the Fractional Calculus. 2. The Contributions of Abel and Liouville. 3. A Longstanding Controversy. 4. Riemann's Contribution, Errors by Noted Mathematicians. 5. The Mid-Nineteenth Century. 6. The Origin of the Riemann-Liouville Definition. 7. The Last Decade of the Nineteenth Century. 8. The Twentieth Century -- II. The Modern Approach. 2. The Iterated Integral Approach. 3. The Differential Equation Approach. 4. The Complex Variable Approach. 5. The Weyl Transform. 6. The Fractional Derivative. 7. The Definitions of Grunwald and Marchaud -- III. The Riemann-Liouville Fractional Integral. 2. Definition of the Fractional Integral. 3. Some Examples of Fractional Integrals. 4. Dirichlet's Formula. 5. Derivatives of the Fractional Integral and the Fractional Integral of Derivatives. 6. Laplace Transform of the Fractional Integral. 7. Leibniz's Formula for Fractional Integrals -- IV. The Riemann-Liouville Fractional Calculus.
• 2. The Fractional Derivative. 3. A Class of Functions. 4. Leibniz's Formula for Fractional Derivatives. 6. The Law of Exponents. 7. Integral Representations. 8. Representations of Functions. 9. Integral Relations. 10. Laplace Transform of the Fractional Derivative -- V. Fractional Differential Equations. 2. Motivation: Direct Approach. 3. Motivation: Laplace Transform. 4. Motivation: Linearly Independent Solutions. 5. Solution of the Homogeneous Equation. 6. Explicit Representation of Solution. 7. Relation to the Green's Function. 8. Solution of the Nonhomogeneous Fractional Differential Equation. 9. Convolution of Fractional Green's Functions. 10. Reduction of Fractional Differential Equations to Ordinary Differential Equations. 11. Semidifferential Equations -- VI. Further Results Associated with Fractional Differential Equations. 2. Fractional Integral Equations. 3. Fractional Differential Equations with Nonconstant Coefficients. 4. Sequential Fractional Differential Equations.
• 5. Vector Fractional Differential Equations. 6. Some Comparisons with Ordinary Differential Equations -- VII. The Weyl Fractional Calculus. 2. Good Functions. 3. A Law of Exponents for Fractional Integrals. 4. The Weyl Fractional Derivative. 5. The Algebra of the Weyl Transform. 6. A Leibniz Formula. 8. An Application to Ordinary Differential Equations -- VIII. Some Historical Arguments. 2. Abel's Integral Equation and the Tautochrone Problem. 3. Heaviside Operational Calculus and the Fractional Calculus. 4. Potential Theory and Liouville's Problem. 5. Fluid Flow and the Design of a Weir Notch -- Appendix A. Some Algebraic Results. 2. Some Identities Associated with Partial Fraction Expansions. 3. Zeros of Multiplicity Greater than One. 4. Complementary Polynomials. 5. Some Reduction Formulas. 6. Some Algebraic Identities -- Appendix B. Higher Transcendental Functions. 2. The Gamma Function and Related Functions. 3. Bessel Functions. 4. Hypergeometric Functions.
• 5. Legendre and Laguerre Functions -- Appendix C. The Incomplete Gamma Function and Related Functions. 2. The Incomplete Gamma Function. 3. Some Functions Related to the Incomplete Gamma Function. 4. Laplace Transforms. 5. Numerical Results -- Appendix D. A Brief Table of Fractional Integrals and Derivatives.

ISBN

0471588849 (acid-free) :

LCCN

93009500

OCLC

• 27428486
• ocm27428486

Owning Institutions

Columbia University Libraries