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Introduction to the Fractional Calculus and Fractional Differential Equations

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ISBN-10: 0471588849

ISBN-13: 9780471588849

Edition: 1993

Authors: Kenneth Miller, Bertram Ross

List price: $145.00
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Book details

List price: $145.00
Copyright year: 1993
Publisher: John Wiley & Sons, Incorporated
Publication date: 6/2/1993
Binding: Hardcover
Pages: 384
Size: 6.50" wide x 9.75" long x 1.00" tall
Weight: 1.584
Language: English

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