## The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary OrderIn this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation; methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; and methods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory. As a result, the book represents a blend of new methods in general computational analysis, and specific, but also generic, techniques for study of systems theory ant its particular branches, such as optimal filtering and information compression. - Best operator approximation, - Non-Lagrange interpolation, - Generic Karhunen-Loeve transform - Generalised low-rank matrix approximation - Optimal data compression - Optimal nonlinear filtering |

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### Contents

1 | |

Chapter 2 DIFFERENTIATION AND INTEGRATION TO INTEGER ORDER | 25 |

DEFINITIONS AND EQUIVALENCES | 45 |

Chapter 4 DIFFERINTEGRATION OF SIMPLE FUNCTlONS | 61 |

Chapter 5 GENERAL PROPERTIES | 69 |

Chapter 6 DIFFERINTEGRATION OF MORE COMPLEX FUNCTIONS | 93 |

Chapter 7 SEMIDERIVATIVES AND SEMIINTEG RALS | 115 |

Chapter 8 TECHNIQUES IN THE FRACTIONAL CALCULUS | 133 |

Chapter 9 REPRESENTATION OF TRANSCENDENTAL FUNCTIONS | 161 |

Chapter 10 APPLICATIONS IN THE CLASSICAL CALCULUS | 181 |

Chapter 11 APPLICATIONS TO DIFFUSION PROBLEMS | 197 |

219 | |

225 | |

### Other editions - View all

The Fractional Calculus: Theory and Applications of Differentiation and ... Keith B. Oldham,Jerome Spanier No preview available - 1974 |

The Fractional Calculus: Theory and Applications of Differentiation and ... Keith B. Oldham,Jerome Spanier No preview available - 2006 |

### Common terms and phrases

Abramowitz and Stegun algorithms analytic functions application arbitrary order basis hypergeometric Bessel function binomial boundary chain rule Chapter coefficients complexity composition rule constant convergence cos(x d-ºf defined denominatorial parameters derivatives and integrals differ differential equations differintegrable function differintegrable series differintegral operators diffusion dºf example exp(x expression finite formula fractional calculus Fractional Derivatives fractional differentiation Fractional Integration fractional operations function f gamma function geometric Heaviside hypergeometric function identity incomplete gamma function infinite series integer integer order inverse Laplace transform Leibniz Leibniz's Leibniz's rule Liouville ln(x logarithm lower limit Math multiplication noninteger Oldham Osler positive integer problem properties replaced resistor result Riemann Riemann–Liouville definition Section 3.1 semiderivative semidifferential equation semiintegral sin(x ſº solution Struve functions summation symbolism synthesis diagram term theorem theory transcendental functions valid variable Vºx zero