## 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 |

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Abramowitz and Stegun algorithms analytic functions application arbitrary order basis hypergeometric Bessel function binomial boundary chain rule Chapter classical calculus coefﬁcients complexity composition rule constant convergence deﬁned deﬁnite integrals deﬁnition denominatorial parameters derivatives and integrals differ differential equations differintegrable series differintegral operators diffusion dQfU evaluation example exp(x exponential expression ﬁnal ﬁnd ﬁrst formula fractional calculus Fractional Derivatives fractional differentiation Fractional Integration fractional operations function f G. H. Hardy geometries Griinwald Heaviside hypergeometric function Id(x identity incomplete gamma function inﬁnite series integer integer order integral equation inverse Laplace transform Legendre functions Liouville ln(x logarithm lower limit Math mathematical multiplication negative integer Oldham Osler positive integer problem properties replaced resistor result Riemann Riemann—Liouville deﬁnition Section 3.1 semiderivative semidifferential equation semiinﬁnite semiintegral sin(x solution solving Struve functions summation symbolism synthesis diagram technique term term-by-term theorem theory tion transcendental functions valid variable zero