The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order

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Elsevier, Sep 5, 1974 - Mathematics - 322 pages
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In 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

Chapter 1 INTRODUCTION
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
References
219
Index
225
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