حساب کسری: نظریه و کاربردهای مشتق‌گیری و انتگرال‌گیری به ترتیب دلخواه (کتاب‌های دوور در ریاضیات)

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 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 The product of a collaboration between a mathematician and a chemist, this text is geared toward advanced undergraduates and graduate students. Not only does it explain the theory underlying the properties of the generalized operator, but it also illustrates the wide variety of fields to which these ideas may be applied. Rather than an exhaustive treatment, it represents an introduction that will appeal to a broad spectrum of students. Accordingly, the mathematics is kept as simple as possible. The first of the two-part treatment deals principally with the general properties of differintegral operators. The second half is mainly oriented toward the applications of these properties to mathematical and other problems. Topics include integer order, simple and complex functions, semiderivatives and semi-integrals, and transcendental functions. The text concludes with overviews of applications in the classical calculus and diffusion problems. Content: Edited by Page iii Copyright page Page iv Preface Pages ix-xii Acknowledgments Page xiii Chapter 1: Introduction Pages 1-24 Chapter 2: Differentiation and Integration to Integer Order Pages 25-44 Chapter 3: Fractional Derivatives and Integrals: Definitions and Equivalences Pages 45-60 Chapter 4: Differintegration of Simple Functions Pages 61-68 Chapter 5: General Properties Pages 69-91 Chapter 6: Differintegration of More Complex Functions Pages 93-113 Chapter 7: Semiderivatives and Semiintegrals Pages 115-131 Chapter 8: Techniques in the Fractional Calculus Pages 133-160 Chapter 9: Representation of Transcendental Functions Pages 161-180 Chapter 10: Applications in the Classical Calculus Pages 181-195 Chapter 11: Applications to Diffusion Problems Pages 197-218 References Pages 219-223 Index Pages 225-234

not Only Does This Text Explain The Theory Underlying The Properties Of The Generalized Operator, But It Also Illustrates The Wide Variety Of Fields To Which These Ideas May Be Applied. Topics Include Integer Order, Simple And Complex Functions, Semiderivatives And Semiintegrals, And Transcendental Functions. 1974 Edition.

Studies theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. This title considers many computing techniques, such as methods of operator approximation with any given accuracy; methods of covariance matrix estimation; and, methods for low-rank matrix approximations.