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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

  • Book
  • © 2015

Overview

Authors:
  1. Sabir Umarov

    1. Department of Mathematics, University of New Haven, West Haven, USA

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  • A novel theory of pseudo-differential operators and equations with symbols singular in dual variables is presented systematically
  • Recent developments in the theory of fractional order differential equations and their various applications are studied
  • Fractional Fokker-Planck-Kolmogorov equations and their connection with the associated stochastic differential equations driven by a time-changed process are discussed in detail

Part of the book series: Developments in Mathematics (DEVM, volume 41)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Function spaces and distributions

    • Sabir Umarov
    Pages 1-67

  3. Pseudo-differential operators with singular symbols (ΨDOSS)

    • Sabir Umarov
    Pages 69-120

  4. Fractional calculus and fractional order operators

    • Sabir Umarov
    Pages 121-168

  5. Boundary value problems for pseudo-differential equations with singular symbols

    • Sabir Umarov
    Pages 169-206

  6. Initial and boundary value problems for fractional order differential equations

    • Sabir Umarov
    Pages 207-247

  7. Distributed and variable order differential-operator equations

    • Sabir Umarov
    Pages 249-283

  8. Fractional order Fokker-Planck-Kolmogorov equations and associated stochastic processes

    • Sabir Umarov
    Pages 285-344

  9. Random walk approximants of mixed and time-changed Lévy processes

    • Sabir Umarov
    Pages 345-371

  10. Complex ΨDOSS and systems of complex differential equations

    • Sabir Umarov
    Pages 373-414

  11. Back Matter

    Pages 415-434
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Keywords

About this book

The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of these theories to various initial and multi-point boundary value problems for pseudo-differential equations. Fractional Fokker-Planck-Kolmogorov equations associated with a large class of stochastic processes are presented. A complex version of the theory of pseudo-differential operators with meromorphic symbols based on the recently introduced complex Fourier transform is developed and applied for initial and boundary value problems for systems of complex differential and pseudo-differential equations.

Reviews

“The book systematically presents the theory of pseudo-differential operators with symbols singular in dual variables. … The book is interesting both for probabilists and for researchers in those areas of functional analysis where pseudo-differential operators arise. The interesting connections are emphasized by a multitude of examples. Additional notes at the end of each chapter provide nice historical insights and help to understand the role of the presented results within the big picture.” (Alexander Schnurr, zbMATH 1331.35005, 2016)

Authors and Affiliations

  • Department of Mathematics, University of New Haven, West Haven, USA

    Sabir Umarov

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