Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

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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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.

• Complex Fourier Transform
• Fokker-Planck-Kolmogorov Equations
• Fractional Differential Equations
• Levy Processes
• Pseudo-Differential Equations
• partial differential equations

“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)

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

  Sabir Umarov