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.
Author(s): Sabir Umarov (auth.)
Series: Developments in Mathematics 41
Edition: 1
Publisher: Springer International Publishing
Year: 2015
Language: English
Pages: XVI, 434
Tags: Partial Differential Equations; Probability Theory and Stochastic Processes; Fourier Analysis; Statistical Physics, Dynamical Systems and Complexity
Front Matter....Pages i-xvi
Function spaces and distributions....Pages 1-67
Pseudo-differential operators with singular symbols (ΨDOSS)....Pages 69-120
Fractional calculus and fractional order operators....Pages 121-168
Boundary value problems for pseudo-differential equations with singular symbols....Pages 169-206
Initial and boundary value problems for fractional order differential equations....Pages 207-247
Distributed and variable order differential-operator equations....Pages 249-283
Fractional order Fokker-Planck-Kolmogorov equations and associated stochastic processes....Pages 285-344
Random walk approximants of mixed and time-changed Lévy processes....Pages 345-371
Complex ΨDOSS and systems of complex differential equations....Pages 373-414
Back Matter....Pages 415-434
