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Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Part of the Developments in Mathematics Book Series (volume 41)

DOI: 10.1007/978-3-319-20771-1

ISBN: 978-3-319-20770-4

Publisher: Springer International Publishing - Springer Journals

Jun 2, 2015

1 - 9 of 9 Chapters

LitStream

Chapter 1Function spaces and distributions

Umarov, Sabir

Pages: 1 –67

[This chapter is devoted to function and distribution spaces. We first recall definitions of some well-known classical function and distribution spaces, simultaneously introducing the terminology and notations used in this book. Then we introduce (see Section 1.10) a new class of test functions...
LitStream

Chapter 2Pseudo-differential operators with singular symbols (ΨDOSS)

Umarov, Sabir

Pages: 69 –120

[We begin Chapter 2 with simple examples of initial and boundary value problems, solution operators of which have singularities of one or another type in the dual variable. The presence of a singularity often causes a failure of well posedness of the problem in the sense of Hadamard. Let A be a...
LitStream

Chapter 3Fractional calculus and fractional order operators

Umarov, Sabir

Pages: 121 –168

[Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models...
LitStream

Chapter 4Boundary value problems for pseudo-differential equations with singular symbols

Umarov, Sabir

Pages: 169 –206

[Let Ω⊂ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varOmega \subset...
LitStream

Chapter 5Initial and boundary value problems for fractional order differential equations

Umarov, Sabir

Pages: 207 –247

[In this chapter we will discuss boundary value problems for fractional order differential and pseudo-differential equations. For methodological clarity we first consider in detail the Cauchy problem for pseudo-differential equations of time-fractional order β,...
LitStream

Chapter 6Distributed and variable order differential-operator equations

Umarov, Sabir

Pages: 249 –283

[In Section 5.6 we studied the existence of a solution to the multi-point value problem for a fractional order pseudo-differential equation with m fractional derivatives of the unknown function. This is an example of fractional distributed order differential equations. Our main purpose in this...
LitStream

Chapter 7Fractional order Fokker-Planck-Kolmogorov equations and associated stochastic processes

Umarov, Sabir

Pages: 285 –344

[This chapter discusses the connection between pseudo-differential and fractional order differential equations considered in Chapters 2–6 with some random (stochastic) processes defined by stochastic differential equations. We assume that the reader is familiar with basic notions of probability...
LitStream

Chapter 8Random walk approximants of mixed and time-changed Lévy processes

Umarov, Sabir

Pages: 345 –371

[Random walks are used to model various random processes in different fields. In this chapter we are only interested in random walks as approximating processes of some basic driving processes of stochastic differential equations discussed in the previous chapter. There is a vast literature (see,...
LitStream

Chapter 9Complex ΨDOSS and systems of complex differential equations

Umarov, Sabir

Pages: 373 –414

[In Chapters 4–7 we discussed pseudo-differential equations of integer and fractional orders with ψDOSS depending on real variables t∈ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...

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