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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.



Chapter 1. Function spaces and distributions

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 and the corresponding space of distributions (generalized functions), which play an important role in the theory of pseudo-differential operators with singular symbols introduced in Chapter 2 By singular symbols we mean, if not otherwise assumed, symbols singular in dual variables.
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Chapter 2. Pseudo-differential operators with singular symbols (ΨDOSS)

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 linear differential operator mapping a function space X into another function space F.
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Chapter 3. Fractional calculus and fractional order operators

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 neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments - distributed and variable fractional order differential operators.
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Chapter 4. Boundary value problems for pseudo-differential equations with singular symbols

Let \(\varOmega \subset \mathbb{R}^{n}\) be a bounded domain with a smooth boundary or \(\varOmega = \mathbb{R}^{n}.\) This chapter discusses well-posedness problems of general boundary value problems for pseudo-differential and differential-operator equations of the form
$$\displaystyle\begin{array}{rcl} L[u]& \equiv & \frac{\partial ^{m}u} {\partial t^{m}} +\sum _{ k=0}^{m-1}A_{ k}(t)\frac{\partial ^{k}u} {\partial t^{k}} = f(t,x),\quad t \in (T_{1},T_{2}),\ x \in \varOmega,{}\end{array}$$
$$\displaystyle\begin{array}{rcl} B_{k}[u]& \equiv & \sum _{j=0}^{m-1}b_{ kj}\frac{\partial ^{j}u(t_{kj},x)} {\partial t^{j}} =\varphi _{k}(x),\quad x \in \varOmega,\,k = 0,\mathop{\ldots },m - 1,{}\end{array}$$
where f(t, x) is defined on \((T_{1},T_{2})\times \varOmega,\) \(-\infty <T_{1} <T_{2} \leq \infty,\) and \(\varphi _{k}(x),\,x \in \varOmega,\,k = 0,\ldots,m - 1,\) are given functions; A k (t) and b kj , \(k = 0,\ldots,m - 1,\,j = 0,\ldots,m - 1,\) are operators acting on some spaces (specified below) of functions defined on Ω; and \(t_{jk} \in [T_{1},T_{2}],\,j,k = 0,\ldots,m - 1.\) For example, when \(\varOmega = \mathbb{R}^{n},\) the latter operators may act as ΨDOSS defined on the space of distributions \(\varPsi _{-G,p^{'}}^{'}(\mathbb{R}^{n})\) with an appropriate \(G \subset R^{n}\).
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Chapter 5. Initial and boundary value problems for fractional order differential equations

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, \(m - 1 <\beta <m,\) (\(m \in \mathbb{N}\))
$$\displaystyle\begin{array}{rcl} D_{{\ast}}^{\beta }u(t,x) = A(D)u(t,x) + h(t,x),\quad t> 0,\ x \in \mathbb{R}^{n},& &{}\end{array}$$
$$\displaystyle\begin{array}{rcl} \frac{\partial ^{k}u(0,x)} {\partial t^{k}} =\varphi _{k}(x),\quad x \in \mathbb{R}^{n},\ k = 0,\ldots,m - 1,& &{}\end{array}$$
where h(t, x) and \(\varphi _{k},\ k = 0,\ldots,m - 1,\) are given functions in certain spaces described later, \(D = (D_{1},\ldots,D_{n})\), \(D_{j} = -i \frac{\partial } {\partial x_{j}},\ j = 1,\ldots,n\), A(D) is a ΨDOSS with a symbol A(ξ) ∈ XS p (G) defined in an open domain \(G \subset \mathbb{R}^{n}\), and \(D_{{\ast}}^{\beta }\) is the fractional derivative of order β > 0 in the sense of Caputo-Djrbashian (see Section 3.5)
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Chapter 6. Distributed and variable order differential-operator equations

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 chapter is the mathematical treatment of boundary value problems for general distributed and variable order fractional differential-operator equations. We will study the existence and uniqueness of a solution to initial and multi-point value problems in different function spaces.
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Chapter 7. Fractional order Fokker-Planck-Kolmogorov equations and associated stochastic processes

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 theory and stochastic processes, such as a random variable, its density function, mathematical expectation, characteristic function, etc. Since we are interested only in applications of fractional order ΨDOSS, we do not discuss in detail facts on random processes that are already established and presented in other sources. For details of such notations and related facts we refer the reader to the book by Applebaum [App09] (or [IW81, Sat99]). We only mention some basic notations directly related to our discussions on fractional Fokker-Planck-Kolmogorov equations.
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Chapter 8. Random walk approximants of mixed and time-changed Lévy processes

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, e.g., [GK54, Don52, Bil99, Taq75, GM98-1, GM01, MS01]) devoted to approximation of various basic stochastic processes like Brownian motion, fractional Brownian motion, Lévy processes, and their time-changed counterparts. In the context of approximation, the question in what sense a random walk approximates (or converges to) an associated stochastic process becomes important. We will be interested only in the convergence in the sense of finite-dimensional distributions, which is equivalent to the locally uniform convergence of corresponding characteristic functions (see, e.g., [Bil99]).
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Chapter 9. Complex ΨDOSS and systems of complex differential equations

In Chapters 4–7 we discussed pseudo-differential equations of integer and fractional orders with ψDOSS depending on real variables \(t \in \mathbb{R}\) and \(x \in \mathbb{R}^{n}\). In this section we will discuss differential and pseudo-differential equations depending on complex variables \(t =\tau +i\sigma \in \mathbb{C}\) and \(z = x + iy \in \mathbb{C}^{n}.\) Consider two simple examples with the one-dimensional “spatial” variable:
“complex wave” equation, and
“complex heat” equation.
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