Pseudodifferential Equations Research Articles

An inverse problem of determining orders of systems of fractional pseudo-differential equations

As it is known various dynamical processes can be modeled through systems of time-fractional order pseudo-differential equations. In the modeling process one frequently faces with the problem of determination of adequate orders of time-fractional derivatives in the sense of Riemann–Liouville or Caputo. This problem is qualified as an inverse problem. The correct (vector) order can be found utilizing the available data. In this paper we offer an new method of solution of this inverse problem for linear systems of fractional order pseudo-differential equations. We prove that the Fourier transform of the vector-solution $$\widehat{U}(t, \xi )$$ evaluated at a fixed time instance, which becomes possible due to the available data, recovers uniquely the unknown vector-order of a system of governing pseudo-differential equations.

Rational interpolation of the one-way Helmholtz propagator

This study is devoted to the higher-order finite-difference numerical methods for solving the pseudo-differential parabolic equation of diffraction theory. The relationship between the pseudo-differential propagation operator, variations of the refractive index, and the maximum propagation angle is established. It is proposed to use a rational approximation of the exponential propagation on an interval instead of the Padé approximation in a vicinity of a point. It is shown that using the approximation on an interval is more natural for this problem and allows using a more sparse computational grid than when using the local Padé approximation. Three variants of rational approximation on interval was considered: Chebyshev–Padé, rational interpolation and AAA method. The advantages and disadvantages of each approach are analyzed in the context of the problem under consideration. The proposed method differs from the existing ones only in the coefficients of the numerical scheme and does not require any significant changes in the implementations of the existing numerical schemas. Stability and error analysis of the proposed approach was carried out. The application of the proposed approach to the tropospheric radio-wave propagation and underwater acoustics is provided. Numerical examples and comparison with alternative methods quantitatively demonstrate the advantages of the proposed approach. Python 3 implementation of the proposed method is freely available. This work is an extended version of the ICCS-2021 conference paper (Lytaev, 2021).

THE NONLOCAL PROBLEM FOR FRACTAL DIFFUSION EQUATION

Over the past few decades, the theory of pseudodifferential operators (PDO) and equations with such operators (PDE) has been intensively developed. The authors of a new direction in the theory of PDE, which they called parabolic PDE with non-smooth homogeneous symbols (PPDE), are Yaroslav Drin and Samuil Eidelman. In the early 1970s, they constructed an example of the Cauchy problem for a modified heat equation containing, instead of the Laplace operator, PDO, which is its square root. Such a PDO has a homogeneous symbol |σ|, which is not smooth at the origin. The fundamental solution of the Cauchy problem (FSCP) for such an equation is an exact power function. For the heat equation, FSCP is an exact exponential function. The Laplace operator can be interpreted as a PDO with a smooth homogeneous symbol |σ|^2, σ ∈ Rn. A generalization of the heat equation is PPDE containing PDO with homogeneous non-smooth symbols. They have an important application in the theory of random processes, in particular, in the construction of discontinuous Markov processes with generators of integro-differential operators, which are related to PDO; in the modern theory of fractals, which has recently been rapidly developing. If the PDO symbol does not depend on spatial coordinates, then the Cauchy problem for PPDE is correctly solvable in the space of distribution-type generalized functions. In this case, the solution is written as a convolution of the FSCP with an initial generalized function. These results belong to a number of domestic and foreign mathematicians, in particular S. Eidelman and Y. Drin (who were the first to define PPDO with non-smooth symbols and began the study of the Cauchy problem for the corresponding PPDE), M. Fedoruk, A. Kochubey, V. Gorodetsky, V . Litovchenko and others. For certain new classes of PPDE, the correct solvability of the Cauchy problem in the space of Hölder functions has been proved, classical FSCP have been constructed, and exact estimates of their power-law derivatives have been obtained [1–4]. Of fundamental importance is the interpretation of PDO proposed by A. Kochubey in terms of hypersingular integrals (HSI). At the same time, the HSI symbol is constructed from the known PDO symbol and vice versa [6]. The theory of HSI, which significantly extend the class of PDO, was developed by S. Samko [7]. We extends this concept to matrix HSI [5]. Generalizations of the Cauchy problem are non-local multipoint problems with respect to the time variable and the problem with argument deviation. Here we prove the solvability of a nonlocal problem using the method of steps. We consider an evolutionary nonlinear equation with a regularized fractal fractional derivative α ∈ (0, 1] with respect to the time variable and a general elliptic operator with variable coefficients with respect to the second-order spatial variable. Such equations describe fractal properties in real processes characterized by turbulence, in hydrology, ecology, geophysics, environment pollution, economics and finance.

Conjugation problem for elliptic pseudodifferential equations on the plane

Conjugation problem for elliptic pseudodifferential equations on the plane

Equations Related to Stochastic Processes: Semigroup Approach and Fourier Transform

The work is devoted to integro-differential equations related to stochastic processes. We study the relationship between differential equations with random perturbations - stochastic differential equations (SDEs) - and deterministic equations for the probabilistic characteristics of processes determined by random perturbations. The resulting deterministic pseudodifferential equations are investigated by semigroup methods and Fourier transform methods.

Abstract McKean–Vlasov and Hamilton–Jacobi–Bellman Equations, Their Fractional Versions and Related Forward–Backward Systems on Riemannian Manifolds

We introduce a class of abstract nonlinear fractional pseudo-differential equations in Banach spaces that includes both the Mc-Kean-Vlasov-type equations describing nonlinear Markov processes and the Hamilton-Jacobi-Bellman(HJB)-Isaacs equation of stochastic control and games thus allowing for a unified analysis of these equations. This leads to an effective theory of coupled forward-backward systems (forward McKean-Vlasov evolution and backward HJB-Isaacs evolution) that are central to the modern theory of mean-field games.

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.

On Discrete Solutions for Elliptic Pseudo-Differential Equations

We consider discrete analogue for simplest boundary value problem for elliptic pseudo-differential equation in a half-space with Dirichlet boundary condition in Sobolev–Slobodetskii spaces. Based on the theory of discrete boundary value problems for elliptic pseudo-differential equations we give a comparison between discrete and continuous solutions for certain model boundary value problem.

On Solvability Conditions for a Certain Conjugation Problem

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

О дискретных эллиптических псевдодифференциальных уравнениях

О дискретных эллиптических псевдодифференциальных уравнениях

Traveling capillary waves on the boundary of a fluid disc

AbstractThe equation for a traveling wave on the boundary of a two‐dimensional droplet of an ideal fluid is derived by using the conformal variables technique for free surface waves. The free surface is subject only to the force of surface tension and the fluid flow is assumed to be potential. We use the canonical Hamiltonian variables discovered and map the lower complex plane to the interior of a fluid droplet conformally. The equations in this form have been originally discovered for infinitely deep water and later adapted to a bounded fluid domain.The new class of solutions satisfies a pseudodifferential equation similar to the Babenko equation for the Stokes wave. We illustrate with numerical solutions of the time‐dependent equations and observe the linear limit of traveling waves in this geometry.

Operators and Equations: Discrete and Continuous

We consider discrete pseudo-differential equations with elliptic symbols and the corresponding discrete boundary-value problems in special canonical domains of multidimensional spaces. The solvability of such equations and boundary-value problems in discrete analogs of Sobolev–Slobodetsky spaces is examined.

Evolutionary Pseudodifferential Equations in the Spaces of Generalized Periodic Functions

Evolutionary Pseudodifferential Equations in the Spaces of Generalized Periodic Functions

Enhanced dynamic homogenization of hexagonally packed granular materials with elastic interfaces

It is well known that the classical energetically consistent micropolar model has limits in simulating the frequency band structure of packed granular materials (see Merkel et al., 2011). It is here shown that if a standard continualization of the difference equation of motion of the discrete model is carried out, an equivalent micropolar continuum is obtained which is able to accurately simulate the optical branches of the discrete model. Nevertheless, this homogenized continuum presents non-positive defined elastic potential energy, a deficiency that limits the reliability of the model and implies instability phenomena (destabilizing effects) in the acoustic branches. This drawback is circumvented here through an high-frequency dynamic homogenization scheme which is based on an enhanced continualization of the discrete governing equations into pseudo-differential equations. Through a formal Taylor expansion of the pseudo-differential operators a higher order differential equation corresponding to the governing equation of a non-local continuum thermodynamically consistent are obtained. The resulting approach allows obtaining an equivalent micropolar continuum characterized by inertial non-locality. Moreover, higher order continua with non-local constitutive and inertial terms may be derived. The proposed continuum models are proved to be able to accurately describe both the static and dynamic behavior of the discrete granular model. Finally, the convergence to the response of the discrete system is shown when increasing the order of the higher order continuum.

On Some Multidimensional Limit Boundary Value Problems

We study solvability for a model elliptic pseudo-differential equation with an additional integral condition in Sobolev–Slobodetskii spaces in certain conical domains in Euclidean space. . Using the wave factorization for an elliptic symbol we construct a solution for this boundary value problem and study the behavior of the solution when parameters of cones tend to their extremal values. It was shown that for such a solvability we need certain additional condition besides presented integral condition.

Estimates of Certain Exit Probabilities for p-Adic Brownian Bridges

For each prime p, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated p-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of p-adic valued paths that is concentrated on the paths originating at the origin. We calculate the first exit probabilities of paths from balls and estimate these probabilities for the Brownian bridges.

A Priori Estimation of Solutions of a Boundary Problem for a Pseudodifferential Equation with Degeneration

A Priori Estimation of Solutions of a Boundary Problem for a Pseudodifferential Equation with Degeneration

Optimal operator preconditioning for pseudodifferential boundary problems

We propose an operator preconditioner for general elliptic pseudodifferential equations in a domain $\Omega$, where $\Omega$ is either in $\mathbb{R}^n$ or in a Riemannian manifold. For linear systems of equations arising from low-order Galerkin discretizations, we obtain condition numbers that are independent of the mesh size and of the choice of bases for test and trial functions. The basic ingredient is a classical formula by Boggio for the fractional Laplacian, which is extended analytically. In the special case of the weakly and hypersingular operators on a line segment or a screen, our approach gives a unified, independent proof for a series of recent results by Hiptmair, Jerez-Hanckes, N\'{e}d\'{e}lec and Urz\'{u}a-Torres. We also study the increasing relevance of the regularity assumptions on the mesh with the order of the operator. Numerical examples validate our theoretical findings and illustrate the performance of the proposed preconditioner on quasi-uniform, graded and adaptively generated meshes.

Properties of single layer potentials for a pseudo- differential equation related to a linear transformation of a rotationally invariant stable stochastic process

This article is aimed at determining existence conditions of single layer potentials for pseudo-differential equations related to some linear transformations of a rotationally invariant stable stochastic process in a multidimensional Euclidean space and investigating their properties as well. The carrier surface of the potential is smooth enough. In this article, we consider two main cases: the first, when this surface is bounded and closed; the second, when it is unbounded, but could be presented by an explicit equation in some coordinate system. The density of this potential is a continuous function. It is bounded with respect to the spatial variable and, probably, has an integrable singularity with respect to the time variable at zero. Classic properties of this potential, including a jump theorem of the action result of some operator (an analog of the co-normal differential) at its surface points, considered.
 A rotationally invariant $\alpha$-stable stochastic process in $\mathbb{R}^d$ is a L\'{e}vy process with the characte\-ristic function of its value in the moment of time $t>0$ defined by the expression $\exp\{-tc|\xi|^\alpha\}$, $\xi\in\mathbb{R}^d$, where $\alpha\in(0,2]$, $c>0$ are some constants. If $\alpha=2$ and $c=1/2$, we get Brownian motion and classic theory of potential. There are many different results in this case. The situation of $\alpha\in(1,2)$ is considered in this paper. We study constant and invertible linear transformations of the rotationally invariant $\alpha$-stable stochastic process. The related pseudo-differential equation is the parabolic equation of the order $\alpha$ of the ``heat'' type in which the operator with respect to the spatial variable is the process generator. The single layer potential is constructed in the same way as the single layer potential for the heat equation in the classical theory of potentials. That is, we use the fundamental solution of the equation, which is the transition probability density of the related process. In our theory, the role of the gradient operator is performed by some vector pseudo-differential operator of the order $\alpha-1$. We have already studied the following main properties of the single layer potentials: the single layer potential is a solution of the relating equation outside of the carrier surface and the jump theorem is held. These properties can be useful to solving initial boundary value problems for the considered equations.

Pseudodifferential Equation of Fluctuations of Nonstationary Gravitational Fields

Developing Holtzmark’s idea, the distribution of nonstationary fluctuations of local interaction of moving objects of the system with gravitational influence, which is characterized by the Riesz potential, is constructed. A pseudodifferential equation with the Riesz fractional differentiation operator is found, which corresponds to this process. The general nature of symmetric stable random Lévy processes is determined.