Pseudodifferential Equations Research Articles

Exact Asymptotic Formulas for the Heat Kernels of Space and Time-Fractional Equations

Abstract This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic formulas for the heat kernels of time-changed Brownian motions and Cauchy processes. As an application, we obtain exact asymptotic formulas for the fundamental solutions to the n-dimensional fractional heat equations in both time and space $$\begin{array}{} \displaystyle \frac{\partial^\beta}{\partial t^\beta}u(t,x) = -(-\Delta_x)^\gamma u(t,x), \quad \beta,\gamma\in(0,1). \end{array}$$

Kinetic evaluation of the hydroformylation of the post-metathesis product 7-tetradecene using a heterobimetallic rhodium-ferrocenyl Schiff base derived precatalyst

Reaction engineering kinetics for the hydroformylation of the post-metathesis product 7-tetradecene using a heterobimetallic rhodium-ferrocenyl Schiff base derived precatalyst was investigated with variation of reaction temperature (85–105 °C), precatalyst loading (0.25–0.52 mM), carbon monoxide partial pressures (20–40 bar) and hydrogen partial pressures (20–40 bar). The experimental product-time distributions for the parallel hydroformylation and isomerization reaction system are well described by four interdependent pseudo first-order differential mole balance equations. The effects of temperature in the Arrhenius equation, precatalyst concentration, carbon monoxide and hydrogen partial pressures have been incorporated into a phenomenological mechanism-based rate equation. The rate of hydroformylation is first order in alkene, carbon monoxide and hydrogen, with fractional dependence in precatalyst concentration. The activation energy for the hydroformylation reaction was calculated to be 62 kJ mol−1, which is comparable to that determined for the commercialized phosphorus-modified catalyst systems.

P-Adic Brownian Motion as a Limit of Discrete Time Random Walks

The p-adic diffusion equation is a pseudo differential equation that is formally analogous to the real diffusion equation. The fundamental solutions to pseudo differential equations that generalize the p-adic diffusion equation give rise to p-adic Brownian motions. We show that these stochastic processes are similar to real Brownian motion in that they arise as limits of discrete time random walks on grids. While similar to those in the real case, the random walks in the p-adic setting are necessarily non-local. The study of discrete time random walks that converge to Brownian motion provides intuition about Brownian motion that is important in applications and such intuition is now available in a non-Archimedean setting.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Adjusted Sparse Tensor Product Spectral Galerkin Method for Solving Pseudodifferential Equations on the Sphere with Random Input Data

An adjusted sparse tensor product spectral Galerkin approximation method based on spherical harmonics is introduced and analyzed for solving pseudodifferential equations on the sphere with random input data. These equations arise from geodesy where the sphere is taken as a model of the earth. Numerical solutions to the corresponding $k$-th order statistical moment equations are found in adjusted sparse tensor approximation spaces which are accordingly designed to the regularity of the data and the equation. Established convergence theorem shows that the adjusted sparse tensor Galerkin discretization is superior not only to the full tensor product but also to the standard sparse tensor counterpart when the statistical moments of the data are of mixed unequal regularity. Numerical experiments illustrate our theoretical results.

Babenko’s Equation for Periodic Gravity Waves on Water of Finite Depth: Derivation and Numerical Solution

The nonlinear two-dimensional problem describing periodic steady gravity waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko’s equation), which is investigated analytically and numerically. This equation has the same form as the equation for waves on infinitely deep water; the latter had been proposed by Babenko and studied in detail by Buffoni, Dancer and Toland. Instead of the $$2 \pi $$ -periodic Hilbert transform $${\mathcal {C}}$$ used in the equation for deep water, the equation obtained here contains a certain operator $${\mathcal {B}}_r$$ , which is the sum of $${\mathcal {C}}$$ and a compact operator depending on a parameter related to the depth of water. Numerical computations are based on an equivalent form of Babenko’s equation derived by virtue of the spectral decomposition of the operator $${\mathcal {B}}_r \mathrm {d}/ \mathrm {d}t$$ . Bifurcation curves and wave profiles of the extreme form are obtained numerically.

Nonlinear pseudo-differential equations defined by elliptic symbols on Lp(ℝn) and the fractional Laplacian

We develop an Lp(ℝn)-functional calculus appropriated for interpreting “non-classical symbols” of the form a(−Δ), and for proving existence of solutions to nonlinear pseudo-differential equations of the form [1 + a(−Δ)]s/2(u) = V (·, u). We use the theory of Fourier multipliers for constructing suitable domains sitting inside Lp(ℝn) on which the formal operator appearing in the above equation can be rigorously defined, and we prove existence of solutions belonging to these domains. We include applications of the theory to equations of physical interest involving the fractional Laplace operator such as (generalizations of) the (focusing) Allen–Cahn, Benjamin–Ono and nonlinear Schrodinger equations.

Acausal quantum theory for non-Archimedean scalar fields

We construct a family of quantum scalar fields over a [Formula: see text]-adic spacetime which satisfy [Formula: see text]-adic analogues of the Gårding–Wightman axioms. Most of the axioms can be formulated in the same way for both the Archimedean and non-Archimedean frameworks; however, the axioms depending on the ordering of the background field must be reformulated, reflecting the acausality of [Formula: see text]-adic spacetime. The [Formula: see text]-adic scalar fields satisfy certain [Formula: see text]-adic Klein–Gordon pseudo-differential equations. The second quantization of the solutions of these Klein–Gordon equations corresponds exactly to the scalar fields introduced here. The main conclusion is that there seems to be no obstruction to the existence of a mathematically rigorous quantum field theory (QFT) for free fields in the [Formula: see text]-adic framework, based on an acausal spacetime.

Operator Ordering and Solution of Pseudo-Evolutionary Equations

The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.

On some distributions associated to boundary value problems

We consider an elliptic pseudo-differential equation in a multidimensional cone in Sobolev–Slobodetskii space and describe its general solution for some cases. This description is based on some distributions supported on a conical surface. For certain conical surfaces we describe such distributions. It permits to use such distributions to refine a general solution.

Comparison of one-way and full-wave linear propagation models in inhomogeneous medium

One-way wave propagation models based on the parabolic approximation or its wide-angle extensions are often used for describing bounded acoustic beams. Such models are highly demanded when solving nonlinear problems that are computationally intensive and thus technically difficult to solve using full-wave approaches. When describing the propagation of a wave beam in a homogeneous medium, the one-way assumption is fulfilled exactly, and therefore, the inaccuracy is caused only by the limitations of the parabolic approximation. Such an error is significantly reduced within the wide-angle approach and completely disappears when using the exact propagator in the framework of the angular spectrum method. The situation is less obvious in the case of a heterogeneous environment, when a part of the wave energy is inevitably reflected becoming a counter-propagating wave and thus is not taken into account in the one-way approximation. The degree this phenomenon affects the accuracy of the one-way approach is still under discussion. In the current paper, a one-way propagator based on the pseudo-differential wide-angle equation is proposed. The propagator is tested for the homogeneous medium and for several configurations of media with regular and random inhomogeneities. The corresponding solutions are compared with those obtained using the k-Wave toolbox. Results of comparison show how the one-way propagator accuracy depends on the contrast and smoothness of the inhomogeneities. [Work supported by RSF No. 18-72-00196.]

A hybrid finite-difference/low-rank solution to anisotropy acoustic wave equations

P-wave extrapolation in anisotropic media suffers from SV-wave artifacts and computational dependency on the complexity of anisotropy. The anisotropic pseudodifferential wave equation cannot be solved using an efficient time-domain finite-difference (FD) scheme directly. The wavenumber domain allows us to handle pseudodifferential operators accurately; however, it requires either smoothly varying media or more computational resources. In the limit of elliptical anisotropy, the pseudodifferential operator reduces to a conventional operator. Therefore, we have developed a hybrid-domain solution that includes a space-domain FD solver for the elliptical anisotropic part of the anisotropic operator and a wavenumber-domain low-rank scheme to solve the pseudodifferential part. Thus, we split the original pseudodifferential operator into a second-order differentiable background and a pseudodifferential correction term. The background equation is solved using the efficient FD scheme, and the correction term is approximated by the low-rank approximation. As a result, the correction wavefield is independent of the velocity model, and, thus, it has a reduced rank compared with the full operator. The total computation cost of our method includes the cost of solving a spatial FD time-step update plus several fast Fourier transforms related to the rank. The accuracy of our method is of the order of the FD scheme. Applications to a simple homogeneous tilted transverse isotropic (TTI) medium and modified BP TTI models demonstrate the effectiveness of the approach.

Stokes waves with constant vorticity: I. Numerical computation

AbstractPeriodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.

Complete Systems of Eigenfunctions of the Vladimirov Operator in L2(Br) and L2(ℚp)

We construct new bases of real functions from L2(Br) and from L2(ℚp). These functions are eigenfunctions of the p-adic pseudo-differential Vladimirov operator, which is defined on a compact set Br ⊂ ℚp of the field of p-adic numbers ℚp or, respectively, on the entire field ℚp. A relation between the basis of functions from L2(ℚp) and the basis of p-adic wavelets from L2(ℚp) is found. As an application, we consider the solution of the Cauchy problem with the initial condition on a compact set for a pseudo-differential equation with a general pseudo-differential operator that is diagonal in the basis constructed.

Intermittency and stochastic pseudo-differential equation with spatially inhomogeneous white noise

In this paper, we study the intermittent property for the following nonlinear stochastic partial differential equation (SPDE in the sequel) in (1+1)-dimension $$\begin{aligned} \left( \frac{\partial }{\partial t}+q(x,D_x)\right) u(t,x)= g(u(t,x))\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x),\quad t>0 \quad \mathrm{and} \quad x\in {\mathbb {R}}, \end{aligned}$$ with $$q(x,D_x)$$ is a pseudo-differential operator which generates a stable-like process. The forcing noise denoted by $$\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x)$$ is a spatially inhomogeneous white noise. Under some mild assumptions on the catalytic measure of the inhomogeneous Brownian sheet $$w_\rho (t,x)$$ , we prove that the solution is weakly full intermittent based on the moment estimates of the solution.

Partial parabolicity of the boundary-value problem for pseudodifferential equations in a layer

A nonlocal boundary-value problem for evolutional pseudodifferential equations in an infinite layer is considered in this paper. The notion of the partially parabolic boundary-value problem is introduced when a solving function decreases exponentially only by the part of space variables. This concept generalizes the concept of a parabolic boundary value problem, which was previously studied by one of the authors of this paper (A. A. Makarov). Necessary and sufficient conditions for the pseudodifferential operator symbol are obtained in which partially parabolic boundary-value problems exist. It turned out that the real part of the symbol of a pseudodifferential operator should increase unboundedly powerfully in some of the spatial variables. In this case, a specific type of boundary conditions is indicated, which depend on a pseudodifferential equation and are also pseudodifferential operators. It is shown that for solutions of partially parabolic boundary-value problems, smoothness in some of the spatial variables increases. The disturbed (excitated) pseudodifferential equation with a symbol which depends on space and temporal variables is also investigated. It has been found for partially parabolic boundary-value problems what pseudodifferential operators are possible to be disturbed in the way that the input equation of this boundary-value problem would remain correct in Sobolev-Slobodetsky spaces. It is also shown that although the properties of increasing the smoothness of solutions in part of the variables for partially parabolic boundary value problems are similar to the property of solutions of partially hypoelliptic equations introduced by L. H\"{o}rmander, these examples show that the partial parabolic boundary value problem does not follow from partial hipoellipticity; and vice versa - an example of a partially parabolic boundary value problem for a differential equation that is not partially hypoelliptic is given.

Pseudo-differential equations and conical potentials: 2-dimensional case

We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given.

Asymptotics of Linear Surface Waves Generated by a Localized Source Moving Along the Bottom of the Basin. I. One-Dimensional Case

In the paper, water waves generated by a source moving along the bottom of the basin are considered. The complete system of equations for water waves is very complicated and does not have an exact solution at present. Assuming that the change of the bottom function is slow, the authors of the paper [S. Yu. Dobrokhotov and V. E. Nazaikinskii, RJMP 25 (1), 1–16 (2018)] showed that the linearized system of hydrodynamic equations is reduced to a pseudodifferential equation on the surface of the liquid, and the unknown function in this equation is the elevation of the free surface. Assume that the source of the waves is localized in the vicinity of a point moving along the bottom of the basin at a speed less than the speed of the long waves. Assume also that the time of motion of the source and the horizontal sizes of the source are sufficiently small, and, during the motion, the source changes its shape and also changes its velocity. We show that, under some conditions, the last assumption leads to the occurrence of waves on the surface of the liquid with a wavelength approximately equal to the product of the typical speed of long waves in the vicinity of the place where the source moves and the time of its motion, which exceeds the size of the source. In this way, one of the possible mechanisms for generating the so-called landslide tsunami waves can be described. In this paper, we consider the situation with a single horizontal spatial variable.

Non-existence of positive solutions for a higher order fractional equation

In this paper, we consider a nonlinear equation involving fractional Laplacian of higher order on the whole space. We establish the equivalence between the pseudo-differential equation and an integral equation by applying the maximum principle and the Liouville theorem. For positive solutions to the equation, we obtained non-existence by applying the method of moving planes.

On Some Discrete Potential Like Operators

Abstract We consider some discrete pseudo-differential equations in discrete Sobolev–Slobodetskii spaces. For a discrete half-space and certain values of an index of periodic factorization for an elliptic symbol we introduce additional potential-like unknowns and prove existence and uniqueness theorem in appropriate discrete Sobolev–Slobodetskii spaces.