Euler solutions of pseudodifferential equations

We consider a homogeneous pseudodifferential equation on a cylinderC=×X over a smooth compact closed manifoldX whose symbol extends to a meromorphic function on the complex plane with values in the algebra of pseudodifferential operators overX. When assuming the symbol to be independent on the variablet , we show an explicit formula for solutions of the equation. Namely, to each non-bijectivity point of the symbol in the complex plane there corresponds a finite-dimensional space of solutions, every solution being the residue of a meromorphic form manufactured from the inverse symbol. In particular, for differential equations we recover Euler's theorem on the exponential solutions. Our setting is model for the analysis on manifolds with conical points sinceC can be thought of as a stretched manifold with conical points att=– andt=. When compared with the general theory, our approach is constructive while highlighting all the features of this latter.     

First-order hyperbolic pseudodifferential equations with generalized symbols

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for the case of differential operators [J. Math. Anal. Appl. 160 (1991) 93-106, J. Math. Anal. Appl. 142 (1989) 452-467].     

Corrected collocation methods for periodic pseudodifferential equations

Summary. We propose an approximation method for periodic pseudodifferential equations, which yields higher convergence rates in Sobolev spaces with negative order than the collocation method. The main idea consists in correcting the usual collocation solution in a certain way by the solution of a small Galerkin system for the same equation. Both trigonometric and spline approximation methods are considered. In most of the cases our convergence result even improves that of the qualocation method. Received January 3, 1994 / Revised version received August 17, 1994     

A spline collocation method for parabolic pseudodifferential equations

The purpose of this paper is to examine a boundary element collocation method for some parabolic pseudodifferential equations. The basic model problem for our investigation is the two-dimensional heat conduction problem with vanishing initial condition and a given Neumann or Dirichlet type boundary condition. Certain choices of the representation formula for the heat potential yield boundary integral equations of the first kind, namely the single layer and the hypersingular heat operator equations. Both of these operators, in particular, are covered by the class of parabolic pseudodifferential operators under consideration. Moreover, the spatial domain is allowed to have a general smooth boundary curve. As trial functions the tensor products of the smoothest spline functions of odd degree (space) and continuous piecewise linear splines (time) are used. Stability and convergence of the method is proved in some appropriate anisotropic Sobolev spaces.     

Spline approximation methods for multidimensional periodic pseudodifferential equations

We investigate several numerical methods for solving the pseudodifferential equationAu=f on the n-dimensional torusT ⁿ . We examine collocation methods as well as Galerkin-Petrov methods using various periodical spline functions. The considered spline spaces are subordinated to a uniform rectangular or triangular grid. For given approximation method and invertible pseudodifferential operatorA we compute a numerical symbol ^C , resp. ^G , depending onA and on the approximation method. It turns out that the stability of the numerical method is equivalent to the ellipticity of the corresponding numerical symbol. The case of variable symbols is tackled by a local principle. Optimal error estimates are established.The second author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant namber Ko 634/32-1.     

Multidimensional system of Diophantine equations

An asymptotics for the number of solutions to a system of three Diophantine equations of additive type in six variables is found. Each additive summand of these equations is a simplest form whose degree in each variable does not exceed 1.     

Some boundary value problems for pseudodifferential equations with degeneration

Boundary value problems for a new class of degenerate pseudodifferential equations containing a variable-symbol degenerate pseudodifferential operator based on a special integral transform and the first derivative with respect to one of the variables are studied. Existence theorems for these problems are proved. A priori estimates for their solutions are obtained in special weighted spaces similar to Sobolev ones.     

Nonlocal time-multipoint problem for a class of evolution pseudodifferential equations

We prove the well-posed solvability of a nonlocal time-multipoint problem for evolution equations with pseudodifferential operators with analytic symbols and initial condition in the space of distributions of the type W′.     

Fast solvers of integral and pseudodifferential equations on closed curves

On the basis of a fully discrete trigonometric Galerkin method and two grid iterations we propose solvers for integral and pseudodifferential equations on closed curves which solve the problem with an optimal convergence order , (Sobolev norms of periodic functions) in arithmetical operations.

     

Strongly elliptic pseudodifferential equations on the sphere with radial basis functions

Spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the Galerkin and collocation methods. A salient feature of the paper is a unified theory for error analysis of both approximation methods.     

Preconditioners for pseudodifferential equations on the sphere with radial basis functions

In a previous paper a preconditioning strategy based on overlapping domain decomposition was applied to the Galerkin approximation of elliptic partial differential equations on the sphere. In this paper the methods are extended to more general pseudodifferential equations on the sphere, using as before spherical radial basis functions for the approximation space, and again preconditioning the ill-conditioned linear systems of the Galerkin approximation by the additive Schwarz method. Numerical results are presented for the case of hypersingular and weakly singular integral operators on the sphere \mathbbS²{\mathbb{S}^2} .     

Perturbation of homogeneous parabolic pseudodifferential equations by locally unbounded vector fields

We construct solutions of homogeneous pseudodifferential equations of parabolic type perturbed by locally unbounded vector fields. We investigate some properties of these solutions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 11, pp. 1561–1566, November, 1997.     

Wavelet approximation methods for pseudodifferential equations: I Stability and convergence

The third author has been supported by a grant of Deutsche Forschungsgemeinschaft undergrant number Ko 634/32-1