Numerical pseudodifferential operator and Fourier regularization

The concept of numerical pseudodifferential operator, which is an extension of numerical differentiation, is suggested. Numerical pseudodifferential operator just is calculating the value of the pseudodifferential operator with unbounded symbol. Many ill-posed problems can lead to numerical pseudodifferential operators. Fourier regularization is a very simple and effective method for recovering the stability of numerical pseudodifferential operators. A systematically theoretical analysis and some concrete examples are provided.     

Chaos of the Differentiation Operator on Weighted Banach Spaces of Entire Functions

Motivated by recent work on the rate of growth of frequently hypercyclic entire functions due to Blasco, Grosse-Erdmann and Bonilla, we investigate conditions to ensure that the differentiation operator is chaotic or frequently hypercyclic on generalized weighted Bergman spaces of entire functions studied by Lusky, whenever the differentiation operator is continuous. As a consequence we partially complete the knowledge of possible rates of growth of frequently hypercyclic entire functions for the differentiation operator.     

Problems for equations with special parabolic operator of fractional differentiation

We establish the well-posedness of the Cauchy problem and the two-point boundary-value problem for an equation with an operator of fractional differentiation that corresponds to the singular parabolic Beltrami – Laplace operator on a surface of the Dini class.     

Differentiation of analytic functions of sectorial operators in noncommuting directions

We find expressions and estimates for integer powers of the differentiation operator in noncommuting directions for analytic functions of sectorial operators on a complex Banach space. We study the exponential of such a differentiation operator and give an application to the theory of perturbations of evolution equations.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 70–74.     

On the extension of a generalized differentiation operator

A generalized differentiation operator introduced by A. F. Leont'ev is considered in this note. The concept of an L_D-operator is introduced, and necessary and sufficient conditions are determined for the extension of a generalized differentiation operator to some L_D-operator.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 425–436, October, 1969.The present work was completed under the direction of A. F. Leont'ev to whom the author expresses much gratitude.     

On the regularization of an equation of the first kind with a multiple integration operator

The zero-order Tikhonov regularization method as applied to an equation of the first kind with a multiple differentiation operator is considered for the case when the solution belongs to a class from the domain of the adjoint operator. An estimate of the error of the approximate solution in the uniform metric is obtained, which is sharp with respect to the order, and the order is established. It is proved that the proposed method is optimal with respect to the order. Unimprovable estimates of the order of the modulus of continuity of the inverse operator are obtained.     

Algebraic operator method for the construction of solitary solutions to nonlinear differential equations

Solutions of the KdV equation are derived by the algebraic operator method based on generalized operators of differentiation. The algebraic operator method based on the generalized operator of differentiation is exploited for the derivation of analytic solutions to the KdV equation. The structure of solitary solutions and explicit conditions of existence of these solutions in the subspace of initial conditions are derived. It is shown that special solitary solutions exist only on a line in the parameter plane of initial and boundary conditions. This new theoretical result may lead to important findings in a variety of practical applications.     

Some remarks on tensor differentiation

Summary The Jacobian of classical tensor analysis is generalized to a transformation operator containing second derivatives. This operator and the Jacobian may be used to formulate a second order tensor calculus. In this calculus, a simple contraction scheme includes as special cases Lie differentiation, the Lie bracket, covariant differentiation, and differentiation with respect to the tensor connections of Bompiani. To Enrico Bompiani on his scientific Jubilee     

The Spectrum of a Parametrized Partial Differential Operator Occurring in Hydrodynamics

A partial differential operator associated with natural oscillationsof an incompressible fluid in the neighbourhood of an ellipticalflow is considered. The differentiation is only taken with respectto the angular variable, and thus the operator becomes a familyof ordinary differential operators parametrized by the radialvariable. It is shown that the spectra of these ordinary differentialoperators completely determine the spectrum of the given operatorwhich turns out to have a kind of skeleton structure.     

Boundary value problem for a first-order partial differential equation with a fractional discretely distributed differentiation operator

We solve a boundary value problem for a first-order partial differential equation in a rectangular domain with a fractional discretely distributed differentiation operator. The fractional differentiation is given by Dzhrbashyan–Nersesyan operators. We construct a representation of the solution and prove existence and uniqueness theorems. The results remain valid for the corresponding equations with Riemann–Liouville and Caputo derivatives. In terms of parameters defining the fractional differential operator, we derive necessary and sufficient conditions for the solvability of the problem.     

On some properties of a class of degenerate pseudodifferential operators

A new class of pseudodifferential operators with degeneration is considered. The operators are constructed using a special integral transform mapping a weighted differentiation operator to a multiplication operator. The composition and boundedness properties of such operators in special weighted spaces are examined. Theorems on commutation of such operators with differentiation operators are obtained. The behavior of these operators as t → 0and t → +∞ is investigated. The properties of adjoint operators are studied, and an analogue of Gårding’s inequality is proved.     

Approximation of differentiation operator in the space L 2 on semiaxis

We establish an upper bound for the error of the best approximation of the first order differentiation operator by linear bounded operators on the set of twice differentiable functions in the space L ₂ on the half-line. This upper bound is close to a known lower bound and improves the previously known upper bound due to E. E. Berdysheva. We use a specific operator that is introduced and studied in the paper.     

ON SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENOUS POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER

In the paper we study questions about solvability of some boundary value problems for a non-homogenous poly-harmonic equation.As a boundary operator we consider differentiation operator of fractional order in Miller-Ross sense.The considered problem is a generalization of well-known Dirichlet and Neumann problems.     

Nonlocal time-multipoint problem for a certain class of evolutionary pseudodifferential equations with variable symbols: I

It has been proved that in generalized spaces of the type S, a pseudodifferential operator constructed based on a variable symbol can be treated as the operator of infiniteorder differentiation if the operator symbol satisfies certain conditions. The properties of the fundamental solution to a nonlocal time-multipoint problem for the evolutionary equation with this operator have been studied.     

A new approach to numerical differentiation

In this paper we consider the numerical differentiation of functions specified by noisy data. A new approach, which is based on an integral equation of the first kind with a suitable compact operator, is presented and discussed. Since the singular system of the compact operator can be obtained easily, TSVD is chosen as the needed regularization technique and we show that the method calls for a discrete sine transform, so the method can be implemented easily and fast. Numerical examples are also given to show the efficiency of the method.     

Approximation of the differentiation operator by linear bounded operators on the class of twice differentiable functions in the space <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(0,∞)

We give an upper estimate for the value of the best approximation of the (firstorder) differentiation operator by linear bounded operators on the class of twice differentiable functions in the space L ₂(0,∞). This upper estimate is close to a known lower estimate and improves previously known upper estimates. To prove the upper estimate, we consider a specific family of operators; in this family, we choose an operator that provides the least estimate for the value of the best approximation.     

Best approximation of the differentiation operator on the class of functions analytic in a strip

On the class of functions analytic in a strip, we study a number of related extremal problems for the differentiation operator: the best approximation of the operator, computation of its modulus of continuity, and optimal recovery of the operator from boundary values of a function given with error on a straight line.     

Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm-Liouville problems

In this paper, using spectral differentiation matrix and an elimination treatment of boundary conditions, Sturm-Liouville problems (SLPs) are discretized into standard matrix eigenvalue problems. The eigenvalues of the original Sturm-Liouville operator are approximated by the eigenvalues of the corresponding Chebyshev differentiation matrix (CDM). This greatly improves the efficiency of the classical Chebyshev collocation method for SLPs, where a determinant or a generalized matrix eigenvalue problem has to be computed. Furthermore, the state-of-the-art spectral method, which incorporates the barycentric rational interpolation with a conformal map, is used to solve regular SLPs. A much more accurate mapped barycentric Chebyshev differentiation matrix (MBCDM) is obtained to approximate the Sturm-Liouville operator. Compared with many other existing methods, the MBCDM method achieves higher accuracy and efficiency, i.e., it produces fewer outliers. When a large number of eigenvalues need to be computed, the MBCDM method is very competitive. Hundreds of eigenvalues up to more than ten digits accuracy can be computed in several seconds on a personal computer.     

A Note on Random Perturbations of a Multiple Eigenvalue of a Hermitian Operator

The problem of a random Hermitian perturbation of a multiple isolated eigenvalue of a Hermitian operator is considered. It is shown that the combined multiplicities of the perturbed eigenvalues converge in probability to the multiplicity of the eigenvalue of the target operator. Also the asymptotic distribution of a certain average of these eigenvalues, centered at the target, is obtained. As a tool differentiation of analytic functions of operators is employed in conjunction with an ensuing “delta-method”. The result is of a probabilistic rather than statistical nature.     

On a generalization of the Neumann problem for the Laplace equation

We investigate solvability of a fractional analogue of the Neumann problem for the Laplace equation. As a boundary operator we consider operators of fractional differentiation in the Hadamard sense. The problem is solved by reduction to an integral Fredholm equation. A theorem on existence and uniqueness of the problem solution is proved.