Bessel Operator Research Articles

Heat equations associated with matrix singular differential operators and spectral theory

The main subject of this paper is the study of the heat functions and the Gauss kernel associated with the operator (Δα + q) where Δα is the Bessel operator with matricial coefficients and q is an n × n matrix valued function. For this purpose we use the properties of its eigenfunctions and some integral transforms.

The Pseudo-differential Operator hμ,a on Hankel Invariant Spaces

Using Hankel transform the symbol 'a' is defined and the pseudo-differential operator (p.d.o.) hμ,a associated with the Bessel operator d 2/dx 2 + (1 − 4μ 2)/4x 2 in terms of this symbol is defined. It is shown that the operator hμ,a is a continuous linear map of a Hankel invariant space into itself. A special pseudo-differential operator called the Hankel potential is defined and some of its properties are investigated.

Asymptotic Series of General Symbol and Pseudo-Differential Operators

An Asymptotic series of general symbol of pseudo-differential operators associated with Bessel operators {S}_\\mu are obtained by using the theory of Hankel convolution.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

On initial boundary value problem with Dirichlet integralconditions for a hyperbolic equation with the Besseloperator

We consider a mixed problem with Dirichlet and integral conditions for a second‐order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

Hankel convolution operators on entire functions and distributions

In this paper we study the Hankel convolution operators on the space of even and entire functions and on Schwartz distribution spaces. We characterize the Hankel convolution operators as those ones that commute with Hankel translations and with a Bessel operator. Also we prove that the Hankel convolution operators are hypercyclic and chaotic on the spaces under consideration.

ON THE CONJUGATE DARBOUX-PROTTER PROBLEMS FOR THE TWO DIMENSIONAL WAVE EQUATIONS IN THE SPECIAL CASE

In the article [2], the conjugate Darboux-Protter problem Dn is formulated for the two dimensional wave equation in the class of unbounded functions and the uniqueness of solutions has been established. In this paper, we shall show the existence of solutions for the hyperbolic equations with Bessel operators in another special case.

Asymptotics of Determinants of Bessel Operators

In this paper we determine the asymptotics of the determinant of Bessel operators for sufficiently smooth generating functions. These operators are similar to Wiener-Hopf operators with the Fourier transform replaced by the Hankel transform and thus the asymptotics of the determinanst are similar to the well-known Szeg\"o-Akhiezer-Kac formula for truncated Wiener-Hopf determinants. In order to compute the above, we also show that the Bessel operators differ from the Wiener-Hopf by a Hilbert-Schmidt operator.

Fuchsian bispectral operators

The aim of this paper is to classify the bispectral operators of any rank with regular singular points (the infinite point is the most important one). We characterise them in several ways. Probably the most important result is that they are all Darboux transformations of powers of generalised Bessel operators (in the terminology of [4]). For this reason they can be effectively parametrised by the points of a certain (infinite) family of algebraic manifolds as pointed out in [4].

Initial inverse problem in heat equation with Bessel operator

We investigate the inverse problem involving recovery of initial temperature from the information of final temperature profile in a disc. This inverse problem arises when experimental measurements are taken at any given time, and it is desired to calculate the initial profile. We consider the usual heat equation and the hyperbolic heat equation with Bessel operator. An integral representation for the problem is found, from which a formula for initial temperature is derived using Picard's criterion and the singular system of the associated operators.

Continuity of Pseudo-differential Operators Associated with the Bessel Operator in Some Gevrey Spaces

The pseudo-differential operator (p.d.o) h w , a associated with the Bessel operator d 2 / dx 2 + (1 m 4 w 2 )/4 x 2 involving the symbol a ( x , y ) whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain Gevrey spaces (ultradifferentiable function spaces). It is shown that the operator h w , a is a continuous linear map of one Gevrey space into another Gevrey space. A special p.d.o. called the Gevrey-Hankel potential is defined and some of its properties are investigated. A variant H w , a of h w , a is also discussed.

Modern Mathematical Methods for Physicists and Engineers

This 763 page book, available as both hardback and paperback, is unique in the sense that it sails across the ocean of contemporary advanced mathematics by an impressive tour de force. In twelve ambitious chapters it covers major topics in set theory, groups, rings and fields, vector spaces, matrix theory, linear functionals and special functions. The depth of coverage, however, is highly uneven. Two chapters on linear functionals, one chapter on inner products and norms, and one chapter on convergence in normed vector spaces contain a wealth of finely tuned knowledge. On the other hand, Chapter 1 on the foundations of computation, and the final Chapter 12 on special functions, just skim the surface. The selection of material and its presentation is evidently aimed at computer scientists and computer engineers. Their colleagues in other disciplines will find little that applies directly to their respective fields in spite of the many (more than 400) exercise problems. The author's approach is fundamentally different from similar titles in the literature, e.g. the time-honoured multi-edition text for physicists by Arfken. It provides a cogent, elegant and often esoteric structural framework for mathematical formulations, but with limited interest in exploring detailed solutions. A case in point is the brief presentation of Weber's Y (or N) function, better known as the Bessel function of the second kind. It appears as an eigenfunction of the Bessel operator (Section 12.4), but without any mention of how this function can be routinely eliminated from solutions where the physical quantity of interest is finite at the origin - a cornerstone of numerous practical problem solutions in physics and engineering. Other omissions are equally vexing. In Chapter 3 (Evaluation of Functions), the bisection method is dismissed in nine lines, and no mention is made of the secant method and Wegstein's method, which are just as important for numerical root finding as the analytical (and temperamental) Newton's method. Classical integration techniques are discussed, but not even one of the modern methods (e.g. Gaussian, Chebyshev, Rodau, Lobatto, Laguerre, Hermite and Filon integration) is given any mention. The classical Euler method for integrating ordinary differential equations is the subject of four pages, but there is nothing at all on the widely used Runge-Kutta techniques, nor the highly efficient Richardson's extrapolation method. Modified Bessel functions and Kelvin functions, essential tools in physics and engineering, are completely missing. Fourier transforms are bestowed a short section (11.8), but Laplace transforms? - nothing. Complex numbers pop up here and there, but complex calculus is ignored. The list continues.... Will this work become `... an ideal textbook for senior undergraduate and graduate students in the physical sciences and engineering...' and `... a valuable reference for working engineers...'? The author's claim for the affirmative is extremely tenuous, inasmuch as only highly motivated students with a broad command of mathematics would be able (if willing) to master its contents. The working engineer would most likely look for pragmatic texts, which provide practical/numerical examples in equally modern packing (e.g. Advanced Engineering Mathematics by Robert J Lopez (Addison Wesley, 2000)). Finally, a number of witty comments throughout the book lighten the task of its reading. Some readers will consider the statement (p 494) that `... eigenvalue is a mongrel word...' (in contrast, presumably, with the purebred Eigenwert and valeur propre) to be amusingly clever. An entire Section 8.2 (albeit consisting only of four lines) tells us that `... there seem to be almost as many ways to spell Chebyshev as there are written human languages...', then gracefully provides the proper Cyrillic spelling. Thomas Z Fahidy

Linear recursive schemes associated with the nonlinear wave equation involving Bessel's operator

We prove the existence and uniqueness of a weak solution of a semilinear wave equation involving Bessel's operator, the nonlinear term being in the form f( r, t, u, ▿ u), while the boundary condition at r = 0 takes the weak form ▪. In the proof, the Faedo-Galerkin method associated with a linear recursive scheme in appropriate Sobolev spaces with weight is used.

Bessel Wavelets and the Galerkin Analysis of the Bessel Operator

We construct a complete orthonormal system {ψj,k}j∈Z,k∈N for L2(R+,dx) such that each Bessel wavelet ψj,k has a compactly supported Hankel transform. We apply this system in the Galerkin solution of the differential equation Lu=f on R+, where L is the Bessel operator. The associated linear system can be preconditioned using a simple diagonal preconditioning matrix to give a system which is sparse and has condition number which is bounded independent of the Galerkin subspace. Thus we obtain the same advantages for the singular operator L that have been obtained for uniformly elliptic operators using standard wavelets. The main step is a characterization of the Bessel–Sobolev norm of a function in terms of its Bessel wavelet coefficients. We also present numerical work which shows that the condition numbers are small enough for this method to be practical.

Weyl–Bessel transforms

In this work, we define and study the Wigner–Bessel transform and prove for this transform an inversion formula. Next, we use this transform to define the Weyl–Bessel transform W σ , where σ is a symbol in S m, m∈ R . Using harmonic analysis associated with the Bessel operator, we study criteria in terms of the symbols σ for the boundedness and compactness of the Weyl–Bessel transform W σ .

Lpμ-Boundedness of the Pseudo-differential Operator Associated with the Bessel Operator

An Lpμ-boundedness result for the pseudo-differential operator associated with the Bessel operator is obtained.

Density properties of Hankel translations of positive definite functions

In this paper we study density properties of the Hankel translations of certain positive definite (in the Hankel sense) functions on \((0,\infty )\). The density is understood in a Frechet space of smooth functions on \((0,\infty )\) whose topology is defined by seminorms involving the Bessel operator \(x^{-2\mu -1}Dx^{2\mu +1}D\).

Mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator

In this paper, we prove the existence, uniqueness and continuous dependence on the data of a solution of a mixed problem with a weighted integral condition for a parabolic equation with the Bessel operator. The proof uses a functional analysis method based on an a priori estimate and on the density of the range of the operator generated by the considered problem.

A three-point boundary value problem with an integral condition for parabolic equations with the Bessel operator

In this paper, we study a three-point boundary value problem with an integral condition for a class of parabolic equation with Bessel operator. The existence and uniqueness of the solution in functional weighted Sobolev space are proved. The proof is based on two sided a priori estimates and the density of the range of the operator generated by the considered problem.

Singular Point-like Perturbations of the Bessel Operator in a Pontryagin Space

The spectral problem for the Bessel equation of order ν on (0, ∞) in the case 0<ν<1 is closely related to the Nevanlinna function Q(z)=−π(−z)ν/(2sinπν). If ν>1 and ν≠2, 3, ..., this function belongs to the generalized Nevanlinna class Nm, m=[ν+12]. A natural question appears: To what spectral problem does this function correspond? We answer this and related questions using Pontryagin space operator realizations of suitable singular point-like perturbations of the Bessel operator. In this paper we discuss the spectra of these realizations and we derive eigenfunction expansions via related wave operators.