Bessel Operator Research Articles

The Problem for a Mixed Equation with Fractional Power of the Bessel Operator

В последнее время особый интерес представляют уравнения с частными производными, содержащими дифференциальный оператор дробного порядка. Подобные уравнения и задачи для них находят применение в теории вязкой упругости, электрохимии, теории управления, моделировании эпидемий и пандемий и в других различных областях. Настоящая работа посвящена решению дифференциальных уравнений, содержащих оператор Бесселя дробной степени. В статье рассматривается прямое и обратное преобразование Мейера, модифицированное для удобства работы с оператором Бесселя дробной степени. Для рассматриваемого преобразования Мейера получена свертка. Используя преобразования Лапласа и Пуассона получены факторизации прямого и обратного преобразований Мейера. С использованием рассмотренного модифицированного преобразования Мейера находится решение обыкновенного дифференциального уравнения с оператором Бесселя дробной степени. Рассматривается нелокальная краевая задача для смешанного параболо-гиперболческого уравнения, содержащего дробной степени оператор Бесселя. Доказывается, что, при выполнении определенных условий гладкости входных функций задачи и выполнения условия сопряжения на линии раздела областей гиперболичности и параболичности, регулярное решение нелокальной краевой задачи для смешанного параболо-гиперболического уравнения с оператором Бесселя дробной степени существует и единственно. Recently, of particular interest are partial differential equations containing a fractional order differential operator. Similar equations and problems for them find application in the theory of viscous elasticity, electrochemistry, control theory, modeling of epidemics and pandemics, and in various other areas. The present work is devoted to the solution of differential equations containing the Bessel operator of fractional degree. The article discusses the direct and inverse Meyer transforms, modified for the convenience of working with the Bessel operator of a fractional degree. For the considered Meyer transformation, a convolution is obtained. Using the Laplace and Poisson transformations, factorizations of the direct and inverse Meyer transformations are obtained. Using the considered modified Meyer transform, we find a solution to an ordinary differential equation with a Bessel operator of fractional degree. A nonlocal boundary value problem for a mixed parabolic-hyperbolic equation containing a fractional degree Bessel operator is considered. It is proved that, under certain conditions of smoothness of the input functions of the problem and the condition of conjugation on the dividing line of the regions of hyperbolicity and parabolicity, a regular solution of a nonlocal boundary value problem for a mixed parabolic-hyperbolic equation with a Bessel operator of fractional degree exists and is unique.

To the Properties of One Fox Function

В работе рассматривается частный случай специальной функции Фокса с четырьмя параметрами, которая возникает в теории краевых задач для параболических уравнений c оператором Бесселя и дробной производной по времени. Целью исследования является получение некоторых рекуррентных соотношений, формул дифференцирования и интегрального преобразования рассматриваемой функции. При получении результатов работы в основном используется представление рассматриваемой функции через интеграл Меллина–Барнса. Также используются её асимптотические разложения при большом и малом значениях аргумента. С помощью указанного интегрального представления и некоторых известных формул для гамма-функции Эйлера, получены рекуррентные соотношения, связывающие функции с разными параметрами, а также функцию с её производной первого порядка. Получена формула дифференцирования n-го порядка. Исследуется несобственный интеграл первого рода, который содержит рассматриваемую функцию с двумя зависимыми параметрами из четырёх. Показывается, что этот несобственный интеграл может быть записан в терминах известной специальной функции Макдональда. При частных значениях параметров рассматриваемой в работе функции получаются некоторые известные элементарные и специальные функции. Результаты работы носят теоретический характер и будут полезны при исследовании краевых задач для вырождающихся параболических уравнений с производными дробного порядка по времени. The paper considers a particular case of a special Fox function with four parameters, which arises in the theory of boundary value problems for parabolic equations with a Bessel operator and a fractional time derivative. The research objective is to obtain some recurrence relations, formulas for differentiation and integral transformation of the function under consideration. The results are obtained through representation of the considered function in terms of the Mellin–Barnes integral. The function asymptotic expansions for large and small values of the argument are also used. Employing the integral representation and some wellknown formulas for the Euler gamma function, recurrent relations are obtained connecting functions with different parameters, as well as a function with its first-order derivative. A formula for differentiation of the nth order is obtained. The paper studies an improper integral of the first kind that includes the considered function with two dependent of the four parameters. We show that the improper integral can be written out in terms of the well-known special Macdonald function. With special values of the parameters of the considered function we obtain some well-known elementary and special functions. The results of the study are theoretical and applicable in the study of boundary value problems for degenerate parabolic equations with fractional time derivatives.

Solution of the Cauchy problem for a hyperbolic equation of the fourth order with the Bessel operator by the method of transmutation operators

Solution of the Cauchy problem for a hyperbolic equation of the fourth order with the Bessel operator by the method of transmutation operators

On Recovery of the Singular Differential Laplace—Bessel Operator from the Fourier–Bessel Transform

This paper is devoted to the problem of the best recovery of a fractional power of the B-elliptic operator of a function on R+N by its Fourier–Bessel transform known approximately on a convex set with the estimate of the difference between Fourier–Bessel transform of the function and its approximation in the metric L∞. The optimal recovery method has been found. This method does not use the Fourier–Bessel transform values beyond a ball centered at the origin.

Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed.

On the generalized Bessel potential and perfect functional completion

he class of generalized Bessel potentials is the main object of study in this article. The generalized Bessel potential is a negative real power of the operator (I − Δγ), where Δγ = (...)∂xk - Laplace - Bessel operator, γ = (γ1,...,γn) - multi-index consisting of positive fixed real numbers. When solving various problems for differential equations, proving embedding theorems for some classes of functions, and inverting integral operators, there is a need to consider functions up to some small, from the point of view of the problem under consideration, set. Often a set of Lebesgue measure zero is taken as such a small set. However, often the sets of Lebesgue measure zero turn out to be too large to be neglected. For example, when solving a boundary problem, the behavior of the solution on the boundary is essential. In this regard, there was a need to construct complete classes of admissible functions suitable for solving specific problems. N. Aronshine and K. Smith presented two stages of constructing a functional completion. The first of these is finding a suitable class of exceptional sets. The second is to find the functions defined modulo these exceptional classes that need to be attached to get a complete functional class. It turns out that there can be infinitely many suitable exceptional classes in a particular problem, but each of them corresponds essentially to one functional completion. It is clear that the most suitable functional completion is the one whose exceptional class is the smallest, since then the functions will be defined with the best possible accuracy. Whenever such a minimal exceptional class exists, the corresponding functional completion is called a perfect completion. In this paper, perfect completions are constructed from the norm associated with the kernel of the generalized Bessel potential.

The continuous generalized wavelet transform associated with q-Bessel operator

The continuous generalized wavelet transform associated with -Bessel operator is defined, which will invariably be called continuous -Bessel wavelet transform . Certain and boundedness results and inversion formula for continuous -Bessel wavelet transform are obtained. Discrete -Bessel wavelet transform is defined and a reconstruction formula is derived for discrete- Bessel wavelet.

Perturbed Bessel operators. Boundary conditions and closed realizations

We study perturbed Bessel operators Lm2=−∂x2+(m2−14)1x2+Q(x) on L2]0,∞[, where m∈C and Q is a complex locally integrable potential. Assuming that Q is integrable near ∞ and x↦x1−εQ(x) is integrable near 0, with ε≥0, we construct solutions to Lm2f=−k2f with prescribed behaviors near 0. The special cases m=0 and k=0 are included in our analysis. Our proof relies on mapping properties of various Green's operators of the unperturbed Bessel operator. Then we determine all closed realizations of Lm2 and show that they can be organized as holomorphic families of closed operators.

FRACTIONAL WEIGHTED SPHERICAL MEAN AND MAXIMAL INEQUALITY FOR THE WEIGHTED SPHERICAL MEAN AND ITS APPLICATION TO SINGULAR PDE

In this paper we establish a mean value property for the functions which is satisfied to Laplace–Bessel equation. Our results involve the generalized divergence theorem and the second Green’s identities relating the bulk with the boundary of a region on which differential Bessel operators act. Also we design a fractional weighted mean operator, study its boundedness, obtain maximal inequality for the weighted spherical mean and get its boundedness. The connection between the boundedness of the spherical maximal operator and the properties of solutions of the Euler–Poisson–Darboux equation with Bessel operators is given as an application.

Strong coupling expansion in \U0001d4a9 = 2 superconformal theories and the Bessel kernel

We consider strong ’t Hooft coupling expansion in special four-dimensional \U0001d4a9 = 2 superconformal models that are planar-equivalent to \U0001d4a9 = 4 super Yang-Mills theory. Various observables in these models that admit localization matrix model representation can be expressed at large N in terms of a Fredholm determinant of a Bessel operator. The latter previously appeared in the study of level spacing distributions in matrix models and, more recently, in four-point correlation functions of infinitely heavy half-BPS operators in planar \U0001d4a9 = 4 SYM. We use this relation and a suitably generalized Szegő-Akhiezer-Kac formula to derive the strong ’t Hooft coupling expansion of the leading corrections to free energy, half-BPS circular Wilson loop, and certain correlators of chiral primaries operators in the \U0001d4a9 = 2 models. This substantially generalizes partial results in the literature and represents a challenge for dual string theory calculations in AdS/CFT context. We also demonstrate that the resulting strong-coupling expansions suffer from Borel singularities and require adding non-perturbative, exponentially suppressed corrections. As a byproduct of our analysis, we determine the non-perturbative correction to the above mentioned four-point correlator in planar \U0001d4a9 = 4 SYM.

Jacobi-type functions defined by fractional Bessel derivatives

ABSTRACT For a class of even weight functions, generalizations of the classical Jacobi and Laguerre polynomials are defined via fractional Rodrigues' type formulas using the Bessel operators in the form . Their properties, including hypergeometric representation, differential recurrence relations and fractional boundary value problem, are investigated.

Hardy spaces meet harmonic weights

We investigate the Hardy space H L 1 H^1_L associated with a self-adjoint operator L L defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an L L -harmonic non-negative function h h such that the semigroup exp ⁡ ( − t L ) \exp (-tL) , after applying the Doob transform related to h h , satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space H L 1 H^1_L in terms of a simple atomic decomposition associated with the L L -harmonic function h h . Our approach also yields a natural characterisation of the B M O BMO -type space corresponding to the operator L L and dual to H L 1 H^1_L in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in R n {\mathbb {R}^n} , Schrödinger operators with certain potentials, and Bessel operators.

On the solvability of nonlocal initial-boundary value problems for a partial differential equation of high even order

In the present paper, two non-local initial-boundary value problems have been formulated for a partial differential equation of high even order with a Bessel operator in a rectangular domain. The correctness of one of the considered problems has been investigated. To do this, applying the method of separation of variables to the problem under consideration, the spectral problem was obtained for an ordinary differential equation of high even order. The self-adjointness of the last problem was proved, which implies the existence of the system of its eigenfunctions, as well as orthonormality and completeness of this system. Further, the Green's function of the spectral problem was constructed, with the help of which it was equivalently reduced to the Fredholm integral equation of the second kind with symmetrical kernel. Using this integral equation and Mercer's theorem, the uniform convergence of some bilinear series depending on found eigenfunctions has been studied. The order of the Fourier coefficients was established. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and also the series obtained from it by term-by-term differentiation was proved. Using the method of spectral analysis, the uniqueness of the solution of the problem was proved. An estimate for the solution of the problem was obtained, from which its continuous dependence on the given functions follows.

Some Upper Bounds for RKHS Approximation by Bessel Functions

A reproducing kernel Hilbert space (RKHS) approximation problem arising from learning theory is investigated. Some K-functionals and moduli of smoothness with respect to RKHSs are defined with Fourier–Bessel series and Fourier–Bessel transforms, respectively. Their equivalent relation is shown, with which the upper bound estimate for the best RKHS approximation is provided. The convergence rate is bounded with the defined modulus of smoothness, which shows that the RKHS approximation can attain the same approximation ability as that of the Fourier–Bessel series and Fourier–Bessel transform. In particular, it is shown that for a RKHS produced by the Bessel operator, the convergence rate sums up to the bound of a corresponding convolution operator approximation. The investigations show some new applications of Bessel functions. The results obtained can be used to bound the approximation error in learning theory.

Inverse Problem for an Equation of Mixed Parabolic–Hyperbolic Type with a Bessel Operator

Inverse Problem for an Equation of Mixed Parabolic–Hyperbolic Type with a Bessel Operator

Riesz Transform Characterizations for Multidimensional Hardy Spaces

We study Hardy space $$H^1_L(X)$$ related to a self-adjoint operator L defined on an Euclidean subspace X of $${{\mathbb {R}}^d}$$ . We continue study from [27], where, under certain assumptions on the heat semigroup $$\exp (-tL)$$ , the atomic characterization of local type for $$H^1_L(X)$$ was proved. In this paper we provide additional assumptions that lead to another characterization of $$H^1_L(X)$$ by the Riesz transforms related to L. As an application, we prove the Riesz transform characterization of $$H^1_L(X)$$ for multidimensional Bessel and Laguerre operators, and the Dirichlet Laplacian on $${\mathbb {R}}^d_+$$ .

Solvability and dynamics study of a weak solution for a nonlinear parabolic problem with the bessel operator

In this paper, we present a class of parabolic equation with nonlinear where we show two part of this study the theoretical part we prove the existence and uniqueness of the solution by energy inequality method. Then we present finite time blow up by using the energy method.

FUNDAMENTAL SOLUTION OF THE CAUCHY PROBLEM FOR PARABOLIC EQUATION OF THE SECOND ORDER WITH INCREASING COEFFICIENTS AND WITH BESSEL OPERATORS OF DIFFERENT ORDERS

The theory of the Cauchy problem for uniformly parabolic equations of the second order with limited coefficients is sufficiently fully investigated, for example, in the works of S.D. Eidelman and S.D. Ivasyshen, in contrast to such equations with unlimited coefficients. One of the areas of research of Professor S.D. Ivasyshen and students of his scientific school are finding fundamental solutions and investigating the correctness of the Cauchy problem for classes of degenerate equations, which are generalizations of the classical Kolmogorov equation of diffusion with inertia and contain for the main variables differential expressions, parabolic according to I.G. Petrovskyi and according to S.D. Eidelman (S.D. Ivasyshen, L.M. Androsova, I.P. Medynskyi, O.G. Wozniak, V.S. Dron, V.V. Layuk, G.S. Pasichnyk and others). Parabolic Petrovskii equations with the Bessel operator were also studied (S.D. Ivasyshen, V.P. Lavrenchuk, T.M. Balabushenko, L.M. Melnychuk). The article considers a parabolic equation of the second order with increasing coefficients and Bessel operators. In this equation, the some of coefficients for the lower derivatives of one group of spatial variables $x\in \mathbb{R}^n $ are components of these variables, therefore, grow to infinity. In addition, the equation contains Bessel operators of different orders in another group of spatial variables $y\in \mathbb{R}^m_+ $, due to which the coefficients in the first derivatives of these variables are unbounded around the point y=0. The paper defines a modified Fourier-Bessel transform that takes into account different orders of Bessel operators on different variables. With the help of this transformation and the method of characteristics, the solution of the Cauchy problem of the specified equation is found in the form of the Poisson integral, and its kernel, which is the fundamental solution of the Cauchy problem, is written out in an explicit form. Some properties of the found fundamental solution, in particular, estimates of its derivatives, have been established. They will be used to establish the correctness of the Cauchy problem.

Non-local problem with an integral condition for a parabolic equation with a Bessel operator

For the parabolic equation with the Bessel operator ∂u/∂t=(∂^2 u)/(∂x^2 )+k/x ∂u/∂x in the rectangular domain 0<x<l, 0<t≤T, we consider a boundary value problem with the non-local integral condition of the first kind ∫_0^l▒〖u(x,t)xdx=0, 0≤t≤T.〗 This problem is reduced to an equivalent boundary value problem with mixed boundary conditions of the first and third kind. It is shown that the homogeneous equivalent boundary value problem has only a trivial zero solution, and hence the original inhomogeneous problem cannot have more than one solution. This proof uses Gronwall’s lemma. Then, by the method of spectral analysis, the existence theorem for a solution to an equivalent problem is proved. This solution is defined explicitly in the form of a Dini series. Sufficient conditions with respect to the initial condition are obtained. These conditions guarantee the convergence of the constructed series in the class of regular solutions.

An initial boundary value problem for a partial differential equation of higher even order with a Bessel operator

In present paper, an initial-boundary value problem is formulated in a rectangle for a higher even order partial differential equation with the Bessel operator. Applying the method of separation of variables to the considered problem a spectral problem is obtained for an ordinary differential equation of higher even order. The self-adjointness of the last problem is proved, which implies the existence of the system of its eigenfunctions, as well as the orthonormality and completeness of this system. The uniform convergence of some bilinear series and the order of the Fourier coefficients, depending on the found eigenfunctions, is investigated. The solution of the considered problem is found as the sum of the Fourier series with respect to the system of eigenfunctions of the spectral problem. The absolute and uniform convergence of this series, as well as the series obtained by its differentiating, have been proved. The uniqueness of the solution of the problem is proved by the method of spectral analysis. An estimate is obtained for the solution of the problem which implies the continuous dependence of the solution on the given functions.