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In this note, we deal with the fractional logarithmic Schrödinger operator (I + (- Δ) s) log and the corresponding energy spaces for variational study. The fractional (relativistic) logarithmic Schrödinger operator is the pseudo-differential operator with logarithmic Fourier symbol, log (1 + | ξ | 2 s) , s > 0 . We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of R N. We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity. [ABSTRACT FROM AUTHOR]

The Fokker–Planck equation with one spatial variable without the lowest term is considered. The diffusion coefficient is assumed to be measurable, bounded, and separated from zero. The existence of a weak fundamental solution of the Fokker–Planck equation is proved and some properties of this solution are established. Under the additional assumption that the leading coefficient is a Hölder function, we consider the first boundary-value problem in a semi-bounded domain. We assume that the right-hand side of the equation and the initial function are zero and the boundary function is continuous. We prove the solvability of this problem in the class of bounded functions. [ABSTRACT FROM AUTHOR]

An integro‐differential equation for the probability density of the generalized stochastic Ornstein–Uhlenbeck process with jump diffusion is considered for a special case of the Laplacian distribution of jumps. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly, as a finite sum. Alternatively, the fundamental solution can be represented as converging power series. The properties of this solution are investigated. The fundamental solution makes it possible to obtain explicit formulas for the density at each instant of time, which is important, for example, for testing numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Semi-Analytical method for Barrier Option (SABO) applied to continuously monitored Arithmetic Asian Options with barrier. The effectiveness and utility of this approach, based on collocation Boundary Element Method (BEM), has already been tested for European style barrier options in other frameworks and in particular with reference to the similar but unusual Geometric Asian Options. European style Arithmetic Asian Options are more common among practitioners, but much more difficult to treat. Hence a deep investigation of the strategies applied to face the numerical challenges involved in their evaluation by SABO is here presented, together with several results that show the performance of the considered approach. [ABSTRACT FROM AUTHOR]

In this work we further develop a nonlocal calculus theory (initially introduced in Bellido et al. (Adv Nonlinear Anal 12:20220316, 2023)) associated with singular fractional-type operators which exhibit kernels with finite support of interactions. The applicability of the framework to nonlocal elasticity and the theory of peridynamics has attracted increased interest and motivation to study it and find connections with its classical counterpart. In particular, a critical contribution of this paper is producing vector identities, integration by part type theorems (such as the Divergence Theorem, Green identities), as well as a Helmholtz–Hodge decomposition. The estimates, together with the analysis performed along the way provide stepping stones for proving additional results in the framework, as well as pathways for numerical implementations. [ABSTRACT FROM AUTHOR]

We consider a traveling wave solution of a two-dimensional transmission problem across the $ y- $axis modeling autoignition of reactive jets, with a limiting version corresponding to large heat loss. A jump of the normal derivative generates a singularity at the origin. We construct an explicit solution to a problem in a distributional sense that verifies the transmission system and closely examine the properties of the solution near the singularity. Using the Implicit Function Theorem, we study the level sets, especially the one through the origin. The numerical illustrations are consistent with our theoretical results. [ABSTRACT FROM AUTHOR]

The paper examines the problem of unique determination of the fundamental solution of a mesh analogue of Laplace's equation within the theory of discrete gravitational potential. The mesh fundamental solution of the finite-difference analogue of Laplace's equation plays a key role in reconstructing a continuously distributed source of gravitational or magnetic field from heterogeneous and different-precision data obtained at points of a certain mesh set. [ABSTRACT FROM AUTHOR]

This article considers the biharmonic equation Δ 2 u = K (x) f (u) in R N. Under suitable assumptions, the existence of positive solutions is obtained. The methods used here contain the integral operator and the Schauder fixed point theory. Since the form of fundamental solution of Δ 2 u = 0 in R N depends on N, we divide our discussions into three cases as (a) N = 2 ; (b) N = 4 ; (c) N > 2 but N ≠ 4 . The fundamental solution of Δ 2 plays an essential role in our results. [ABSTRACT FROM AUTHOR]

Let \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential \Upsilon belonging to the reverse Hölder class B_{Q/2}, where Q is the homogeneous dimension of the stratified Lie group \mathbb {G}. Inspired by Shen's pioneer work and Li's work, we study fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for \mathcal {L}-harmonic functions on \mathbb {G}. [ABSTRACT FROM AUTHOR]

Asymptotic behavior at infinity is investigated for fundamental solutions of a hypoelliptic partial differential operator 퐏 ⁢ (i ⁢ ∂ x) = (P 1 ⁢ (i ⁢ ∂ x)) m 1 ⁢ ⋯ ⁢ (P l ⁢ (i ⁢ ∂ x)) m l with the characteristic polynomial that has real multiple zeros. Based on asymptotic expansions of fundamental solutions, asymptotic classes of functions are introduced and existence and uniqueness of solutions in those classes are established for the equation 퐏 ⁢ (i ⁢ ∂ x) ⁢ u = f in ℝ n . The obtained results imply, in particular, a new uniqueness theorem for the classical Helmholtz equation. [ABSTRACT FROM AUTHOR]

Deep operator networks (DeepONets) have demonstrated their capability of approximating nonlinear operators for initial- and boundary-value problems. One attractive feature of DeepONets is their versatility since they do not rely on prior knowledge about the solution structure of a problem and can thus be directly applied to a large class of problems. However, training the parameters of the networks may sometimes be slow. In order to improve on DeepONets for approximating the wave equation, we introduce the Green operator networks (GreenONets), which use the representation of the exact solution to the homogeneous wave equation in term of the Green's function. The performance of GreenONets and DeepONets is compared on a series of numerical experiments for homogeneous and heterogeneous media in one and two dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we present a technique for constructing generalized solutions of the Cauchy problem for abstract integro-differential equations with degeneration in Banach spaces. A generalized solution is constructed as the convolution of the fundamental operator function (fundamental solution, influence function) of the integro-differential operator of the equation with a generalized function of a special form, which involves all input data of the original problem. Based on the analysis of the representation for the generalized solution, we obtain sufficient solvability conditions for the original Cauchy problem in the class of functions of finite smoothness. Under these sufficient conditions, the generalized solution constructed turns out to be a classical solution with the required smoothness. The abstract results obtained in the paper are applied to the study of applied initial-boundary-value problems from the theory of oscillations in viscoelastic media. [ABSTRACT FROM AUTHOR]

We introduce Besov space $ \dot {B}_{p}^{\alpha,q}(\partial \Omega _k) $ B ˙ p α , q (∂ Ω k) and Triebel-Lizorkin space $ \dot {F}^{\alpha,q}_{p}(\partial \Omega _k) $ F ˙ p α , q (∂ Ω k) on a family of model domains $ \partial \Omega _k=\left \{(\mathbf{z},z_{n+1})=(z_1,z_2,\ldots, z_{n+1}):\right. \left. \mbox {Im} (z_{n+1})=\phi (|\mathbf{z}|^2)\right \} $ ∂ Ω k = { (z , z n + 1) = (z 1 , z 2 , ... , z n + 1) : Im (z n + 1) = ϕ (| z | 2) } with $ \phi (x)=x^{k} $ ϕ (x) = x k in $ \mathbf {C}^{n+1} $ C n + 1 which can be considered as a space of homogeneous type in the sense of Coifman RR, Weiss G.[Analyse Harmonique Non-commutative and Certains Espaces Homogǹes. Berlin: Springer; 1971. (Lecture Notes in Math.; 242).], Coifman R, Weiss G. [Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc. 1977;83:569–645.]. We study the sharp estimates on the fundamental solution for the Kohn Laplacian and Cauchy-Szegö projection on $ \partial \Omega _k $ ∂ Ω k in these spaces. [ABSTRACT FROM AUTHOR]

We present several methods for constructing the asymptotics of the fundamental solution of Fokker–Planck–Kolmogorov-type parabolic equations with a small parameter both for small and finite positive times. [ABSTRACT FROM AUTHOR]

The fundamental solution of generalized magneto thermo viscoelasticity (MTVE) with two relaxation times for perfect isotropic conduction is obtained. An infinitely long circular cylinder application has been studied. The solution is obtained by the potential method. In the transform domain Laplace and Hankel are used to obtain the solution. The inverse process is performed by asymptotic expansions appropriate for limited time values. Numerical results are displayed and graphically explained for the temperature and stress distributions. A comparison is made with the results obtained in the presence and absence of a magnetic field, for different values of time and for two theories (MTE and MTVE). [ABSTRACT FROM AUTHOR]

In this article, we study the approximate controllability for a semilinear second-order stochastic neutral evolution equation with infinite delay in Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. Finally, an example is given to illustrate our main conclusion. [ABSTRACT FROM AUTHOR]

In [6, 7], it was shown that mixed problems can be both ill-posed for Petrovsky well-conditioned equations and well-posed for ill-conditioned equations. In the present paper we study the existence and uniqueness of the solution of a mixed problem for a class of equations with complex-valued coefficients that behave as parabolic ones, despite the fact that they can change over "time" from parabolic type to Schrödinger type, or even to antiparabolic type. [ABSTRACT FROM AUTHOR]

We construct the fundamental solution of system of differential equations in the theory of functionally graded non local couple stress thermoelastic solid with voids in case of steady oscillations in terms of elementary functions. Some properties of fundamental solution are also established. Some special cases are also derived. [ABSTRACT FROM AUTHOR]

We consider a class of degenerate equations in non-divergence form satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin diffusions. [ABSTRACT FROM AUTHOR]

The paper considers Kolmogorov–Feller equation for the probability density of a Markov process on a half-axis, which arises in important problems of biology. This process consists of random jumps distributed according to the Laplace law and a deterministic return to zero. It is shown that the Green function for such an equation can be found both in the form of a series and in explicit form for some ratios of the parameters. This allows finding explicit solutions to the Kolmogorov–Feller equation for many initial data. [ABSTRACT FROM AUTHOR]

This paper examines a three-component differential equation model with a self-control mechanism in vision as a slight extension of the lateral inhibition model proposed by Peskin (Partial differential equations in biology: Courant Institute of Mathematical Sciences Lecture Notes, New York, 1976). First, we derive a condition under which the exact solution of our differential equation model for time-dependent input I = I (t) is described by the convolution integral with a temporal biphasic function. Second, we analyze the model with the input signal I = I (t , x) depending on time t and position x ∈ R , and we prove that the solution can be represented in convolution integral form and that t 1 > 0 exists such that the spatiotemporal integral kernel K u (t 1 , x) is positive for x ∈ R and t ∈ (0 , t 1) . Moreover, we numerically demonstrate that there exists t 2 (> t 1) such that K u (t 2 , x) includes the Mexican-hat function and a temporal biphasic function under certain parameter conditions. From these mathematical and numerical results, we find that there is a time lag before the Mexican-hat function appears in K u (t , x) , and the shape of K u (t , x) is similar to the receptive field structure observed in experiments in the field of neurophysiology. We conclude that the partial differential equations for visual information processing can be used to analytically determine the shape of the spatiotemporal kernel indicating the self-control mechanism. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and ℓ p estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on ℓ p spaces. [ABSTRACT FROM AUTHOR]

A multipole expansion of the fundamental solution of the fractional power of the Laplace operator is constructed in terms of the Gegenbauer polynomials. Based on the decomposition constructed and the idea of the fast multipole method, we propose a numerical algorithm for solving the fractional differential generalization of the Poisson equation in the two-dimensional and three-dimensional spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the approximate controllability of a class of semi-linear neutral integro-differential equation with nonlocal condition. The basic tools of this study are the theory of resolvent operators of linear neutral integro-differential equation and fractional powers. The fundamental solution of linear neutral integro-differential equations is also constructed to deal with the non-uniform boundedness of the extra linear term. Sufficient conditions of approximate controllability are then obtained through the resolvent condition. Finally, an example is provided to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

An analytical solution of unsaturated poroelastic transversely isotropic (TI) half‐space under surface time‐harmonic load with consideration of compressibility, viscous and inertial coupling of soil skeleton, pore‐water and pore‐air, as well as the capillary pressure is investigated in this paper. Firstly, the governing equations of unsaturated TI soil are established in cylindrical coordinate system. Then, the displacement function and operator theory are introduced to decouple the general solution. Secondly, Fourier expansion and Hankel transform are applied to the circumferential and radial coordinates, and the integral general solutions of soil skeleton displacement, pore‐water displacement and pressure, pore‐air displacement and pressure, and the total stress components are derived in the transformed domain. Combined with boundary conditions and continuity conditions, the fundamental solution of unsaturated TI half‐space is obtained. Finally, the proposed solution is verified by benchmarking with two existing solutions. The influence of the TI parameters, vibration frequency and the degree of saturation on the dynamic response of unsaturated poroelastic half‐space is investigated. [ABSTRACT FROM AUTHOR]

This work focused on establishing sufficient conditions to guarantee the well-posedness of the following nonlinear fractional semidiscrete model: D t β u (n , t) = B u (n , t) + f (n - c t , u (n , t)) , n ∈ Z , t > 0 , u (n , 0) = φ (n) , n ∈ Z , under the assumptions that β ∈ (0 , 1 ] , c > 0 some constant, and B is a discrete convolution operator with kernel b ∈ ℓ 1 (Z) , which is the infinitesimal generator of the Markovian C 0 -semigroup and suitable nonlinearity f. We present results concerning the existence and uniqueness of solutions, as well as establishing a comparison principle of solutions according to the respective initial values. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to the study of anisotropic parabolic equation related to the p i $p_{i}$ -Laplacian with a source term f ( u ) $f(u)$ . If f ( u ) = 0 $f(u)=0$ , then the fundamental solution of the equation is constructed. If there are some restrictions on the growth order of u in the source term, the initial energy E ( 0 ) $E(0)$ is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time.

We are addressing a parabolic equation with fractional derivatives in time and space that governs the scaling limit of continuous-time random walks with anomalous diffusion. For these equations, the fundamental solution represents the probability density of finding a particle released at the origin at time 0 at a given position and time. Using some estimates of the asymptotic behaviour of the fundamental solution, we evaluate the probability of the process returning infinite times to the origin in a heuristic way. Our calculations suggest that the process is always recurrent.

We consider a pseudodifferential equation of parabolic type with a fractional power of the Laplace operator of order $\alpha\in(0;1)$ acting with respect to the spatial variable. This equation naturally generalizes the well-known fractal diffusion equation. It describes the local interaction of moving objects in the Riesz gravitational field. A simple example of such system of objects is stellar galaxies, in which interaction occurs according to Newton's gravitational law. The Cauchy problem for this equation is solved in the class of continuous bounded initial functions. The fundamental solution of this problem is the Polya distribution of probabilities $\mathcal{P}_\alpha(F)$ of the force $F$ of local interaction between these objects. With the help of obtained solution estimates the correct solvability of the Cauchy problem on the local field fluctuation coefficient under certain conditions is determined. In this case, the form of its classical solution is found and the properties of its smoothness and behavior at the infinity are studied. Also, it is studied the possibility of local strengthening of convergence in the initial condition. The obtained results are illustrated on the $\alpha$-wandering model of the Lévy particle in the Euclidean space $\mathbb{R}^3$ in the case when the particle starts its motion from the origin. The probability of this particle returning to its starting position is investigated. In particular, it established that this probability is a descending to zero function, and the particle "leaves" the space $\mathbb{R}^3$.

We establish the Caccioppoli inequality, a reverse Hölder inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order 2m2m with certain lower order terms.

Последние десятилетия количество работ, посвященных исследованию задач для дифференциальных уравнений дробного порядка, заметно растет. Интерес исследователей вызван тем, что количество областей науки, в которых используются уравнения, содержащие дробные производные, варьируется от биологии и медицины до теории управления, инженерии, финансов, а также оптики, физики и так далее. Включение запаздывания в уравнение дробного порядка существенно влияет на ход процесса, описываемого этим уравнением, так как неизвестная функция задается при различных значениях аргумента, что вносит эффект предыстории в уравнение. Поэтому, математические модели, содержащие дробный оператор и запаздывающий аргумент, более точны, чем модели, содержащие производные целого порядка. В данной работе исследуется задача Коши для линейного обыкновенного дифференциального уравнения с запаздывающим аргументом c оператором дробного дифференцирования Джрбашяна – Нерсесяна, обобщающим известные дробные операторы Римана – Лиувилля и Герасимова – Капуто. Результаты работы получены с использованием методов теории целого и дробного исчислений, методов теории дифференциальных уравнений с запаздывающим аргументом, метода специальных функций. В работе доказывается теорема о справедливости аналога формулы Лагранжа. Также доказано, что специальная функция Wγm​​(t), которая, в свою очередь, определяется через обобщенную функцию Миттаг – Леффлера (или функция Прабхакара), удовлетворяет уравнению и условиям, сопряженным исследуемому, и является фундаментальным решением рассматриваемого уравнения. Сформулирована и доказана теорема существования и единственности решения начальной задачи. Решение поставленной задачи выписано в терминах специальной функции Wν​(t).

We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L ∞ norm of these solutions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of anisotropic parabolic equation related to the p i -Laplacian with a source term f (u) . If f (u) = 0 , then the fundamental solution of the equation is constructed. If there are some restrictions on the growth order of u in the source term, the initial energy E (0) is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time. [ABSTRACT FROM AUTHOR]

This study considers the wave phenomena and fundamental solution in swelling porous elastic solids consisting of a mixture of solid, liquid, and gas. There exist three dilatational and two transversal waves which vibrate at distinct speeds. The basic characteristics of waves (phase velocity and attenuation coefficient) are computed numerically and displayed in the form of graphs. Also, a fundamental solution is explored for steady oscillation and some preliminary features are explored in this model. The obtained results for both waves are analyzed with and without swelling porous elastic solid. This new model enables us to improve the efficacy of swelling porous elastic solid which involves solid and fluid and find it applicable in exploration industries and to investigate various wave propagation problems. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions and the approximate controllability for a class of systems governed by neutral equations of second-order with infinite delay in infinite-dimensional Hilbert spaces. The mild solution and approximate controllability are achieved by constructing the fundamental solution for the associated linear equation and assuming that the linear system is approximately controllable. The discussion is based on the fundamental solution theory and Rothe's fixed point theorem. In addition, an example is given to illustrate our main conclusion. [ABSTRACT FROM AUTHOR]

The Laplace equation in three-dimensional Euclidean space is R -separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé–Wangerin functions called internal and external bi-cyclide harmonics. An expansion for a fundamental solution of Laplace's equation in products of internal and external bi-cyclide harmonics is derived. In limiting cases this expansion reduces to known expansions in bi-spherical and prolate spheroidal coordinates. [ABSTRACT FROM AUTHOR]

We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative partial differential equations (PDEs) based on the spirit of LY algorithm [T.-P. Liu and S.-H. Yu, Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. 7 (2012) 477–543]. The new method can handle more general domains than that of LYs'. We convert the evolutionary PDEs into the elliptic PDEs by the Laplace transformation. Using the Laplace transformation of the fundamental solutions of the evolutionary PDEs and the image method, we can construct Green's functions for the corresponding elliptic PDEs. Finally, we obtain Green's functions for the evolutionary PDEs by inverting the Laplace transformation. As a consequence, we establish Green's functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. On the other hand, the structure of hyperbolic-dissipative PDEs means its fundamental solution is non-symestric and hence the image method does not work. We utilize the idea of Laplace wave train introduced by Liu and Yu in [Navier–Stokes equations in gas dynamics: Greens function, singularity and well-posedness, Comm. Pure Appl. Math. 75(2) (2022) 223–348] to generalize the image method. Combining this with the notions of Rayleigh surface wave operators introduced in [S. J. Deng, W. K. Wang and S.-H. Yu, Green's functions of wave equations in ℝ + n × ℝ + , Arch. Ration. Mech. Anal. 216 (2015) 881–903], we are able to obtain the complete representations of Green's functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions. [ABSTRACT FROM AUTHOR]

In this article, we study Lie symmetries to fundamental solutions to the Leutwiler-Weinstein equation L u : = Δ u + k x n ∂ u ∂ x n + ℓ (x n) 2 u = 0 in the upper half-space ℝ + n . Starting from the infinitesimal generators of the equation Lu = 0, we deduce symmetries of the equation Lu = δ(x − x0), and using its invariant solutions, we construct a fundamental solution. As an application, we study a Green functions of the operator in the hyperbolic unit ball. [ABSTRACT FROM AUTHOR]

A -hyperbolic equation with a Bessel operator in time and with an operator in space is considered. The operator includes singular differential Bessel operators with non-negative parameters and derivatives of these operators. The found fundamental solution is expressed in terms of the Bessel function of the first kind of negative order. [ABSTRACT FROM AUTHOR]

In this paper, we study the heat equation with non-smooth perturbation caused by a delay applied to the time variable through a piecewise constant function, but keeping the time positive, which does not merit changing the phase space as usually occurs when the delay makes the time negative. To do so, we first prove the existence of solutions using Fourier's transform. Next, we prove the uniqueness of solutions by applying the maximum principle method. After that, we study the stability of these solutions. Finally, we propose some problems that can be solved with this or similar techniques. [ABSTRACT FROM AUTHOR]

This is a continuation of [5] which is concerned about the regularity property of stochastic convolutions for abstract linear stochastic retarded differential equations with unbounded operators on delay terms. In this work, we improve and generalize the main results in [5] by considering those delay operators which may have the same order as the infinitesimal generator of the system under consideration. To this end, we need restrict the weight function of distributed delay term to be Hölder continuous type in this system. [ABSTRACT FROM AUTHOR]

The singular differential Bessel operator with negative parameter is considered. Solutions to the singular differential Bessel equation are represented by linearly independent functions and , . Some properties of the functions , which are expressed in terms of the properties of the Bessel–Levitan j-function, are studied. Direct and inverse Bessel -transforms are introduced. Based on the -pseudoshift operator introduced earlier, a generalized -shift operator belonging to the Levitan class of generalized shifts commuting with the Bessel operator is constructed. A fundamental solution is found for the singular differential operator with a singularity at an arbitrary point on the semiaxis . [ABSTRACT FROM AUTHOR]