Erdelyi-Kober Operators Research Articles

Сompactness of one class of fractional integrating operator with variable upper limit

The paper establishes a characterization of the compactness for fractional operators of a general class, including the Riemann-Liouville, Hadamard and Erdelyi-Kober operators. The paper considers an integral fractional integration operator of Hardy type with nonnegative kernels and a variable limit of integration (a function as the upper limit of integration) and under certain conditions on the kernel, a criterion of the compactness in weighted Lebesgue spaces is obtained for this operator, when the parameters of the spaces satisfy the conditions Moreover, more general results are obtained for the weighted differential inequality of Hardy type on the set of locally absolutely continuous functions that vanish and infinity at the ends of the interval, covering the previously known results, and more precise estimates for the best constant are given. The localization method, Schauder’s theorem, the Kantorovich test, and the theorem on the uniform limit of compact operators were used in the proof of the main theorem. The obtained results of the study the compactness of fractional integration operators can be used in the estimation of solutions of differential equations that model various processes in mathematics. In particular, these results yield new results in the theory of Hardy-type inequalities.

Gellerstedt Type Problem for the Loaded Parabolic-Hyperbolic Type Equation with Caputo and Erdelyi–Kober Operators of Fractional Order

The work is devoted to the proof of the uniqueness and existence of solution to local and nonlocal problems with an integral gluing condition for a loaded parabolic-hyperbolic type equation with differential and integral operators of fractional order, in which the trace of the solution appears in the Erdelyi–Kober integral operator. Using the method of energy integrals, the uniqueness of the solution is proved, and the existence of the solution is proved by the method of integral equations.

Invariant analysis, conservation laws, and some exact solutions for (2+1)-dimension fractional long-wave dispersive system

The present study is devoted to investigate the (2+1)-dimensional time-fractional long-wave dispersive (fLWD) model, which describes the dynamical behaviors of shallow water waves propagate along two horizontal directions with fast memories. We first establish the symmetries and conservation laws for this model by improving the fractional Lie group approach and fractional Noether’s formulae. Then, using the general fractional Erdelyi–Kober operator, we reduce the original equations into a time-fractional integro-differential system. Finally, the invariant subspace method is applied to this model to obtain more explicit solutions and the dynamical behaviors are analyzed through numerical simulations. This work extends the basic method to solve the higher dimensional system with mixed order of integer and fractional derivatives.

The Cauchy Problem for the Iterated Klein–Gordon Equation with the Bessel Operator

An analogue of the Cauchy problem for an iterated multidimensional Klein–Gordon equation with a time-dependent Bessel operator is investigated. Applying the generalized Erdelyi–Kober operator of fractional order, we reduce the formulated problem to the Cauchy problem for the polywave equation. Applying a spherical mean method, we construct an explicit formula to solve this problem for the polywave equation; then, basing on this solution, we find an integral representation of the solution of the formulated problem. The obtained formula allows one to immediately discern the character of the dependence of the solution on the initial functions and, in particular, to establish conditions for the smoothness of the classical solution. The paper will be useful for specialists engaged in the resolving of problems of higher spin theory.

On the Cauchy Problem for the Iterated Generalized Two-axially Symmetric Equation of Hyperbolic Type

The Cauchy problem for an iterated generalized two-axially symmetric equation of hyperbolic type is investigated. Unlike traditional methods, the generalized Erdelyi–Kober operator of fractional order is applied to solve the problem. The application of this operator allows us to reduce the equations with the lowest term and with the singular Bessel operator acting in one of the variables to an equation without the Bessel operator acting on this variable and without the lowest term. The proposed approach makes it possible to construct an explicit formula for solving the formulated problem. These formulas express solutions of the Cauchy problem in terms of the sum of the initial functions under the action of powers of the Bessel operator. They allow us to directly see the character of the dependence of the solution on the initial functions.

Задачи типа Геллерстедта для нагруженного уравнения параболо-гиперболического типа с операторами Капуто и Эрдейли-Кобера дробного порядка

Задачи типа Геллерстедта для нагруженного уравнения параболо-гиперболического типа с операторами Капуто и Эрдейли-Кобера дробного порядка

Symmetry properties, conservation laws, reduction and numerical approximations of time-fractional cylindrical-Burgers equation

In this paper the Lie group analysis of the time-fractional cylindrical Burgers equation (time-FCB), which is a fundamental PDE occurring in various areas of applied mathematics, such as fluid mechanics, non-linear acoustics, gas dynamics, traffic flow and etc. is given. For this purpose the Riemann–Liouville derivative is used to implement the Lie algorithm for finding the symmetry operators. A reduced form of the equation is given by using the similarity variables obtained from a symmetry and Erdelyi–Kober operator. In the next step conservation laws are derived via a generalization of Noether’s theorem. Finally the Chebyshev wavelets for time-fractional differential equations (FDEs) is applied for solving the considered equation.

On One Method for the Solution of an Analog of the Cauchy Problem for a Polycaloric Equation with Singular Bessel Operator

We study an analog of the Cauchy problem for an inhomogeneous singular polycaloric (polyparabolic) equation with Bessel operator. By using the Erdelyi–Kober operator of the fractional order, we deduce an explicit formula for the solution of the formulated problem.

Solvability of a non-local problem with integral gluing condition for mixed type equation with Erdelyi-Kober operators

Solvability of a non-local problem with integral gluing condition for mixed type equation with Erdelyi-Kober operators

Boundedness and compactness of a class of Hardy type operators

We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out.

Abstract Euler–Poisson–Darboux equation with nonlocal condition

We find conditions for the unique solvability of nonlocal problems for abstract differential equation of the Euler–Poisson–Darboux equation. Nonlocal conditions contain either Erdelyi–Kober operator or Riemann–Liouville fractional integration operator.

On certain new CAUCHY–TYPE fracitioanl integral inequalities and OPIAL–TYPE fractional derivative inequalities

The aim of this paper is to establish several new fractional integral and derivative inequalities for non-negative and integrable functions. These inequalities related to the extension of general Cauchy type inequalities and involving Saigo, Riemann-Louville type fractional integral operators together with multiple Erdelyi-Kober operator. Furthermore the Opial-type fractional derivative inequality involving H-function is also established. The generosity of H-function could leads to several new inequalities that are of great interest of future research.

Thin-film superconducting rings in the critical state: the mixed boundary value approach

In this paper, we describe the critical state of a thin superconducting ring (and of a perfectly conducting ring as a limiting case) as a mixed boundary value problem. The disc is characterized by a three-part boundary division of the positive real axis, so this work is an extension of the procedure used in a previous work of ours for describing superconducting discs and strips, which are characterized by a two-part boundary division of the real axis. Here, we present the mathematical tools to solve this kind of problems—the Erdelyi–Kober operators—in a frame that can be immediately used. Contrary to the two-part problems considered in our previous work, three-part problems do not generally have analytical solutions and the numerical work takes on a significant heaviness. Nevertheless, this work is remunerated by three clear advantages: firstly, all the cases are afforded in the same way, without the necessity of any brilliant invention or ability; secondly, in the case of superconducting rings, the penetration of the magnetic field in the internal/external rims is a result of the method itself and does not have to be imposed, as it is commonly done with other methods presented in the literature; thirdly, the method can be extended to investigate even more complex cases (four-part problems). In this paper, we consider the cases of rings in uniform field and with transport current, with or without flux trapping in the hole and the case without net current, corresponding to a cut ring (washer), as used in some SQUID applications.

On extended M−series

This paper deals with extended M-series, which is extension of the generalized M-series [12]. Mittag-Leffler function, $\omega$ - hypergeometric function, generalized $\omega-$ Gauss hypergeometric function, $\omega-$ confluent hypergeometric function, Bessel-Maitland function can be deduced as special cases of our finding. Moreover, we obtain some theorem for extended M-series by using fractional calculus operators and many results associated with Riemann-Liouville, Weyl and ErdelyiKober operators. We begin our study from the following definitions.

A generalized multiparameter function of Mittag-Leffler type

In this paper, we define and study a generalized multiparameter function of Mittag-Leffler type. Some special cases and basic properties including its Mellin–Barnes-type integral representation, recurrence relation, and expansion are given. The Laplace transform and the first Sonine integral of the generalized function are evaluated. Differential and integral equations with the Erdelyi–Kober operator are established. The Cauchy problem for the multiparameter differential equation is formulated and its solution is obtained.

Fractional differintegral transformations of univalent Meijer's G-functions

In this article, the univalent Meijer's G-functions are classified into three types. Certain integral, differential or differintegral transformations preserving the univalence of the Meijer's G-functions, have been discussed. This classification and transformations are based on Kiryakova's studies in representing the generalized hypergeometric functions as fractional differintegral operators of three basic elementary functions. In fact, these transformations are the Erdelyi-Kober operators (m = 1) or their two-tuple compositions (for m = 2) known also as hypergeometric fractional differintegrals. A number of new univalent Meijer's G-functions can be obtained by successive applications of such transformations, being operators of the generalized fractional calculus (GFC). Some new relations are then interpreted for the starlike, convex, and positive real part functions in terms of Meijer's G-functions.

Formal Solutions of Certain Dual and Triple Series Involving the Fox H-Functions

It this paper we show how certain classes of dual (triple) series equations involving the Fox H-functions can be reduced to some dual (triple) integral equations involving the Fox H-functions. Also, using the Erdelyi-Kober operators of fractional integration exact solutions of these dual (triple) integral equations are obtained.

Stochastic processes and special functions: On the probabilistic origin of some positive kernels associated with classical orthogonal polynomials

We describe and investigate a class of Markovian models based on a form of “dynamic occupancy problem” originating in statistical mechanics. The most fundamental of these gives rise to a transition-probability matrix over ( N + 1) discrete states, which proves to have the Hahn polynomials as eigenvectors. The structure of this matrix, which is a convolution of two negative hypergeometric distributions, leads to a factorization into finite-difference sumoperators having forms analogous to the Erdelyi-Kober operators for the continuous variable. These make possible the exact solution of the corresponding eigenvalue problem and hence the spectral representation of the transition matrix. By taking suitable limits, further families of Markov processes can be generated having other classical polynomials as eigenvectors; these, like the polynomials, inherit their properties from the original Hahn system. The Meixner, Jacobi and Laguerre systems arise in this way, having their origin in variants of the basic model. In the last of these cases, the spectral resolution of the continuous transition kernel proves to be identical with Erdelyi's (1938) bilinear formula, which is thus both generalized and given a physical interpretation. Various symmetry and “duality” properties are explored and a number of interesting formulas are obtained as by-products. The use of statistical models to generate kernels, which are thereby guaranteed to be both positive and positive definite, appears to be mathematically fruitful, while the models themselves seem likely to have application to a variety of topics in applied probability.

A theory of fractional integration for generalised functions II

SynopsisIn a previous paper [2], a theory of fractional integration was developed for certain spaces Fp,μ of generalised functions. In this paper we extend this theory by relaxing some of the restrictions on the various parameters involved. In particular we show how a generalised Erdelyi-Kober operator can be defined on Fʹp,μ for 1 ≦ p ≦ ∞ and for all complex numbers μ except for those lying on a countable number of lines of the form Re μ = constant in the complex μ-plane. Mapping properties of these generalised operators are obtained and several applications mentioned.

Some triple integral equations

φ(s>> *} = 0, 0 Φ(s); x] = Ux a η, δ > 0, 0, are real parameters, /2O&) is a known function, Φ(s) is to be determined and (3) m{h(x); s} = H(s), Tl^His); x) = h(x) , denote the Mellin transform of h(x) and its inversion formula respectively. The above equations are an extension of the dual integral equations solved in a recent paper by Erdelyi [2] by means of a systematic application of the Erdelyi-Kober operators of fractional integration [4]. Using the properties of some slightly extended forms of the Erdelyi-Kober operators we show, in a purely formal manner, that the solution of the triple integral equations can be expressed in terms of the solution of a Fredholm integral equation of the second kind. Srivastav and Parihar [5] have solved a very special case of the equations by a completely different method from that used in this paper. The method of solution employed here will be seen to follow closely that used by Cooke [1] to obtain the solution to some triple integral equations involving Bessel functions; indeed Cooke's equations may be regarded as a special case of equations (1) and (2) and it is shown that a solution of his equations can be readily obtained from that presented in this paper.