We aim to give closed integral form expressions for the Mathieu-type (a, A; α) -series and their alternating counterparts, which contain fractional powers special functions as generalized Fox-Wright hypergeometric function a_pΨ_q in their terms, together with their contiguous representations with respect to the additional parameters. Particular cases of a_pΨ _q are specified, including the Le Roy–type functions and their multi–index analogues and other special functions. The corresponding results are illustrated with important special cases.
In this work, delay stochastic fractional reaction-diffusion equations driven by standard Brownian motion and fractional Brownian motion are considered. The first result is devoted to the existence, uniqueness, and regularity of the solution in different functional spaces, where the solution is considered in the generalized space L^p(Ω ,U_γ ) instead of L^2(Ω ,U) as usual. Besides considering the common case of the Hurst parameter, namely h ∈ (1/2,1), , we extend the result to the case h ∈ (0, 1/2). The second result is devoted to the convergence of the solution of the fractional model to the solution of the corresponding classical model when the fractional order α→ 1^- , which is of importance in showing the connection of the two models. Due to the aforementioned novelties, different techniques and methods used for the innovative points are required, including designing new boundedness and continuity properties for the fractional solution operator, applying the Burkholder-Davis-Gundy inequality instead of the Itô isometry, and estimating the Wiener integral directly from its explicit representation instead of relying on common tools like those used in the case h ∈ (1/2,1).
Contact problems involving fractional-order derivatives are an emerging area of research in contact mechanics because of their wide range of applications in science and engineering. The incorporation of fractional-order derivatives into contact models not only provides a more accurate description of the phenomena occurring at contact surfaces but also offers greater flexibility in modeling the mechanical behavior of materials. In this article, we first introduce the basic preliminary results that are essential for the analysis of fractional contact problems and discuss the various types of boundary conditions at the contact surface. We then present a brief overview of recent developments in the theory and analysis of fractional contact problems. These include fractional viscoelastic contact problems with adhesion, fractional thermo-viscoelastic contact problems, and fractional electro-viscoelastic contact problems. Finally, we present concluding remarks and draw conclusions based on the existing literature.
This study extends the known isomorphism associated with integer-order Laguerre-type operators to the fractional-order setting via the Riemann-Liouville integral operator. The resulting framework yields a Rodrigues-type operational representation for Prabhakar functions of the Le Roy type and leads to two new polynomial families, namely the Le Roy-type Laguerre and Konhauser polynomials. Several properties of these polynomials and the associated functions are investigated, including operational and integral representations, generating functions, Laplace transforms, and transformation formulas under the iterated fractional integral and derivative operators constructed from the Riemann-Liouville operators. A multidimensional iterated integral operator whose kernel involves the same function family is then introduced. Its action on power and exponential functions, semigroup property, and a corresponding left-inverse fractional derivative operator are established. Finally, the developed operator is applied to a multidimensional Abel-type fractional integral equation, and its solution is obtained in an explicit operator form through a Neumann-series expansion.
