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This study aims to investigate the properties of fractional calculus theory (FCT) in the complex domain. We focus on the relationship between the theories of special functions (SFT) and FCT, which have seen recent advancements and have led to various successful applications in fields such as engineering, mathematics, physics, biology, and other allied disciplines. Our main contribution is the development of a special function, specifically the confluent hypergeometric function (CHF) on the complex domain. By deriving various implementations of fractional order derivatives and integral operators using this function, we present a new class of special functions combining certain cases of Mittag-Leffler and confluent hypergeometric functions. Moreover, a new numerical technique for solving linear and nonlinear multi-order fractional differential equations has been developed using the proposed class of functions and the point collocation method. Graphical results are shown to demonstrate the efficacy of this proposed technique and its applicability. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of fractional functions based on Chelyshkov wavelets for solving one- and two-variable distributed-order fractional differential equations. The concept of fractional derivative is utilized in the Caputo sense. The idea of solving these problems is based on fractional integral operator of fractional-order Chelyshkov wavelets with composite collocation method. This operator and collocation method are utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations. The convergence of the fractional-order Chelyshkov wavelets bases is discussed. The efficiency and the applicability of the new methodology are illustrated by eight examples, in addition our findings in comparison with the existing results show the advantage of our method. [ABSTRACT FROM AUTHOR]

Abstract Unsteady nonlinear boundary layer convection flows and heat transfer of a fractional Maxwell fluid near a vertical plate with constant thermal flux are studied. The fractional constitutive equations for the shear stress and thermal flux are formulated for the first time with the integral time-fractional operator of type Caputo–Fabrizio. These types of fractional relationships provide a weighted average exponential to the velocity gradient and temperature gradient. The studied model is constituted from a system of nonlinear partial differential equations coupled with fractional differential equations with initial and boundary conditions. The solutions for the shear stress, velocity, thermal flux and temperature are obtained by using the Laplace transform along with a suitable transformation of variables. In order to finding solutions of the studied model, two inverse Laplace transforms of exponential type are obtained. The particular cases when only the stress or thermal flux are damped, as well as the convection flows of the ordinary fluid are obtained as particular cases when the stress/thermal fractional parameter tends to 1. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the definition of a generalized fractional mixed Morrey space L p → , η , φ (R n) , where p → = (p 1 , p 2 , ⋯ , p n) , 1 < p → ≤ ∞ and φ : (0 , ∞) → (0 , ∞) is an increasing function satisfying certain doubling conditions. Second, we prove that the Calderón-Zygmund integral operator T and its commutator [ b , T ] which is generated by T and b ∈ BMO (R n) are bounded from spaces L p → , η , φ (R n) into spaces L p → , η , φ (R n) ; furthermore, we prove that the fractional integral operator I α and its commutator [ b , I α ] formed by b ∈ BMO (R n) and I α are bounded from the spaces L p → , η , φ (R n) into spaces L q → , η − α , φ (R n) , where 1 < p → < q → < ∞ , ∑ j = 1 n 1 q j = ∑ j = 1 n 1 p j − α and ∑ j = 1 n 1 p j ≥ η > α > 0. Finally, the boundedness for the fractional maximal operator M α , the commutator [ b , M α ] associated with BMO functions and the commutator [ b , T ] formed by T and b ∈ Lip β (R n) on spaces L p → , η , φ (R n) is established, respectively. [ABSTRACT FROM AUTHOR]

Image denoising is an important task in image processing. The interest in using a fractional mask window operator based on fractional calculus has grown for image denoising. This paper mainly introduces the concept of fractional calculus and proposes a new mathematical method in using fractional Alexander polynomials for image denoising. The structures of n × n fractional mask windows on eight directions of this algorithm are constructed. Finally, we measure the denoising performance by employing experiments based on visual perception and by using peak signal-to-noise ratios. The experiments illustrate that the improvements achieved are compatible with other standard smoothing filters. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, we introduce the class of -valent prestarlike functions of order and type with negative coefficients defined by a Sălăgean operator. Extreme points, integral operators and distortion theorems of this class are obtained. We also obtain several results for the radius of starlikeness, convexity and modified Hadamard products of functions belonging to this class. [Copyright &y& Elsevier]

Abstract: In this paper we give an analytical treatment of a fractional wave equation with Caputo time fractional derivative and frictional memory kernel of Mittag-Leffler type. This problem generalizes a recently solved problem of a wave equation for a vibrating string in presence of a fractional friction with power-law memory kernel. Such equations can be used in the context of modeling processes in complex and viscoelastic media. [Copyright &y& Elsevier]

Abstract: In this paper we give an analytical treatment of a wave equation for a vibrating string in the presence of a fractional friction with power-law memory kernel. The exact solution is obtained in terms of the Mittag-Leffler type functions and a generalized integral operator containing a four parameter Mittag-Leffler function in the kernel. The method of separation of the variables, Laplace transform method and Sturm–Liouville problem are used to solve the equation analytically. The asymptotic behaviors of the solution of a special case fractional differential equation are obtained directly from the analytical solution of the equation and by using the Tauberian theorems. The proposed model may be used for describing processes where the memory effects of complex media could not be neglected. [Copyright &y& Elsevier]

Abstract: Recently, Youssef constructed a new theory of fractional order generalized thermoelasticity by taking into account the theory of heat conduction in deformable bodies, which depends upon the idea of the Riemann–Liouville fractional integral operator. In this paper, the variational theorem is obtained for the generalized thermoelasticity model for a homogeneous and isotropic body. [Copyright &y& Elsevier]