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The prime purpose of this article aims to establish connections between various sub-classes of harmonic univalent mappings by applying a convolution operator associated with the well-known Wright function. In particular, the inclusion relations among Goodman–Rønning-type harmonic univalent mappings, harmonic k-uniformly convex mappings and harmonic k-uniformly starlike mappings in the plane are established. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present integrals form of Wright function associated with algebraic function, special functions. Results are key to the analysis of the Wright function in various type of integrals. [ABSTRACT FROM AUTHOR]

In this present work we derived integral transforms such as Euler transform, Laplace transform, and Whittaker transform of K4-function. The results are given in generalized Wright function. Some special cases of the main result are also presented here with new and interesting results. We further extended integral transforms derived here in terms of Gauss Hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, we consider fractional diffusion equation involving the Bessel operator acting with respect to a spatial variable and the Riemann-Liouville fractional differentiation operator acting with respect to a time variable. When the order of the fractional derivative is unity, and the singularity of the Bessel operator is absent, this equation coincides with the classical heat equation. Earlier, a solution of the Cauchy problem has been considered for the considered equation and a uniqueness theorem has been proved for a class of functions satisfying the analog of the Tikhonov condition. In this paper, we have constructed an example to show that the exponent (power) at the condition of the uniqueness of the solution to the Cauchy problem cannot be raised under. Its increase leads to a non-uniqueness of the solution. Using the well-known properties of the Wright function, we have obtained estimates for constructed function, which satisfies the homogeneous equation and the zero Cauchy condition.

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book Fractional Calculus and Waves in Linear Viscoelasticity (2010)], Fσ(x) = ∑n=0∞(−x)n/n!Γ(−nσ), Mσ(x) = ∑n=0∞ (−x)n/n!Γ(−n σ + 1 − σ) (0 < σ < 1) for x → ± ∞ are presented. The situation corresponding to the limit σ → 1− is considered, where Mσ(x) approaches the Dirac delta function δ(x − 1). Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as σ → 1−. [ABSTRACT FROM AUTHOR]

In this paper, we give some sufficient conditions for the univalence of some integral operators. For this, we use the Becker's and generalized version of the well known Ahlfor's and Becker's univalence criteria. [ABSTRACT FROM AUTHOR]

Many authors have established various integral and differential formulas involving different special functions in recent years. In continuation, we explore some image formulas associated with the product of Srivastava’s polynomials and extended Wright function by using Marichev–Saigo–Maeda fractional integral and differential operators, Lavoie–Trottier and Oberhettinger integral operators. The obtained outcomes are in the form of the Fox–Wright function. It is worth mentioning that some interesting special cases are also discussed.

In this paper, we introduced the Mainardi beam and indicated its attributes under the Fractional Fourier transform for power variations of Fractional Fourier transform. The results represent that the behavior of the Mainardi beam is similar to that of the Airy beam. The obtained formula is a very powerful tool to describe propagation of a Mainardi beam through the FFT and the FrFT systems. An analytical expression of the Mainardi beam passing through an Fractional Fourier transform system presented. The influences of the Fractional Fourier transform, rational order of the Mittag-Leffler function (Fourier transform of the Mainardi function) on the normalized intensity distribution and characteristics of the Mainardi beam in the Fractional Fourier transform system examined. Power of the Fractional Fourier transform (p) and rational order of the Mittag-Leffler function (q) control characteristics of the Mainardi beam such as effective beam size, number, width, height, and orientation of the beam spot.

In the present paper, we investigate certain geometric properties and inequalities for the Wright function and mention a few important consequences of our main results. A nonlinear differential equation involving the Wright function is also investigated.

In this paper, we examine properties of an entire function that is a generalization of the Wright function. Various representations, estimates, and differentiation formulas are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we find an integral representation for the fundamental solution of the fractional Ostrovsky equation in terms of the Airy and Bessel-Wright functions. The equation is studied in the sense of the Weyl fractional derivative and the solution is presented as the Airy transforms of Wright functions. Using the asymptotic expansion of Wright function the asymptotic behavior of solution is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we are finding a solution of the fractional Wave-Schrodinger equation by Laplace transform in the sense of Caputo fractional derivative. It was found that the fundamental solution of the equation is related to Wright function. [ABSTRACT FROM AUTHOR]

We consider the partial differential equation of a mathematical model proposed by Sharmaet al. [1] to describe the concentration of nutrients in blood, a factor which influences erythrocytesedimentation rate. Introducing in it a fractional derivative in the Caputo sense, we create a new, timefractionalmathematical model which contains, as a particular case, the original model. We obtainan analytic solution of this time-fractional partial differential equation in terms of Mittag-Leffler andWright functions and to show that our model is more realistic than the Sharma model.

The asymptotic expansion for x→±∞ of the entire function Fn,σ(x;μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk),γk=(k+1)π2n for μ>0, 0<σ<1 and n=1,2,… is considered. In the special case σ=α/(2n), with 0<α<1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn,σ(x;μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained.

The theory of diffusive stresses based on the time-fractional diffusion equation with the Liouville–Caputo derivative is discussed. The diffusion process is characterized by the chemical potential tensor and the concentration tensor. Eliminating the chemical potential tensor and the concentration tensor from the constitutive equations for the stress tensor, the space–time-nonlocal equations for the mean stress and the deviatoric stress are obtained with the kernel being the Wright function. [ABSTRACT FROM AUTHOR]

In this paper, using the Mellin transform of Wright function we derive an addition formula for the Wright function. In some special cases, addition formulas for the Hermite, Bessel and Mittag-Leffler functions are also given and the Green's function of two-dimensional time-fractional diffusion equation is presented in the whole plane. [ABSTRACT FROM AUTHOR]

In this investigation our main aim is to determine the radii of uniform convexity of selected normalized q -Bessel and Wright functions. Here we consider six different normalized forms of q -Bessel functions and we apply three different kinds of the normalization of the Wright function. We also show that the obtained radii are the smallest positive roots of some functional equations. [ABSTRACT FROM AUTHOR]

The Wright function is a generalization of the exponential function and the Bessel functions. Integral relations between the Mittag–Leffler functions and the Wright function are presented. The applications of the Wright function and the Mainardi function to description of diffusion, heat conduction, thermal and diffusive stresses, and nonlocal elasticity in the framework of fractional calculus are discussed.

This paper deals with GPU computing of special mathematical functions that are used in Fractional Calculus. The graphics processing unit (GPU) has grown to be an integral part of nowadays’s mainstream computing structures. The special mathematical functions are an integral part of Fractional Calculus. This paper deals with a novel parallel approach for computing special mathematical functions used in Fractional Calculus. NVIDIA’s GPU hardware is used to speed up the parallel algorithm. A comparison of the sequential code, vectorized code and GPU code is performed. We have successfully reduced the computation time of special mathematical functions using the parallel computing capabilities of GPU.

In this paper, a fractional model for the computation of temperature and heat flux distribution in a semi-infinite solid is discussed which is subjected to spatially decomposing, time-dependent laser source. The apt dimensionless parameters are identified and the reduced temperature and heat flux as a function of these parameters are presented in a numerical form. Some special cases of practical interest are also discussed. The solution is derived by the application of the Laplace transform, the Fourier sine transform and their derivatives. Also, we developed an alternative solution of it by using the Sumudu transform, the Fourier transform and their derivatives. These results are received in compact and graceful forms in terms of the generalized Mittag-Leffler function, which are suitable for numerical computation.

In this paper Cauchy problem for a parabolic equation with Bessel operator and with Riemann-Liouville partial derivative is considered. The representation of the solution is obtained in terms of integral transform with Wright function in the kernel. It is shown that when this equation becomes the fractional diffusion equation, obtained solution becomes the solution of Cauchy problem for the corresponding equation. The uniqueness of the solution in the class of functions that satisfy the analogue of Tikhonov condition is proved.

Fundamental solution of the model equation of anomalous diffusion with Riemann-Liouville operator is constructed. Using the properties of the integral transformation with Wright function in kernel, we give estimates for the fundamental solution. When the considered equation transformes into the diffusion equation of fractional order, constructed fundamental solution goes into the corresponding fundamental solution of the diffusion equation of fractional order. General solution of the model equation of anomalous diffusion of fractional order is constructed.

We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0; t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting times are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N(t) and the Erlang process t(N) yields as diffusion limits the inverse stable and the stable subordinator, respectively.

Background and objectives The MRI brain tumors segmentation is challenging due to variations in terms of size, shape, location and features’ intensity of the tumor. Active contour has been applied in MRI scan image segmentation due to its ability to produce regions with boundaries. The main difficulty that encounters the active contour segmentation is the boundary tracking which is controlled by minimization of energy function for segmentation. Hence, this study proposes a novel fractional Wright function (FWF) as a minimization of energy technique to improve the performance of active contour without edge method. Method In this study, we implement FWF as an energy minimization function to replace the standard gradient-descent method as minimization function in Chan–Vese segmentation technique. The proposed FWF is used to find the boundaries of an object by controlling the inside and outside values of the contour. In this study, the objective evaluation is used to distinguish the differences between the processed segmented images and ground truth using a set of statistical parameters; true positive, true negative, false positive, and false negative. Results The FWF as a minimization of energy was successfully implemented on BRATS 2013 image dataset. The achieved overall average sensitivity score of the brain tumors segmentation was 94.8 ± 4.7%. Conclusions The results demonstrate that the proposed FWF method minimized the energy function more than the gradient-decent method that was used in the original three-dimensional active contour without edge (3DACWE) method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a space fractional negative binomial process (SFNB) by time-changing the space fractional Poisson process by a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Lévy process and the corresponding Lévy measure is given. Extensions to the case of distributed order SFNB, where the fractional index follows a two-point distribution, are investigated in detail. The relationship with space fractional Polya-type processes is also discussed. Moreover, we define and study multivariate versions, which we obtain by time-changing a d-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications to population’s growth and epidemiology models are explored. Finally, we discuss algorithms for the simulation of the SFNB process. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to find the radii of starlikeness and convexity of the normalized Wright functions for three different kinds of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for Wright function and properties of real zeros of the Wright function and its derivative. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized Wright functions. The main results of the paper are natural extensions of some known results on classical Bessel functions of the first kind. Some open problems are also proposed, which may be of interest for further research. AMS subject classifications: 30C45, 30C15, 33C10 [ABSTRACT FROM AUTHOR]

In this note, using the Mellin transform and fractional exponential operators we find a closed-form solution for space-fractional integro-differential Whitham equation. Moreover, by the proposed method we get the solution of space-fractional Benjamin-Ono equation. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to present analytical solution of fractional differential equations. The fractional derivative is considered in Caputo sense. The results are derived by the application of Sumudu transform in term of Mittag-Leffler function, which are suitable for numerical computation. The results obtain by the Sumudu transform method indicate that the approach is easy to implement and computationally very attractive.

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized Erdélyi-Kober fractional differential and integral operators is demonstrated and discussed.

Poisson-Tweedie mixtures are the Poisson mixtures for which the mixing measure is generated by those members of the family of Tweedie distributions whose support is non-negative. This class of non-negative integer-valued distributions is comprised of Neyman type A, back-shifted negative binomial, compound Poisson-negative binomial, discrete stable and exponentially tilted discrete stable laws. For a specific value of the 'power' parameter associated with the corresponding Tweedie distributions, such mixtures comprise an additive exponential dispersion model. We derive closed-form expressions for the related variance functions in terms of the exponential tilting invariants and particular special functions. We compare specific Poisson-Tweedie models with the corresponding Hinde-Demétrio exponential dispersion models which possess a comparable unit variance function. We construct numerous local approximations for specific subclasses of Poisson-Tweedie mixtures and identify Lévy measure for all the members of this three-parameter family. [ABSTRACT FROM AUTHOR]

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions. [ABSTRACT FROM AUTHOR]

We investigate the second boundary-value problem in the half-strip for a parabolic equation with the Bessel operator and Riemann-Liouville partial derivative. In terms of the integral transformation with theWright function in the kernel, we find the representation of a solution in the case of zero edge condition. We prove the uniqueness of a solution in the class of functions satisfying an analog of the Tikhonov condition. [ABSTRACT FROM AUTHOR]

In this paper our main aim is to give some sufficient conditions for functions represented with normalized Wright functions to be univalent in the open unit disk. The key tools in our proofs are the Becker's and the generalized version of the well-known Ahlfor's and Becker's univalence criteria. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to investigate some characterization for the functions represented with normalized Wright functions to be in the subclasses S*(α,β) and (α,β)αβ∊[0,1)of analytic functions in the open unit disk. Sufficient conditions are obtained for these functions to be in these classes. [ABSTRACT FROM AUTHOR]

An operator associated with the Wright function is introduced in the open unit disk. Differential subordination and superordination, results associated with this operator are obtained by investigating appropriate classes of admissible functions. In particular, some inequalities for modified Bessel functions are also obtained. [ABSTRACT FROM AUTHOR]

The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0< r < R) and a matrix (R

In this paper, we provide solutions to the general fractional Caputo-type differential equation models for the dynamics of a sphere immersed in an incompressible viscous fluid and oscillatory process with fractional damping using Laplace transform method. We study the effects of fixing one of the fractional indices while varying the other as particular examples. We conclude this article by explaining the dynamics of the solutions of the models.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

This paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory. We establish new asymptotic properties of this function under numerous assumptions on the ordering of the parameter and variable. Several representations involving well-known special functions are given. We also provide some integral representations and structural properties involving the 'reduced' Wright function. Some of the properties established imply a reflection principle that connects the 'reduced' Wright functions with the opposite values of the parameter and certain Bessel functions. Several asymptotic relations for both particular cases of this function are also given. [ABSTRACT FROM AUTHOR]