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We investigate the log-concavity on the half-line of the Wright function ϕ (- α , β , - x) , in the probabilistic setting α ∈ (0 , 1) and β ≥ 0. Applications are given to the construction of generalized entropies associated to the corresponding Mittag-Leffler function. A natural conjecture for the equivalence between the log-concavity of the Wright function and the existence of such generalized entropies is formulated. The problem is solved for β ≥ α and in the classical case β = 1 - α of the Mittag-Leffler distribution, which exhibits a certain critical parameter α ∗ = 0.771667... defined implicitly on the Gamma function and characterizing the log-concavity. We also prove that the probabilistic Wright functions are always unimodal, and that they are multiplicatively strongly unimodal if and only if β ≥ α or α ≤ 1 / 2 and β = 0. [ABSTRACT FROM AUTHOR]

This research investigates specific classes of fractional integro-differential equations using a straightforward fractional calculus technique. The employed methodology yields a variety of compelling outcomes, including a generalized version of the well-established classical Frobenius method. The approach presented in this study primarily relies on fundamental theorems concerning the specific solutions of fractional integro-differential equations, utilizing the Mohand transform and binomial series extension coefficients. Additionally, advanced techniques for solving fractional integro-differential equations effectively are showcased. [ABSTRACT FROM AUTHOR]

Using the Hadamard factorization, the exponential radii of starlikeness and convexity for various special functions like Wright function, Lommel function, Struve function, Ramanujan type entire function, cross product and product of Bessel function have been investigated. For certain ranges of the parameters appearing in these special functions, the precise values of the exponential radii of starlikeness and convexity are calculated as the solutions of transcendental equations. The interlacing property of the zeros of special functions and their derivatives is the fundamental technique utilized to demonstrate these results. [ABSTRACT FROM AUTHOR]

A boundary value problem of the Bitsadze-Samarskii type is studied in the article for a fractionalorder diffusion equation and a degenerate hyperbolic equation with singular coefficients at lower terms in an unbounded domain. The article considers a mixed domain where the parabolic part of the domain under consideration coincides with the upper half-plane and the hyperbolic part is bounded by two characteristics of the equation under consideration and a segment of the abscissa axis. The uniqueness of the solution to the problem under consideration is proven by the method of energy integrals. The existence of a solution to the problem under consideration is reduced to the concept of solvability of a fractional-order differential equation. An explicit form of the solution to the modified Cauchy problem is given in the hyperbolic part of the mixed domain under consideration. Using this solution, due to the boundary condition of the problem, the main functional relationship between the traces of the unknown function brought to the interval of the degeneracy line of the equation is obtained. Further, using the representation of the solution of the diffusion equation of fractional order, the second main functional relationship between the traces of the sought-for function on the interval of the abscissa axis from the parabolic part of the considered mixed domain is obtained. Through the conjugation condition of the problem under study, an equation with fractional derivatives is obtained from two functional relationships by eliminating one unknown function; its solution is written out in explicit form. In the study of the boundary value problem, generalized fractional integro-differentiation operators with the Gauss hypergeometric function are employed. The properties of the Wright and Mittag-Leffler type functions are extensively utilized in the study.

A new insight is presented into the solution of the erythrocyte sedimentation rate (ESR) model, based on fractional derivative with respect to time, with non‐zero uniform velocity of blood by incorporating the concentration gradient of the blood nutrients. An analytical solution is acquired for the modified ESR fractional model in addition to presenting some new interesting results. The best possible suitable range for the concentration gradient is found for the model whose use will be helpful in approximating the clinical data from laboratory tests in a profound and accurate manner and also in diagnosing the ESR rate more accurately. Further, an appropriate range is proposed for the uniform velocity of blood as well as the fractional order of the time derivative to construct the feasible model. In addition, it is also shown what value of the fractional order gives a closer resemblance to the clinical data. Validation and verification of the obtained results against earlier results demonstrate the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass O μ (Δ 1 , Δ 2 , Δ 3 , Δ 4) of univalent functions defined in the unit disk Λ = { τ ∈ C : τ < 1 } . More specifically, we investigate some sufficient conditions for Rabotnov functions, Poisson functions, Bessel functions and Wright functions to be in this subclass. Some corollaries of our main results are given. The novelty and the advantage of this research could inspire researchers of further studies to find new sufficient conditions to be in the subclass O μ (Δ 1 , Δ 2 , Δ 3 , Δ 4) not only for the aforementioned special functions but for different types of special functions, especially for hypergeometric functions, Dini functions, Sturve functions and others. [ABSTRACT FROM AUTHOR]

In this article, a representation of the solution of the Cauchy problem for an equation of high even order with a fractional derivative in the Riemann-Liouville sense of order greater than one is obtained. The uniqueness of the solution is proved. [ABSTRACT FROM AUTHOR]

In the article, the boundary value problem for the wave equation with a fractional time derivative and with initial conditions specified in the form of a fractional derivative in the Riemann-Liouville sense is solved. The definition domain of the desired function is the upper half-plane (x,t). To solve the problem, the Fourier transform with respect to the spatial variable was applied, then the Laplace transform with respect to the time variable was used. After applying the inverse Laplace transform, the solution to the transformed problem contains a two-parameter Mittag-Leffler function. Using the inverse Fourier transform, a solution to the problem was obtained in explicit form, which contains the Wright function. Next, we consider limiting cases of the fractional derivative’s order which is included in the equation of the problem.

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function. [ABSTRACT FROM AUTHOR]

In the present paper we have given an extension of the Generalized Hurwitz-Lerch Zeta Function фα,γ;δ(z, s, p). Using the Generalized Wright Function Wα,βγ,δ(z) we have obtained two integral representations, a summation formula and a differential formula for the newly introduced function. All results are given in the form of theorems. we have also discussed the corollaries of one of our main theorems. To strengthen our main results, we have also considered some important special cases. [ABSTRACT FROM AUTHOR]

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass Oμ(Δ1,Δ2,Δ3,Δ4) of univalent functions defined in the unit disk Λ={τ∈C:τ<1}. More specifically, we investigate some sufficient conditions for Rabotnov functions, Poisson functions, Bessel functions and Wright functions to be in this subclass. Some corollaries of our main results are given. The novelty and the advantage of this research could inspire researchers of further studies to find new sufficient conditions to be in the subclass Oμ(Δ1,Δ2,Δ3,Δ4) not only for the aforementioned special functions but for different types of special functions, especially for hypergeometric functions, Dini functions, Sturve functions and others.

We prove estimates at infinity of convolutions fn★$f^{n\star }$ and densities of the corresponding compound Poisson measures for a class of radial decreasing densities on Rd$\mathbb {R}^d$, d≥1$d \ge 1$, which are not convolution equivalent. Existing methods and tools are limited to the situation in which the convolution f2★(x)$f^{2\star }(x)$ is comparable to initial density f(x)$f(x)$ at infinity. We propose a new approach, which allows one to break this barrier. We focus on densities, which are products of exponential functions and smaller order terms—they are common in applications. In the case when the smaller order term is polynomial, estimates are given in terms of the generalized Bessel function. Our results can be seen as the first attempt to understand the intricate asymptotic properties of the compound Poisson and more general infinitely divisible measures constructed for such densities. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to obtain inclusion relations between various subclasses of harmonic univalent mappings by applying a convolution operator involving generalized Wright functions. To be more precise, we investigate such connections with Goodman-Rønning-type harmonic univalent functions, k-uniformly harmonic convex functions and k-uniformly harmonic starlike functions in the open unit disc U. Some of our results generalize and correct the results of Maharana and Sahoo [11]. [ABSTRACT FROM AUTHOR]

In this paper, we examine the number of ways to partition an integer n into k th powers when n is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large n , and the stronger conjectures of Ulas are proved. The asymptotics of Wright's generalized Bessel functions are also treated. [ABSTRACT FROM AUTHOR]

This paper analyzes the system describing the absorption solar collector as an application of nanoparticles for the storage of solar energy. The system involves two fractional partial differential equations (FPDEs) utilizing the Caputo fractional definition (CFD). Explicit solutions are determined for the temperature and velocity in terms of the Wright function (WrFn) via Laplace transform (LT). In addition, the solutions are expressed in terms of some well–known special functions at selected values of the fractional–order. Moreover, the behaviors of the temperature and velocity are investigated graphically using the thermo–physical data of the Cu/Al2O3–nanoparticles. Furthermore, some numerical results are conducted about the performance of the Cu and Al2O3. The results reveal the higher efficiency/performance of the Cu–nanoparticles over the Al2O3 and thus copper enjoys higher capability than alumina for the purpose of solar energy storage. In addition, it is found that 1.41% in the enhancement of heat transfer is achieved by adding 1% of the Cu–nanoparticles while the corresponding enhancement rate of the other three nanoparticles was less. Besides, the enhancement rate reaches 7.1% by increasing the volume fraction of the Cu–nanoparticles to 5%. Furthermore, the enhancement rates for the TiO2, Ag, and Al2O3 are 5.41%, 6.47%, and 6.53%, respectively when using 5% of these nanoparticles. Finally, the advantages/effectiveness of the current approach over a previous work in the literature are discussed in detail.

In this paper we consider a boundary value problem for a loaded fractional diffusion equation. The loaded term has the form of the Riemann-Liouville fractional derivative or integral. The BVP is considered in the open right upper quadrant. The problem is reduced to an integral equation that, in some cases, belongs to the pseudo-Volterra type, and its solvability depends on the order of differentiation in the loaded term and the behavior of the support line of the load in a neighborhood of the origin. All these cases are considered. In particular, we establish sufficient conditions for the unique solvability of the problem. Moreover, we give an example showing that violation of these conditions can lead to nonuniqueness of the solution. [ABSTRACT FROM AUTHOR]

Our main goal in this article is to investigate the Hyers–Ulam–Rassias stability (HURS) for a type of integral equation called Volterra integral equation with delay (VIE-D). First, by considering special functions such as the Wright function (WR), Mittag–Leffler function (ML), Gauss hypergeometric function (GH), H -Fox function (H -F), and also by introducing the aggregation function, we select the best control function by performing numerical calculations to investigate the stability of the desired equation. In the following, using the selected optimal function, i.e., the minimum function, we prove the existence of a unique solution and the HURS of the VI-D equation in the matrix-valued fuzzy space (MVFS) with two different intervals. At the end of each section, we provide a numerical example of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we apply some special functions to introduce a new class of control functions that help us define the concept of multi-stability. Further, we investigate the multi-stability of homomorphisms in C ∗ $C^{*}$ -algebras and Lie C ∗ $C^{*}$ -algebras, multi-stability of derivations in C ∗ $C^{*}$ -algebras, and Lie C ∗ $C^{*}$ -algebras for the following random operator equation via fixed point methods: μ f ( ð , x + y 2 ) + μ f ( ð , x − y 2 ) = f ( ð , μ x ) . $$ \mu f \biggl(\eth , \frac{x+y}{2} \biggr) + \mu f \biggl(\eth , \frac{x-y}{2} \biggr) = f(\eth , \mu x) . $$ In particular, for μ = 1 $\mu = 1$ , the above equation turns out to be Jensen’s random operator equation.

В работе исследуется параболическое уравнение в частных производных с дробным дифференцированием по одной из двух независимых переменных, ассоциируемой со временем. Такие уравнения принято относить к классу уравнений дробной диффузии. Оператор дробного дифференцирования представляет собой линейную комбинацию двух операторов Джрбашяна-Нерсесяна. Основным результатом работы является теорема об общем представлении регулярных решений исследуемого уравнения в бесконечной полосе. В терминах функции Райта построено фундаментальное решение и изучены его основные свойства. В частности, доказаны формулы дробного дифференцирования, исследовано асимптотическое поведение и получены оценки для фундаментального решения и его производных при больших и малых значениях автомодельной переменной, доказана его положительность. Для построения общего решения использован метод функции Грина, адаптированный к уравнениям, содержащим операторы Джрбашяна-Нерсесяна. К частным случаям рассматриваемого уравнения относятся уравнения с производными Римана-Лиувилля и Герасимова-Капуто. Поэтому полученные результаты остаются справедливыми и для уравнений с этими операторами дробного дифференцирования и их комбинациями.

Nanofluids are gaining extensive attention due to their thermo‐physical properties in technological and industrial fields for controlling the effects of heat transfer. Classical nanofluid studies are generally confined to models described by partial differential equations of an integer order where the memory effect and hereditary properties of materials are neglected. In order to overcome these downsides, the present work focuses on studying nanofluids with fractional derivative formed by differential equations with Caputo time derivatives that provides memory effect on nanofluid characteristics. Also, investigation on natural convective flow, heat, and mass transfer of nanofluids formed by different base fluids with different shapes of copper nanoparticles past an infinite vertical plate with radiation effect is carried out. The governing fractional differential equations are solved by employing Laplace transform technique with suitable boundary conditions. The different base fluids—water (H2O), SA:sodium alginate (C6H9Na O7), and EG:ethylene glycol (C2H6O2) and different shapes of nanoparticles—blade, brick, platelet, and cylinder are considered for the study. The exact solutions are obtained for the temperature, concentration, and velocity distributions and the respective Nusselt number, Sherwood number, and skin‐friction coefficient. The influence of non‐dimensional parameters provides physical interpretations of temperature, concentration, and velocity fields, Nusselt number, Sherwood number, and skin friction in detail with the help of graphical representations. From the results, it is found that nanofluid with water‐based blade‐shaped nanoparticle exhibits more velocity and temperature distributions. Also, strengthen of fluid flow, temperature, and concentration of nanofluids are inversely correlate with fractional order derivatives. [ABSTRACT FROM AUTHOR]

We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of 3 F 2 hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function. [ABSTRACT FROM AUTHOR]

The topic of this work is the supercritical geometric reproduction of particles in the model of a Markov branching process. The solution to the Kolmogorov equation is expressed by the Wright function. The series expansion of this representation is obtained by the Lagrange inversion method. The asymptotic behavior is described by using two different equivalent forms for the Laplace transform. They include the computation of the limit distribution and its moments. The exact formula for the asymptotic density is written in terms of the reduced Wright function. In particular, when the ultimate extinction probability q = 1/2, the density of the limit random variable is given by the incomplete gamma function. [ABSTRACT FROM AUTHOR]

Research on Casson fluid is very important due to its applicability in the progress of industrial and engineering industries. Here, a fractional order model of the Casson fluid over an oscillating plate in the presence of thermal radiation with constant wall temperature and concentration has been considered. The solution of this fractional model is obtained with the help of Laplace transform technique in terms of Wright function. The graphical analysis is also done by making several variations in parametric values including fractional parameter, mass Grashoff number, Prandtl number, velocity, temperature, concentration profiles etc. [ABSTRACT FROM AUTHOR]

This paper analyzes the system describing the absorption solar collector as an application of nanoparticles for the storage of solar energy. The system involves two fractional partial differential equations (FPDEs) utilizing the Caputo fractional definition (CFD). Explicit solutions are determined for the temperature and velocity in terms of the Wright function (WrFn) via Laplace transform (LT). In addition, the solutions are expressed in terms of some well–known special functions at selected values of the fractional–order. Moreover, the behaviors of the temperature and velocity are investigated graphically using the thermo–physical data of the Cu/Al 2 O 3 –nanoparticles. Furthermore, some numerical results are conducted about the performance of the Cu and Al 2 O 3. The results reveal the higher efficiency/performance of the Cu–nanoparticles over the Al 2 O 3 and thus copper enjoys higher capability than alumina for the purpose of solar energy storage. In addition, it is found that 1.41% in the enhancement of heat transfer is achieved by adding 1% of the Cu–nanoparticles while the corresponding enhancement rate of the other three nanoparticles was less. Besides, the enhancement rate reaches 7.1% by increasing the volume fraction of the Cu–nanoparticles to 5%. Furthermore, the enhancement rates for the TiO 2 , Ag, and Al 2 O 3 are 5.41%, 6.47%, and 6.53%, respectively when using 5% of these nanoparticles. Finally, the advantages/effectiveness of the current approach over a previous work in the literature are discussed in detail. [ABSTRACT FROM AUTHOR]

The two-parameter Wright special function is an interesting mathematical object that arises in the theory of the space and time-fractional diffusion equations. Moreover, many other special functions are particular instantiations of the Wright function. The article demonstrates finite representations of the Wright function in terms of sums of generalized hypergeometric functions, which in turn provide connections with the theory of the Gaussian, Airy, Bessel, and Error functions, etc. The main application of the presented results is envisioned in computer algebra for testing numerical algorithms for the evaluation of the Wright function.

In this paper, we apply some special functions to introduce a new class of control functions that help us define the concept of multi-stability. Further, we investigate the multi-stability of homomorphisms in C ∗ -algebras and Lie C ∗ -algebras, multi-stability of derivations in C ∗ -algebras, and Lie C ∗ -algebras for the following random operator equation via fixed point methods: μ f (ð , x + y 2) + μ f (ð , x − y 2) = f (ð , μ x). In particular, for μ = 1 , the above equation turns out to be Jensen's random operator equation. [ABSTRACT FROM AUTHOR]

At the edge of nineteenth and twentieth century several new special functions were introduced. Among them we point out the celebrated Mittag–Leffler function which got a central role in Fractional Calculus and its applications. Less known is the so called Le Roy function used at the beginning to solve similar problems to that for Mittag–Leffler function. An interest to the Le Roy function has been revisited in the last two decades due to different questions of Analysis and Probability Theory. In our paper we introduce multi-parametric generalization of the Le Roy function and study it from the point of view of an intermediate position between the Mittag–Leffler function and another important function of Fractional Calculus, namely the Wright function. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Faxén integral and present some alternative integral representations using the suitable changing of functions in the integrand. These integral representations are derived in terms of the Wright, generalized Wright and Mittag–Leffler functions. An identity is also given for the addition formula of this function. [ABSTRACT FROM AUTHOR]

This article discusses a new solution to the time-fractional ESR model, taking into account the non-zero uniform average blood velocity. We not only obtain an analytic solution to the generalized model of Sharma et al. (Comput Biol Med 26(1):1–7, 1996) and da Sousa et al. (AIMS Math 2(4):692–705, 2017), but also we present some new results which establish that the developed fractional order model is a better-suited one through which prediction of the ESR can take place more accurately. The same is established through graphs and tables which point toward a more practical ESR model. Furthermore, through numerical experiments, the appropriate values of the uniform blood velocity and the order of the fractional derivative are recommended. [ABSTRACT FROM AUTHOR]

Introduction/purpose: The aim of this article is to establish integral transforms of the generalized Lommel-Wright function. Methods: These transforms are expressed in terms of the Wright Hypergeometric function. Results: Integrals involving the trigonometric, generalized Bessel function and the Struve functions are obtained. Conclusions: Various interesting transforms as the consequence of this method are obtained.

We develop and analyze a fully discrete method, in order to solve numerically time-fractional diffusion equations driven by additive tempered fractional Gaussian noise. The stochastic model involves a Caputo fractional derivative of order α ∈ (0 , 1). We discuss the regularity results of the solution by using the inversion of Laplace transform, the Wright function and covariance function of stochastic process. A spectral Galerkin method is applied to approximate the stochastic model in space. Time discretization is achieved by using firstly a Riemann–Liouville derivative to reformulate the spatial semi-discrete form, and then applying the Grünwald–Letnikov formula. We derive the error estimates of the discrete methods, based on regularity results of the solution. Finally, extensive numerical experiments are presented to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this article, we deal with the efficient computation of the Wright function in the cases of interest for the expression of solutions of some fractional differential equations. The proposed algorithm is based on the inversion of the Laplace transform of a particular expression of the Wright function for which we discuss in detail the error analysis. We also present a code package that implements the algorithm proposed here in different programming languages. The analysis and implementation are accompanied by an extensive set of numerical experiments that validate both the theoretical estimates of the error and the applicability of the proposed method for representing the solutions of fractional differential equations. [ABSTRACT FROM AUTHOR]

We introduce a new class of fuzzy control functions and define the concept of matrix-valued fuzzy Mittag–Leffler–Hypergeometric–Wright stability. Then, we apply Radu–Mihet method derived from an alternative fixed point theorem to investigate existence, uniqueness and matrix-valued fuzzy Mittag–Leffler–Hypergeometric–Wright stability of a a class of P -Hilfer fractional differential equations in matrix-valued fuzzy Banach space. Next, we show the main results for unbounded domains. An example is given to illustrate the Mittag–Leffler–Hypergeometric–Wright stability for a fractional system. [ABSTRACT FROM AUTHOR]

Partial solutions with a singularity which are expressed in terms of the Wright functions for equations with Riemann–Liouville fractional derivative are constructed. [ABSTRACT FROM AUTHOR]

Various extensions of classical gamma, beta, Gauss hypergeometric and confluent hypergeometric functions have been proposed recently by many researchers. In this paper, further generalized extended beta function with some of its properties like summation formulas, Integral representations, connections with other special functions such as incomplete gamma, classical beta, classical Wright, hypergeometric, error, Fox-H, Fox-Wright, Meijer-G functions are obtained. Beta distribution together with-it corresponding moment, mean, variance, moment generating function and cumulative distribution are also presented. Moreover, the generalized beta function is used to generalized classical and other related extended Gauss, Kumar confluent, Appell’s and Lauricella’s hypergeometric functions with their integral representations, differential, difference, summation and transformations formulas.

This paper formulates and solves a generalized boundary value problem for a linear ordinary differential equation with a discretely distributed fractional differentiation operator. The fractional derivative is understood as the Gerasimov-Caputo derivative. The boundary conditions are given in the form of linear functionals, which makes it possible to cover a wide class of linear local and non-local conditions. A representation of the solution is found in terms of special functions. A necessary and sufficient condition for the solvability of the problem under study is obtained, as well as conditions under which the solvability condition is certainly satisfied. The theorem of existence and uniqueness of the solution is proved.

We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of 3F2 hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function.

By making use of the both Mittag-Leffler and Wright function, we establish a new subfamily of the class Sϑ of ϑ-spirallike functions. The main object of the paper is to provide sufficient conditions for a function to be in this newly established class and to discuss subordination outcomes. [ABSTRACT FROM AUTHOR]

Introduction/purpose: The aim of this article is to establish integral transforms of the generalized Lommel-Wright function. Methods: These transforms are expressed in terms of the Wright Hypergeometric function. Results: Integrals involving the trigonometric, generalized Bessel function and the Struve functions are obtained. Conclusions: Various interesting transforms as the consequence of this method are obtained. [ABSTRACT FROM AUTHOR]

В работе исследуется вторая краевая задача в полуполосе для параболического уравнения с оператором Бесселя, действующим по пространственной переменной, и частной производной Герасимова–Капуто по времени. Доказаны теоремы существования и единственности решения рассматриваемой задачи. Представление решения найдено в терминах интегрального преобразования с функцией Райта в ядре. Единственность решения доказана в классе функций быстрого роста. При частных значениях параметров, содержащихся в рассматриваемом уравнении, последнее совпадает с классическим уравнением диффузии.

В работе для трехмерного гиперболического уравнения высокого порядка с доминирующей смешанной производной исследуется разрешимость нелокальной задачи с интегральными условиями. Поставленная задача сводится к интегральному уравнению и с помощью априорных оценок доказывается существование единственного решения.

A boundary value problem for a fractionally loaded heat equation is considered in the first quadrant. The loaded term has the form of the Riemann-Liouville’s fractional derivative with respect to the time variable, and the order of the derivative in the loaded term is less than the order of the differential part. The study is based on reducing the boundary value problem to a Volterra integral equation. The kernel of the obtained integral equation contains a special function, namely, the Wright function. The kernel is estimated, and the conditions for the unique solvability of the integral equation are obtained.

In this paper, four parameters Wright function is considered. Certain geometric properties such as starlikeness, convexity, uniform convexity and close-to-convexity are discussed for this function. Further, certain geometric properties of normalized Bessel function of the first kind and two parameters Wright function are studied as a consequence. Interesting corollaries and examples are provided to support that these results are better than the existing ones and improve several results available in the literature. [ABSTRACT FROM AUTHOR]

In this article the inverse problem of determining the time depending coefficient in the Cauchy problem for a time-fractional diffusion-wave equation with a single observation at the point is studied. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion-wave equation is used and properties of this solution are investigated. The fundamental solution contains a Wright function, which is widely used in the theory of diffusion-wave equation. In particular, using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient. This estimate by norm is used in further in studying inverse problem. The inverse problem is reduced to the equivalent integral equation. In the proof of one valued solvability of this integral equation the contracted mapping principle is applied. The local existence and global uniqueness of the solution of the inverse problem are proven. The stability estimate also is obtained. [ABSTRACT FROM AUTHOR]