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We extend the results obtained in [14] by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain Ω with non-local dynamic conditions on the boundary ∂ Ω . Due to the pioneering nature of the present research, we propose here the apparently simple case of Ω = (0 , ∞) with boundary { 0 } of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models. [ABSTRACT FROM AUTHOR]

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties. [ABSTRACT FROM AUTHOR]

The main goal of this study is to use the ψ -Caputo fractional derivative of order ϑ ∈ (0 , 1) to construct the criteria for controllability in non-instantaneous impulsive dynamical systems. To obtain the necessary and sufficient conditions for the controllability of linear fractional systems by incorporating the positiveness of the Grammian matrices. To obtain sufficient conditions for controllability criteria for nonlinear systems, we have used Schaefer’s fixed point theorem. To enhance comprehension of the theoretical findings, several numerical examples have been provided. [ABSTRACT FROM AUTHOR]

Direct and two inverse problems for a Legendre equation involving integral convolution in time are studied. The inverse problems are ill-posed in the sense of Hadamard. The analytical series solutions of the problems are constructed by using method of variable separation. The determination of only u(x, y) is studied in the direct problem, the recovery of a pair of functions, i.e., { u (x , y) , f (x) } with appropriate addition data at some T is investigated in the 1st inverse problem while the identification of a pair of functions, i.e., { u (x , y) , q (y) } with an integral type data is considered in the 2nd inverse problem. By imposing certain regularity conditions, the unique existence of series solutions is developed. We provided some numerical examples to illustrate our results for the inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, we derive several fuzzy differential subordination and fuzzy differential superordination results for analytic functions $ \mathcal{M}_{\xi, \beta}^{s, \gamma} $, which involve the extended Mittag-Leffler function and the Pascal distribution series. We also investigate and introduce a class $ \mathcal{MB}_{\xi, \beta}^{F, s, \gamma}(\rho) $ of analytic and univalent functions in the open unit disc $ \mathcal{D} $ by employing the newly defined operator $ \mathcal{M}_{\xi, \beta}^{s, \gamma} $. We determine a specific relationship of inclusion for this class. Further, we establish prerequisites for a function role in serving as both the fuzzy dominant and fuzzy subordinant of the fuzzy differential subordination and superordination, respectively. Some novel results that are sandwich-type can be found here.

In this study, we derive multiple incomplete matrix Mittag-Leffler (ML) functions. We systematically investigate several properties of these incomplete matrix ML functions, which include some general properties and distinct representations of integral transforms. We further study the properties of the Riemann–Liouville fractional integrals and derivatives related to the incomplete matrix ML functions. Additionally, some interesting special cases of this work are highlighted. Finally, we establish the solution to the kinetic equations as an application. [ABSTRACT FROM AUTHOR]

In this paper, we derive several fuzzy differential subordination and fuzzy differential superordination results for analytic functions Ms,γξ,β, which involve the extended Mittag-Leffler function and the Pascal distribution series. We also investigate and introduce a class MBF,s,γξ,β (ρ) of analytic and univalent functions in the open unit disc D by employing the newly defined operator Ms,γξ,β. We determine a specific relationship of inclusion for this class. Further, we establish prerequisites for a function role in serving as both the fuzzy dominant and fuzzy subordinant of the fuzzy differential subordination and superordination, respectively. Some novel results that are sandwich-type can be found here. [ABSTRACT FROM AUTHOR]

Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations. [ABSTRACT FROM AUTHOR]

The paper describes the method of symbolic evaluation that serves as a useful tool to extend the studies of certain special functions including their properties and capabilities. In the paper, we exploit certain symbolic operators to introduce a new family of special polynomials, which is called the Mittag–Leffler–Gould–Hopper polynomials. We obtain the generating function, series definition and symbolic operational rule for these polynomials. This approach gives a wide platform to explore the study of classical and hybrid special polynomials. We establish summation formulae and certain identities for these polynomials. Further, we derive the multiplicative and derivative operators to study the quasi-monomiality property of these polynomials. Graphical representations and concluding remarks are also given. [ABSTRACT FROM AUTHOR]

Recently, Kovács et al. considered a Mittag‐Leffler Euler integrator for a stochastic semilinear Volterra integral‐differential equation with additive noise and proved the strong convergence error estimates [see SIAM J. Numer. Anal. 58(1) 2020, pp. 66‐85]. In this article, we shall consider the Mittag‐Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise. The mild solutions of our models involve four different Mittag‐Leffler functions. We first consider the existence, uniqueness and the regularities of the solutions. We then introduce the full discretization schemes for solving the problems. The temporal discretization is based on the Mittag‐Leffler integrators and the spatial discretization is based on the spectral method. The optimal strong convergence error estimates are proved under the reasonable assumptions for the semilinear term and for the regularity of the noise. Numerical examples are given to show that the numerical results are consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we propose some unified integral formulas for the Mittag-Leffler confluent hypergeometric function (MLCHF), and our findings are assessed in terms of generalized special functions. Additionally, certain unique cases of confluent hypergeometric function have been corollarily presented. [ABSTRACT FROM AUTHOR]

The study focuses on thermoelastic behavior of the time-fractional heat conduction equation in a thick hollow cylinder by quasi-static approach. The utilization of the Caputo fractional derivative has been employed. The upper and lower surfaces of the cylinder have been assigned arbitrary temperature values, while the boundaries of the cylinder remain at a constant temperature of zero. The integral transform technique has been utilized to solve the conduction equation. A mathematical model has been prepared for Copper material. The fractional-order thermoelasticity for distinct fractional values of time variable has been discussed and the results are presented graphically. [ABSTRACT FROM AUTHOR]

In this paper, some properties related to the 3 m -parametric Mittag-Leffler (M-L) functions are investigated. More precisely, three families of such functions with different kinds of indices are examined. Appropriate estimates and asymptotic formulae are obtained and which are further used in proving the convergence of series of functions of these families in the whole complex plane, as well as in its compact subsets. Using these auxiliary results, different relations referring to the discussed families are obtained; generalizing the ones already known for particular cases of the parameters. Those are considered different representations of the 3 m -parametric M-L functions pertaining to the families discussed, and with which various equalities are established relating different sums that involve matrix coefficients. Some interesting properties are reached for particular choices of the matrices. Finally, bounds of the 3 m -parametric M-L function are given under a perturbed matrix argument. [ABSTRACT FROM AUTHOR]

In this article, we are interested to study the elliptic equation under the Caputo derivative. We obtain several regularity results for the mild solution based on various assumptions of the input data. In addition, we derive the lower bound of the mild solution in the appropriate space. The main tool of the analysis estimation for the mild solution is based on the bound of the Mittag-Leffler functions, combined with analysis in Hilbert scales space. Moreover, we provide a regularized solution for our problem using the Fourier truncation method. We also obtain the error estimate between the regularized solution and the mild solution. Our current article seems to be the first study to deal with elliptic equations with Caputo derivatives on the unbounded domain.

In this paper, we defined a new family of meromorphic functions whose analytic characterization was motivated by the definition of the multiplicative derivative. Replacing the ordinary derivative with a multiplicative derivative in the subclass of starlike meromorphic functions made the class redundant; thus, major deviation or adaptation was required in defining a class of meromorphic functions influenced by the multiplicative derivative. In addition, we redefined the subclass of meromorphic functions analogous to the class of the functions with respect to symmetric points. Initial coefficient estimates and Fekete–Szegö inequalities were obtained for the defined function classes. Some examples along with graphs have been used to establish the inclusion and closure properties.

In this study, we derive multiple incomplete matrix Mittag-Leffler (ML) functions. We systematically investigate several properties of these incomplete matrix ML functions, which include some general properties and distinct representations of integral transforms. We further study the properties of the Riemann–Liouville fractional integrals and derivatives related to the incomplete matrix ML functions. Additionally, some interesting special cases of this work are highlighted. Finally, we establish the solution to the kinetic equations as an application.

Fractional calculus and fractional-order calculus are arranged in lineage as regards the mathematical models with complexity-theoretical tenets capable of capturing the subtle molecular dynamics by the integration of power-law convolution kernels into time- and space-related derivatives emerging in equations concerning the Magnetic Resonance Imaging (MRI) phenomena to which the fractional models of diffusion and relaxation are applied. Endowed with an intricate level of complexity and a unique physical and structural scaffolding at molecular and cellular levels with numerous synapses forming elaborate neural networks which entail in-depth probing and computing of patterns and signatures in individual cells and neurons, human brain as a heterogeneous medium is constituted of tissues with cells of different sizes and shapes, distributed across an extra-cellular space. Characterization of the unique brain cells is sought after to unravel the connections between different cells and tissues for accurate, reliable, robust and optimal models and computing. Accordingly, Diffusion Magnetic Resonance Imaging (DMRI), as a noninvasive and experimental imaging technique with clinical and research applications, provides a measure related to the diffusion characteristics of water in biological tissues, particularly in the brain tissues. Compatible with these aspects and beyond the diffusion coefficients' measurement, DMRI technique aims to exceed the spatial resolution of the MRI images and draw inferences from the microstructural properties of the related medium. Thus, novel tools become essential for the description of the biological (organelles, membranes, macromolecules and so on) and neurological (axons, dendrites, neurons and so forth) tissues' complexity. Mathematical model-based computational analyses with multifaceted methods to extract information from the DMRI with SpinDoctor into neuronal dynamics can provide quantitative parametric instruments in order to reflect the tissue properties focusing on the precise link between the tissue microstructure and signals acquired by employing advanced medical imaging technologies. Coalesced with accurate neuron geometry models as well as numerical DMRI simulations, a novel extended and multifaceted predictive mathematical model based on SpinDoctor and Bloch–Torrey partial differential equation (BTPDE) with the Caputo fractional-order derivative (FOD) with three-parameter (α , β , δ) Mittag-Leffler function (MLF) has been proposed and developed in our study by extending for the application on Brain Neuron Spin Unit dataset with the relevant multi-stage application-related steps. The feedforward neural networks (FFNNs) with BFGS Quasi-Newton equation, as one of the artificial neural network (ANN) algorithms, are applied on BTPDE with Caputo fractional-order derivative for the neurons and their algorithmic complexity is computed by building a BTPDE with Caputo FOD Neuron model based on different fractional orders. The fractional-order degree of the proposed and developed model is applied in relation to their corresponding complexity degrees. Consequently, experimentation and observations from the simulation-driven FFNN (with BFGS Quasi-Newton equation) learning scheme applied to the Bloch–Torrey PDE–Caputo FOD with MLF Neuron model (named as FFNN–BTPDE–CFODMLF Neuron model) proposed in this study, are made. Thus, by investigating whether the mathematical models based on the accurate neuron geometry models obtained can be optimized by comparing the errors in order to define the order parameter and identify to what optimal extent the errors are in relation to the prediction results with a particular focus on the neuron model, we have been able to estimate and predict brain microstructure through DMRI, accentuating mathematical and medical contributions based on the exploitation and corroboration of powerful modeling as well as computational capabilities. [ABSTRACT FROM AUTHOR]

In this paper, we address a matrix analog of the four‐parameter Mittag‐Leffler function and demonstrate its convergence on |z|=1$$ \mid z\mid =1 $$. Some properties, such as integral representations, derivative formulae, and finite summation formulae, have been determined. Additionally, we provide the addition and multiplication formulas for the addition and multiplication of two arguments of the four‐parameter Mittag‐Leffler matrix function. Further, we discuss the composition of generalized fractional calculus operators with respect to a function with the four‐parameter Mittag‐Leffler matrix function. [ABSTRACT FROM AUTHOR]

The key aim of the present work is to apply the Yang-Abdel-Cattani operator of fractional order derivative with Rabotnov exponential kernel to the mass-heat transfer of the generalized Casson fluid with generalized Fick's and Fourier's laws. The flow is analyzed in a porous medium under the effects of chemical molecular diffusivity, heat generation and slip wall condition on the fluid velocity. The medium through which the fluid is flowing is subjected to the effects of Newtonian heating. The YAC operator of fractional order can describe the generalized memory effects more suitably than the other fractional operators. So for the sake of a better interpretation of the rheological behavior of the Casson fluid we have used the fractional operator of YAC with Rabotnov exponential kernel. The technique of the Laplace transform is used to acquire the exact analytical solution of the problem in the Mittag-Leffler form. Through MathCAD software physical behaviors of various parameters are investigated on the heat, mass and velocity of the fluid during the flow in a porous medium.

This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful methods to help with the learning of key mathematical ideas. The kernel of the general family of weighted fractional integral operators is related to a wide variety of extensions and generalizations of the Mittag-Leffler function and the Hurwitz-Lerch zeta function. It delves into the applications of fractional-order integral and derivative operators in mathematical and engineering sciences. Furthermore, this article derives specific cases for selected functions and presents various applications to illustrate the obtained results. Additionally, novel applications involving the Digamma function are introduced. [ABSTRACT FROM AUTHOR]

Umbral operational techniques offer sturdy mechanism in the studies of special functions and special polynomials. The techniques of umbral calculus are employed to derive properties of families of exponential-like functions and their hyperbolic forms. The generalized forms of Mittag-Leffler functions are used to solve technical problems concerning transport of a charged beam in a solenoid magnet. The proposed method is flexible and has many advantages over standard computational techniques. [ABSTRACT FROM AUTHOR]

Various types of functions, their generalizations and extentions have been widely explored, especially for their applications in various fields of research. In this article, we show that the use of methods of an operational nature, such as umbral calculus, allows us to acquire the hybrid form of the Bessel and Tricomi functions in terms of Mittag-Leffler functions. Certain novel identities such as generating functions, series representations, differential equations, derivative formulae, summation formula and Jacobi-Anger expansion for Mittag-Leffler-Bessel functions are obtained. Some integral formulae for the Oth-order Mittag-Leffler-Bessel functions are established. In addition, the Mittag-Leffler-Tricomi functions are constructed and some captivating properties of these polynomials are explored. The natural transforms of the Mittag-Leffler-Bessel functions and Mittag-Leffler-Tricomi functions are investigated and Laplace and Sumudu transforms are obtained as special cases. The graphical representations of these hybrid functions are given for special values of the parameters. [ABSTRACT FROM AUTHOR]

This article addresses the issue of drive–response synchronization in fractional-order multi-link memristive neural networks (FMMNN) with multiple delays, under hybrid impulsive feedback control. To address the impact of multiple delays on system synchronization, an extended fractional-order delayed comparison principle incorporating impulses is established. By leveraging Laplace transform, Mittag–Leffler functions, the generalized comparison principle, and hybrid impulsive feedback control schemes, several new sufficient conditions are derived to ensure synchronization in the addressed FMMNN. Unlike existing studies on fractional-order single-link memristor-based systems, our response network is a multi-link model that considers impulsive effects. Notably, the impulsive gains α i are not limited to a small interval, thus expanding the application range of our approach ( α i ∈ (− 2 , 0) ∪ (− ∞ , − 2) ∪ (0 , + ∞) ). This feature allows one to choose impulsive gains and corresponding impulsive intervals that are appropriate for the system environment and control requirements. The theoretical results obtained in this study contribute to expanding the relevant theoretical achievements of fractional-order neural networks incorporating memristive characteristics. [ABSTRACT FROM AUTHOR]

An inverse source problem (ISP) of recovering space‐dependent source term along with diffusion concentration for a two‐dimensional diffusion equation involving integral convolution of arbitrary memory kernel in time is considered. The unique existence of the solution is proved using a bi‐orthogonal system of functions obtained from the associated non‐self‐adjoint spectral problem and its adjoint problem which form Riesz basis in L2[(0,1)×(0,1)]$$ {L}^2\left[\left(0,1\right)\times \left(0,1\right)\right] $$. In addition, some particular cases of ISP are described as special cases of our results. [ABSTRACT FROM AUTHOR]

The major objective of this article is to establish a new function called which is generalization of the generalized hypergeometric function and Mittag-Leffler function and obtain its properties particular differential property, integral representation, derivative formula and some integral transform, Euler transform, Laplace transform, Whittaker transform. We also derived the relations that exist between function with well known special functions. This article also deals with some fractional integral properties of the function using the Riemann–Liouville fractional integrals and derivatives operators. [ABSTRACT FROM AUTHOR]

Navier-Stokes (NS) equations dealing with gravitational force with time-fractional derivatives are discussed in this paper. These equations can be used to predict fluid velocity and pressure for a given geometry. This paper investigates the local and global existence and uniqueness of mild solutions to NS equations for the time fractional differential operator. We also work on the regularity effects of such types of equations were caused by orthogonal flow.

It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding integer model, we need to compute the solutions of the fractional differential equations and compare them with an integer model relative to the data available. In this work, we will illustrate that the linear nq-order sequential Caputo fractional differential equations, which are sequential of order q where q<1 with fractional initial conditions and/or boundary conditions, can be solved. The reason for choosing sequential fractional dynamic equations is that linear non-sequential Caputo fractional dynamic equations with constant coefficients cannot be solved in general. We used the Laplace transform method to solve sequential Caputo fractional initial value problems. We used fractional boundary conditions to compute Green’s function for sequential boundary value problems. In addition, the solution of the sequential dynamic equations yields the solution of the corresponding integer-order differential equations as a special case as q→1.

Navier-Stokes (NS) equations dealing with gravitational force with time-fractional derivatives are discussed in this paper. These equations can be used to predict fluid velocity and pressure for a given geometry. This paper investigates the local and global existence and uniqueness of mild solutions to NS equations for the time fractional differential operator. We also work on the regularity effects of such types of equations were caused by orthogonal flow. [ABSTRACT FROM AUTHOR]

Neural networks and fractional order calculus are powerful tools for system identification through which there exists the capability of approximating nonlinear functions owing to the use of nonlinear activation functions and of processing diverse inputs and outputs as well as the automatic adaptation of synaptic elements through a specified learning algorithm. Fractional-order calculus, concerning the differentiation and integration of non-integer orders, is reliant on fractional-order thinking which allows better understanding of complex and dynamic systems, enhancing the processing and control of complex, chaotic and heterogeneous elements. One of the most characteristic features of biological systems is their different levels of complexity; thus, chaos theory seems to be one of the most applicable areas of life sciences along with nonlinear dynamic and complex systems of living and non-living environment. Biocomplexity, with multiple scales ranging from molecules to cells and organisms, addresses complex structures and behaviors which emerge from nonlinear interactions of active biological agents. This sort of emergent complexity is concerned with the organization of molecules into cellular machinery by that of cells into tissues as well as that of individuals to communities. Healthy systems sustain complexity in their lifetime and are chaotic, so complexity loss or chaos loss results in diseases. Within the mathematics-informed frameworks, fractional-order calculus based Artificial Neural Networks (ANNs) can be employed for accurate understanding of complex biological processes. This approach aims at achieving optimized solutions through the maximization of the model's accuracy and minimization of computational burden and exhaustive methods. Relying on a transdifferentiable mathematics-informed framework and multifarious integrative methods concerning computational complexity, this study aims at establishing an accurate and robust model based upon integration of fractional-order derivative and ANN for the diagnosis and prediction purposes for cancer cell whose propensity exhibits various transient and dynamic biological properties. The other aim is concerned with showing the significance of computational complexity for obtaining the fractional-order derivative with the least complexity in order that optimized solution could be achieved. The multifarious scheme of the study, by applying fractional-order calculus to optimization methods, the advantageous aspect concerning model accuracy maximization has been demonstrated through the proposed method's applicability and predictability aspect in various domains manifested by dynamic and nonlinear nature displaying different levels of chaos and complexity. [ABSTRACT FROM AUTHOR]

Transient heat conduction problems are systematically applied to the fading memory formalism with different Mittag-Leffler-type memory kernels. With such an approach, using various memories naturally results in definitions of various fractional operators. Six examples are given and interpreted from a common perspective, covering the most well-liked versions of the Mittag-Leffler function. The fading memory approach was used as a template and demonstrated that, if the constitutive equations are correctly built, it is also possible to directly determine where the hereditary terms are located in the models. [ABSTRACT FROM AUTHOR]

We investigate the applicability and efficiency of the invariant subspace method to (2 + 1)-dimensional time-fractional nonlinear PDEs. We show how to find various types of invariant subspaces and reductions for the (1 + 1) and (2 + 1)-dimensional generalized nonlinear time-fractional thin-film equations which arise from the motion of liquid film on a solid surface under the influence of surface tension. We construct several kinds of exact solutions for the above-mentioned equations depending on arbitrary functions as either a combination of trigonometric, polynomial, Mittag–Leffler, and exponential type functions or any of these forms. Also, we demonstrate the applicability of the invariant subspace method to solve the initial and boundary value problem of nonlinear time-fractional PDEs for the first time and illustrate it through the physically important generalized time-fractional thin-film and linear time-fractional heat equations. Finally, we present some of the obtained exact solutions graphically. [ABSTRACT FROM AUTHOR]

In this paper, we concern about a modified version of the Keller-Segel model. The Keller-Segel is a system of partial differential equations used for modeling Chemotaxis in which chemical substances impact the movement of mobile species. For considering memory effects on the model, we replace the classical derivative with respect to time variable by the time-fractional derivative in the sense of Caputo. From this modification, we focus on the well-posedness of the Cauchy problem associated with such the model. Precisely, when the spatial variable is considered in the space Rd{{\mathbb{R}}}^{d}, a global solution is obtained in a critical homogeneous Besov space with the assumption that the initial datum is sufficiently small. For the bounded domain case, by using a discrete spectrum of the Neumann Laplace operator, we provide the existence and uniqueness of a mild solution in Hilbert scale spaces. Moreover, the blowup behavior is also studied. To overcome the challenges of the problem and obtain all the aforementioned results, we use the Banach fixed point theorem, some special functions like the Mainardi function and the Mittag-Leffler function, as well as their properties.

Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs) with time-FDs of order β ∈ (0 , 1) are investigated. To facilitate anomalous diffusion in fractal media, mild solutions and Mittag-Leffler functions are used. In H δ , r , the existence, and uniqueness of local and global mild solutions are proved, as well as the symmetric structure created. Moderate local solutions are provided in J r . Moreover, the regularity and existence of classical solutions to the equations in J r . are established and presented. [ABSTRACT FROM AUTHOR]

The gravitational effect is a physical phenomenon that explains the motion of a conductive fluid flowing under the impact of an exterior gravitational force. In this paper, we work on the Navier–Stokes equations (NSES) of the fluid flowing under the impact of an exterior gravitational force inclined at an angle of 45 ∘ with A time-fractional derivative of order β ∈ (0 , 1) . To encourage anomalous diffusion in fractal media, we apply these equations. In H δ , r , we prove the existence and uniqueness of local and global mild solutions. Additionally, we provide moderate local solutions in J r . Additionally, we establish the regularity and existence of classical solutions to these equations in J r . [ABSTRACT FROM AUTHOR]

In this paper we establish the existence, uniqueness and some regularity results for the solution of differential evolution equations of fractional order involving an unbounded linear maximal monotone operator in a Hilbert space. The proof relies on spectral properties of the operator. An explicit representation formula of the solution is given, allowing the computation of a sequence of approximate solutions. [ABSTRACT FROM AUTHOR]

In this work, we discuss the derivatives of the Wright functions (of the first and the second kinds) with respect to parameters. The differentiation of these functions leads to infinite power series with the coefficients being the quotients of the digamma (psi) and gamma functions. Only in few cases is it possible to obtain the sums of these series in a closed form. The functional form of the power series resembles those derived for the Mittag-Leffler functions. If the Wright functions are treated as generalized Bessel functions, differentiation operations can be expressed in terms of the Bessel functions and their derivatives with respect to the order. In many cases, it is possible to derive the explicit form of the Mittag-Leffler functions by performing simple operations with the Laplacian transforms of the Wright functions. The Laplacian transform pairs of both kinds of Wright functions are discussed for particular values of the parameters. Some transform pairs serve to obtain functional limits by applying the shifted Dirac delta function. We expect that the present analysis would find several applications in physics and more generally in applied sciences. These special functions of the Mittag-Leffler and Wright types have already found application in rheology and in stochastic processes where fractional calculus is relevant. Careful readers can benefit from the new results presented in this paper for novel applications. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of Cauchy problem for a class of non-autonomous fractional evolution equation with Caputo's fractional derivative of order α ∈ (1 , 2) , which can be applied to model the time dependent coefficients fractional differential systems. We first introduce an operator family and analyze its properties, by the iterative method, we construct a solution to an operator-valued Volterra equation, which is the most critical ingredient to prove solvability of the problem. Finally, based on the solution operators we establish the existence and uniqueness of classical solutions. [ABSTRACT FROM AUTHOR]

It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding integer model, we need to compute the solutions of the fractional differential equations and compare them with an integer model relative to the data available. In this work, we will illustrate that the linear n q -order sequential Caputo fractional differential equations, which are sequential of order q where q < 1 with fractional initial conditions and/or boundary conditions, can be solved. The reason for choosing sequential fractional dynamic equations is that linear non-sequential Caputo fractional dynamic equations with constant coefficients cannot be solved in general. We used the Laplace transform method to solve sequential Caputo fractional initial value problems. We used fractional boundary conditions to compute Green's function for sequential boundary value problems. In addition, the solution of the sequential dynamic equations yields the solution of the corresponding integer-order differential equations as a special case as q → 1 . [ABSTRACT FROM AUTHOR]

We carry out an analysis of the existence of solutions for a class of nonlinear fractional partial differential equations of parabolic type with nonlocal initial conditions. Sufficient conditions for the solvability of the desired problem are presented by transforming it into an abstract non-autonomous fractional evolution equation, and constructing two families of solution operators based on the Mittag-Leffler function, the Mainardi Wright-type function and the analytic semigroup generated by the closed densely defined operator − A (⋅). The discussions are based on the fractional power theory as well as the Banach fixed point theorem in the interpolation space X p υ (0 ⩽ υ < 1 , 1 < p < ∞). [ABSTRACT FROM AUTHOR]

We derive the model for fractional beam equations by making use of a modified constitutive assumption, that is the relationship between stress and strain depending on the creep compliance given by a fractional power-type function. [ABSTRACT FROM AUTHOR]

Suitable families of random variables having power series distributions are considered, and their asymptotic behavior in terms of large (and moderate) deviations is studied. Two examples of fractional counting processes are presented, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in [Integral Transforms Spec. Funct. 27 (2016), 783–793], the second one is inspired by a model studied in [J. Appl. Probab. 52 (2015), 18–36].

It has been obtained that the n-th derivative of the 2-parametric Mittag–Leffler function is a 3-parametric Mittag–Leffler function, with exactness to a constant. Following the analogy, the author later obtained the n-th derivative of the 2 m -parametric multi-index Mittag–Leffler function. It turns out that this is expressed via the 3 m -parametric Mittag–Leffler function. In this paper, upper estimates of the remainder terms of these derivatives are found, depending on n. Some asymptotics are also found for "large" values of the parameters. Further, the Taylor series of the 2 and 2 m -parametric Mittag–Leffler functions around a given point are obtained. Their coefficients are expressed through the values of the corresponding n-th order derivatives at this point. The convergence of the series to the represented Mittag–Leffler functions is justified. Finally, the Bessel-type functions are discussed as special cases of the multi-index ( 2 m -parametric) Mittag–Leffler functions. Their Taylor series are derived from the general case as corollaries, as well. [ABSTRACT FROM AUTHOR]

In this paper, a boundary value problem for an equation with a partial Riemann–Liouville fractional derivative is studied. The unique solvability of the problem is proved. [ABSTRACT FROM AUTHOR]

The inverse problem of recovering a source term along with diffusion concentration for a generalized diffusion equation has been considered. The so-called 2nd level fractional derivative in time of order between 0 and 1 is used by fixing the parameters involved in 2nd level fractional derivative some well known fractional derivatives such as Riemann-Liouville, Caputo and Hilfer can be obtained. We investigated existence, uniqueness results and discussed some particular cases of the considered inverse problem. [ABSTRACT FROM AUTHOR]