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This paper deals with the computation of the two parameter Mittag-Leffler function of operators by exploiting its Stieltjes integral representation and then by using a single exponential transform together with the sinc rule. Whenever the parameters of the function do not allow this representation, we resort to the Dunford-Taylor one. The error analysis is kept in the framework of unbounded accretive operators in order to make it a useful tool for the solution of fractional differential equations. The theory is also used to design a rational Krylov method. [ABSTRACT FROM AUTHOR]

The main task of the present research is a non‐local boundary value problem for a time‐fractional differential equation involving Hilfer fractional derivative on a metric star graph. The main goal of this work is to stud the uniqueness and the existence of solution of the formulated problem. Using by the method of separation of variables, we find a solution of the investigated problem in the form of the Fourier series. Sufficient classes of given functions, which provide the existence and uniqueness of solution of the considered problem, are defined. Uniqueness of solution is proved, by using a priori estimatesy for the solution. [ABSTRACT FROM AUTHOR]

In this study, we investigate novel fractional tuberculosis model with Caputo fractional derivative. The computational solution of the fractional tuberculosis disease is obtained with the help of the generalized Euler’s method (GEM). This article considers two treatment strategies: protective treatment for latent populations and main treatment for infected ones. The compartmental structure of the six-dimensional model includes the susceptible, latent, infected, recovered, and treatment classes. To examine the nature of differential equations of fractional order that arise in biological illness, the findings produced utilizing the method under consideration are more accurate and straightforward. Additionally, the stability of the equilibrium point is also investigated. The fractional model has been graphically simulated using MATLAB22. [ABSTRACT FROM AUTHOR]

The Helmholtz equation is an important differential equation. It has a wide range of uses in physics, including acoustics, electro-statics, optics, and quantum mechanics. In this article, a hybrid approach called the Shehu transform decomposition method (STDM) is implemented to solve space-fractional-order Helmholtz equations with initial boundary conditions. The fractional-order derivative is regarded in the Caputo sense. The solutions are provided as series, and then we use the Mittag-Leffler function to identify the exact solutions to the Helmholtz equations. The accuracy of the considered problem is examined graphically and numerically by the absolute, relative, and recurrence errors of the three problems. For different values of fractional-order derivatives, graphs are also developed. The results show that our approach can be a suitable alternative to the approximate methods that exist in the literature to solve fractional differential equations. [ABSTRACT FROM AUTHOR]

Fractional Kinetic equations (FKEs) including a wide variety of special functions are widely and successfully applied in describing and solving many important problems of physics and astrophysics. In this work, the solutions of the FKEs of the generalized Hurwitz-Lerch Zeta function using the Laplace transform are derived and examined. [ABSTRACT FROM AUTHOR]

The article investigates a Dirichlet-type problem with conjugation conditions for a degenerate equation of high even order with variable coefficients, including the Riemann–Liouville fractional derivative, in a rectangular region consisting of two subdomains ( y > 0 and y < 0 ), in each of which the equation has various kind.The solution is constructed as a series in terms of eigenfunctions of the one-dimensional problem, the existence of eigenfunctions is proved by the method of the theory of integral equations with symmetric kernels. The theorem of expansion in terms of the system of obtained eigenfunctions is proved. Sufficent conditions are found for boundary functions under which the solution in the form of a series converges uniformly. When justifying the convergence of a series, the problem of "small denominators" arises.This problem has been successfully solved in the article. The uniqueness of the solution is proved by the spectral method. [ABSTRACT FROM AUTHOR]

Malaria remains a substantial public health challenge and economic burden globally. Currently, malaria has been declared as endemic in 85 countries. In this study, we developed and analyzed a fractional-order mathematical model for malaria transmission dynamics that incorporates variability of temperature and rainfall using Caputo-type AB operators. The existence and uniqueness of the model's solutions were established using the Banach fixed-point theorem. The model system's equilibria (both disease-free and endemic) were identified, and lemmas and theorems were developed to prove their stability. Furthermore, we used different temperature ranges and rainfall data, validating them against existing literature. Numerical simulations using the Toufik-Atangana schemes with various fractional-order alpha values revealed that as the value of alpha approaches 1, the behavior of the fractional-order model converges to that of the classical model. The numerical results are promising and are expected to be valuable for future research related to fractional-order models. [ABSTRACT FROM AUTHOR]

The paper is devoted to presenting a novel closed-form representation of the resonant thermonuclear functions and the nonrelativistic Voigt function, which are essential tools in nuclear physics. Understanding thermonuclear fusion reaction rates within solar analogs is crucial for understanding stellar evolution and energy production mechanisms. Initially, this paper focuses on evaluating fusion reaction rates, particularly emphasizing resonant reactions, which play pivotal roles in stellar evolution phases. A key challenge lies in solving reaction rate integrals in closed form. The Buschman–Fox H -function of two variables is employed to address this issue. Conventionally, it is assumed that the plasma particles' velocity follows the Maxwell–Boltzmann distribution. However, it is acknowledged that particles may deviate from this assumed equilibrium state in actual scenarios, leading to nonequilibrium situations. The study also aims to address these nonequilibrium situations by utilizing appropriate velocity models from the existing literature. Utilizing the Mellin transform technique, we achieve the closed-form representation of the resonant reaction rate integral. Furthermore, we address the nonrelativistic Voigt profile and, in particular, Voigt function. The Voigt profile, resulting from the convolution of Gaussian and Lorentzian distributions, effectively captures the intricate shapes of spectral lines encountered in spectroscopy. Apart from its significance in spectroscopy, the Voigt function finds application in various areas such as plasma nuclear studies, acoustics, and radiation transfer. Many approximations of the Voigt function can be found in the literature, yet currently, there is no existing closed-form expression. This paper also sets out to fill this gap by deriving the exact closed-form expressions for the Voigt function and its conjugate in terms of Buschman–Fox H -function, employing the Mellin convolution theorem. This paper marks the first instance in the literature where the applications of Buschman Fox's H -function has been documented. [ABSTRACT FROM AUTHOR]

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]

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions). [ABSTRACT FROM AUTHOR]

We introduce new extension of the extended Pochhammer symbol and gamma function by using the extended Mittag-Leffler function. We also present extension of the generalized hypergeometric function as well as some of their special cases by using this extended Pochhammer symbol. Further, we define the extension of the τ-Gauss hypergeometric function. Integral and derivative formulas involving the Mellin transform and fractional calculus techniques associated with this extended τ-Gauss hypergeometric function are also given. Also, new extended τ-Gauss hypergeometric function also provides a few more interesting and well-known results. This enriches the theory of special functions. The obtained results are believed to be newly presented. [ABSTRACT FROM AUTHOR]

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

This paper introduces the ψ-formable integral transform, discusses the several essential properties and results--Convolution, ψ-formable transform of tth derivative, ψ-Riemann Liouville fractional integration and differentiation, ψ-Caputo fractional differentiation, ψ-Hilfer fractional differentiation, ψ-Prabhakar fractional integration and differentiation, and ψ-Hilfer-Prabhakar fractional derivatives. Next, we use the Fourier integral and Modifiable conversions to solve some Cauchy-type fractional differential equations using the generalized three-parameter Mittag-Leffler function and ψ-Hilfer-Prabhakar fractional derivatives. [ABSTRACT FROM AUTHOR]

AbstractThis article introduces a new operator defined by the convolution of the Poisson distribution series and the generalized Mittag-Leffler function. Applying this operator, we study a novel subclass of analytic functions using the technique of fuzzy differential subordination. Also, we prove some significant results related to the fuzzy differential relation for this novel class. These results are applied to particular functions to showcase the significance of our findings.

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

The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.

Different types of fractional calculus have been defined, grouped into categories based on their properties. Two types particularly studied are Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two types by expressing one in terms of the other, using the theory of fuzzy fractional calculus with respect to functions. By naturally extending this connection, a generalization is developed that unifies several existing fractional operators, including Riemann–Liouville, Caputo, ψ -Caputo, classical Hadamard, Hadamard-type, tempered, and all these operators considered with respect to functions via the Hadamard ψ -Caputo tempered fractional derivative (CTFD) using the Mittag-Leffler function. In addition, we explore the Cauchy problem for fuzzy fractional differential equations (FDEs) with this kind of derivative and demonstrate various results of existence and uniqueness using fixed-point theorems of Krasnoselskii, Schaüder's, and Banach. Applications further serve as proof of the utility of our findings, which converge, expand, and generalize several previously reported results in research. [ABSTRACT FROM AUTHOR]

The authors develop and substantiate an algorithm for finding an ordinary differential equation of minimum order with polynomial coefficients over the field of rational numbers, whose solutions are a given system of polynomials (in particular, a system of modified Laguerre–Cayley polynomials). [ABSTRACT FROM AUTHOR]

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize this result, calculating the inverse Laplace transform of a given function F α , β s by using two different approaches: the Bromwich integral and the decomposition of F α , β s in simple fractions. From both calculations, we obtain a set of novel Laplace and Stieltjes transforms. [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]

A super-structure system for probability densities, covering not just typical types but also fractional ones, was developed using the time scale theory. From a mathematical point of view, we discover duals of the Poisson process on the time scale T = R for the time scales T = Z and T = q Z , evaluating ∇ - calculus and Δ - calculus. Also, we search the fractional extensions of the Poisson process on these time scales and detect duals of them. A simulation allows for comparing the nabla and delta types of the observed distributions, not just typical types but also fractional ones. As an application, we also propose new substitution boxes (S-boxes) using the proposed stochastic models and compare the performance of S-boxes created in this way. Given that the S-box is the core for confusion in Advanced Encryption Standard (AES), the formation of these new S-boxes represents an interesting application of these stochastic models. [ABSTRACT FROM AUTHOR]

We consider the initial problem for one class of fractional stochastic differential equations with white noise in time and space for solutions of distribution type. Existence and singularity theorems for solutions are established and explicit representations for them are found. [ABSTRACT FROM AUTHOR]

The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology. [ABSTRACT FROM AUTHOR]

Electrokinetic flows driven by electro-osmotic forces are especially relevant in micro and nano-devices, presenting specific applications in medicine, biochemistry, and miniaturized industrial processes. In this work, we integrate analytical solutions with numerical methodologies to explore the fluid dynamics of viscoelastic electro-osmotic/pressure-driven fluid flows (described by the generalized Phan–Thien–Tanner (gPTT) constitutive equation) in a microchannel under asymmetric zeta potential conditions. The constitutive equation incorporates the Mittag–Leffler function with two parameters (α and β ), which regulate the rate of destruction of junctions in a network model. We analyze the impact of the various model parameters on the velocity profile and observe that our newly proposed model provides a more comprehensive depiction of flow behavior compared to traditional models, rendering it suitable for modeling complex viscoelastic flows. [ABSTRACT FROM AUTHOR]

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer's G-function and Fox's H-functions and their utilization in mathematical physics are the prime focus of this paper. [ABSTRACT FROM AUTHOR]

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 the framework of the theory of linear viscoelasticity, we derive an analytical expression of the relaxation modulus in the Andrade model G α t for the case of rational parameter α = m / n ∈ (0 , 1) in terms of Mittag–Leffler functions from its Laplace transform G ˜ α s . It turns out that the expression obtained can be rewritten in terms of Rabotnov functions. Moreover, for the original parameter α = 1 / 3 in the Andrade model, we obtain an expression in terms of Miller-Ross functions. The asymptotic behaviours of G α t for t → 0 + and t → + ∞ are also derived applying the Tauberian theorem. The analytical results obtained have been numerically checked by solving the Volterra integral equation satisfied by G α t by using a successive approximation approach, as well as computing the inverse Laplace transform of G ˜ α s by using Talbot's method. [ABSTRACT FROM AUTHOR]

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk D. For these new classes, the authors establish initial coefficient bounds of | a 2 | and | a 3 | . Furthermore, the famous Fekete–Szegö coefficient inequality is also obtained for these new classes of functions. [ABSTRACT FROM AUTHOR]

The tracking control of a non-linear fractional-order system (FOS) is highly tedious since cumulative approximation errors are encountered during the control process. Hence, to cope with that problem, several controllers have been adopted previously, such as backstepping controllers, sliding mode controllers, dynamic surface controllers, etc. These controllers have successfully reduced the approximation errors in the integer order non-linear system. However, the performance of those effective controllers in fractional-order non-linear systems is evaluated in this paper. The controllers such as the backstepping controller, sliding mode controller, dynamic surface controller and fuzzy dynamic surface controller are examined on FOS to verify the stability of control systems. As a result, the backstepping controller has performed better than other controllers in the case of actuator errors but still has the problem of the increased number of derivative terms and greater complexity in repeated calculations. The tracking error of the backstepping controller has been reduced to 0.0358 with 1.58 s of response time. [ABSTRACT FROM AUTHOR]

This article uses the Mittag-Leffler transformation to explore a specific category of harmonic meromorphic functions. The Mittag-Leffler transformation is a crucial tool for analysing meromorphic functions and provides essential properties and insights into their behavior. The main focus of this study is on harmonic meromorphic functions that can be represented by the MittagLeffler transformation. Furthermore, this research introduces an innovative derivative operator that incorporates this transformation into the domain of harmonic meromorphic functions. The Mittag-Leffler transformation is widely recognised as a powerful technique for analysing various mathematical functions, especially those with fractional order derivatives. It improves our understanding of harmonic meromorphic functions and their inherent characteristics. The research findings highlight the effectiveness of this new derivative operator in unraveling the complexities of these functions. They provide valuable insights into their behavior and fundamental traits. Additionally, the study offers coefficient inequalities, the distortion theorem, distortion bounds, extreme points, convex combinations, and convolution analyses specifically tailored to functions within this particular class. [ABSTRACT FROM AUTHOR]

In this paper, an SIRS epidemic model using Grunwald–Letnikov fractional-order derivative is formulated with the help of a nonlinear system of fractional differential equations to analyze the effects of fear in the population during the outbreak of deadly infectious diseases. The criteria for the spread or extinction of the disease are derived and discussed on the basis of the basic reproduction number. The condition for the existence of Hopf bifurcation is discussed considering fractional order as a bifurcation parameter. Additionally, using the Grunwald–Letnikov approximation, the simulation is carried out to confirm the validity of analytic results graphically. Using the real data of COVID-19 in India recorded during the second wave from 15 May 2021 to 15 December 2021, we estimate the model parameters and find that the fractional-order model gives the closer forecast of the disease than the classical one. Both the analytical results and numerical simulations presented in this study suggest different policies for controlling or eradicating many infectious diseases. [ABSTRACT FROM AUTHOR]

In this paper, we study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Lefflertype function to study oscillatory-type integrals appearing in the analysis of time-fractional partial differential equations. Several generalisations of the first and second van der Corput lemmas are proved. Optimal estimates on decay orders for particular cases of the Mittag-Leffler functions are also obtained. As an application of the above results, the generalised Riemann–Lebesgue lemma and the Cauchy problem for the time-fractional evolution equation are considered. [ABSTRACT FROM AUTHOR]

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard's iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. [ABSTRACT FROM AUTHOR]

In this paper, the time-resolved model of the photoacoustic signal for samples with a complex inner structure is derived including local non-equilibrium of structural elements with multiple degrees of freedom, i.e., structural entropy of the system. The local non-equilibrium is taken into account through the fractional operator. By analyzing the model for two types of time-dependent excitation, a very short pulse and a very long pulse, it is shown that the rates of non-equilibrium relaxations in complex samples can be measured by applying the derived model and time-domain measurements. Limitations of the model and further directions of its development are discussed. [ABSTRACT FROM AUTHOR]

This paper introduces a modified version of the Variational Iteration Method, incorporating P-transformation. We propose a novel semi-analytical technique named the modified variational iteration method for addressing fractional difierential equations featuring tempered Liouville- Caputo derivatives. The modified variational iteration method emerges as a highly eficient and powerful mathematical tool, ofiering exact or approximate solutions for a diverse range of real-world problems in en- gineering and the natural sciences, specifically those expressed through difierential equations. To assess its efiectiveness and accuracy, we scru- tinize the modified variational iteration method by applying it to three problems related to the heat-like multidimensional difiusion equation with a fractional time derivative in a tempered Liouville-Caputo form [ABSTRACT FROM AUTHOR]

We targeted a new kind of inverse problem for the wave-diffusion equation. Namely, using the given data we have found not only a solution of the wave-diffusion equation satisfying certain boundary and initial conditions but also an unknown time-moment starting from which the wave process transmitted into the diffusion process. In the practical point of view, even finding the first such time-moment seems very interesting. Our main tool of investigation is a spectral expansion method. In addition, we will use the solution of the corresponding Cauchy problem for differential equation in time-variable. [ABSTRACT FROM AUTHOR]

The objective of this paper is to explore certain properties involving the generalized hypergeometric function ..., as introduced by Khan et al. [27]. Several fractional derivatives and integral formulas involving the generalized hypergeometric function are computed. Further, we derive the solution of generalized fractional kinetic equations involving generalized hypergeometric function. The respective solution is given in terms of Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

Motivated by the recent investigations of several authors, the main aim of this article is to derive several functional inequalities for a class of functions related to the incomplete Fox-Wright functions that were introduced and studied recently. Moreover, new functional bounds for functions related to the Fox-Wright function are deduced. Furthermore, a class of completely monotonic functions related to the Fox-Wright function is given. The main mathematical tools used to obtain some of the main results are the monotonicity patterns and the Mellin transform for certain functions related to the two-parameter Mittag-Leffler function. Several potential applications for this incomplete special function are mentioned. [ABSTRACT FROM AUTHOR]

Signal processing plays a crucial role in biomedical applications, facilitating accurate health monitoring and clinical diagnoses. This study presents a comparative analysis of Gaussian, Mittag-Leffler, and Savitzky-Golay filters, evaluating their effectiveness in noise reduction and signal enhancement for electrocardiogram (ECG) signals. These filters offer adjustable parameters, making them adaptable to various applications. Our findings demonstrate that the Savitzky-Golay smoothing filter outperforms the others in smoothing data and computing derivatives of noisy data, despite its limitations in suppressing noise at higher frequencies. On the other hand, the adaptive Gaussian and Mittag-Leffler filters excel in noise reduction but may compromise fine signal details. Through MATLAB simulations and mean squared error (MSE) comparisons as well as Signal to Nosie Ratio (SNR), we evaluate the filters' performance in denoising real-world ECG signals. The results indicate that both the Savitzky-Golay smoothing and Mittag-Leffler filters hold promise for noise reducing in other biomedical signals, such as medical EEG and medical EMG signals. This research serves as a foundational exploration of the application and enhancement of these filters in biomedical signal processing. [ABSTRACT FROM AUTHOR]

In this paper, we shall find the order of starlikeness and convexity for integral operators Fαj,βj,λj,ζ (z) = {ζ ∫0z tζ−1 πnj=1 (Eαj,βj(t)/t )1/λj dt}1/ζ, where the functions Eαj,βj are the normalized Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

In this note, we suggest a generalization of gamma function to be 3D-gamma function. As a consequence, some special functions are generalized including the Mittag–Leffler function. Moreover, we utilize the generalized Mittag–Leffler to extend the AB-fractional calculus. Examples are introduced to cover our theory. The solvability of abstract Riccati equation is considered, and Hyers–Ulam stability is discovered in the sequel. Based on the new study, we design new stable of Self-organization in complex networks (SOCNs).

This paper focuses on the analysis of a class of stochastic differential equations with tempered φ-Caputo fractional derivative (φ-CFD) and Lévy noise. We propose comprehensive mathematical techniques to address the existence, uniqueness and stability of solution to this equation. For existence and uniqueness, the Picard–Lindelof successive approximation technique is used analyze the results. Also, We use Mittag-Leffler (M-L) function to investigate the stability of the solution. This research applies the broad understanding of stochastic processes and fractional differential equations, as well as known results, to the analysis of systems with tempered φ-CFD. These equations capture complex phenomena in the field of financial assets, making their investigation on the stock prices particularly valuable.

ABSTRACTIn this study, we investigate novel fractional tuberculosis model with Caputo fractional derivative. The computational solution of the fractional tuberculosis disease is obtained with the help of the generalized Euler's method (GEM). This article considers two treatment strategies: protective treatment for latent populations and main treatment for infected ones. The compartmental structure of the six-dimensional model includes the susceptible, latent, infected, recovered, and treatment classes. To examine the nature of differential equations of fractional order that arise in biological illness, the findings produced utilizing the method under consideration are more accurate and straightforward. Additionally, the stability of the equilibrium point is also investigated. The fractional model has been graphically simulated using MATLAB22.

Malaria remains a substantial public health challenge and economic burden globally. Currently, malaria has been declared as endemic in 85 countries. In this study, we developed and analyzed a fractional-order mathematical model for malaria transmission dynamics that incorporates variability of temperature and rainfall using Caputo-type AB operators. The existence and uniqueness of the model's solutions were established using the Banach fixed-point theorem. The model system's equilibria (both disease-free and endemic) were identified, and lemmas and theorems were developed to prove their stability. Furthermore, we used different temperature ranges and rainfall data, validating them against existing literature. Numerical simulations using the Toufik-Atangana schemes with various fractional-order alpha values revealed that as the value of alpha approaches 1, the behavior of the fractional-order model converges to that of the classical model. The numerical results are promising and are expected to be valuable for future research related to fractional-order models.