Leffler Function Research Articles

Solutions of a Nonlinear Diffusion Equation with a Regularized Hyper-Bessel Operator

We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our current work is to show the existence and uniqueness of mild solutions. Our desired goal is achieved using the Picard iteration method, and our analysis is based on properties of Mittag–Leffler functions and embeddings between Hilbert scales spaces and Lebesgue spaces.

Transient and passage to steady state in fluid flow and heat transfer within fractional models

PurposeThe classical integer derivative diffusionmodels for fluid flow within a channel of parallel walls, for heat transfer within a rectangular fin and for impulsive acceleration of a quiescent Newtonian fluid within a circular pipe are initially generalized by introducing fractional derivatives. The purpose of this paper is to represent solutions as steady and transient parts. Afterward, making use of separation of variables, a fractional Sturm–Liouville eigenvalue task is posed whose eigenvalues and eigenfunctions enable us to write down the transient solution in the Fourier series involving also Mittag–Leffler function. An alternative solution based on the Laplace transform method is also provided.Design/methodology/approachIn this work, an analytical formulation is presented concerning the transient and passage to steady state in fluid flow and heat transfer within the diffusion fractional models.FindingsFrom the closed-form solutions, it is clear to visualize the start-up process of physical diffusion phenomena in fractional order models. In particular, impacts of fractional derivative in different time regimes are clarified, namely, the early time zone of acceleration, the transition zone and the late time regime of deceleration.Originality/valueWith the newly developing field of fractional calculus, the classical heat and mass transfer analysis has been modified to account for the fractional order derivative concept.

Applications of a q-Differential Operator to a Class of Harmonic Mappings Defined by q-Mittag–Leffler Functions

Many diverse subclasses of analytic functions, q-starlike functions, and symmetric q-starlike functions have been studied and analyzed by using q-analogous values of integral and derivative operators. In this paper, we define a q-analogous value of differential operators for harmonic functions with the help of basic concepts of quantum (q-) calculus operator theory; and introduce a new subclass of harmonic functions associated with the Janowski and q-Mittag–Leffler functions. We obtain several useful properties of the new class, such as necessary and sufficient conditions, criteria for analyticity, compactness, convexity, extreme points, radii of starlikeness, radii of convexity, distortion bounds, and integral mean inequality. Furthermore, we discuss some applications of this study in the form of some results and examples and highlight some known corollaries for verifying our main results presented in this investigation. Finally, the conclusion section summarizes the fact about the (p,q)-variations.

Stability of Boundary Value Discrete Fractional Hybrid Equation of Second Type with Application to Heat Transfer with Fins

The development in the qualitative theory of fractional differential equations is accompanied by discrete analog which has been studied intensively in recent past. Suitable fixed point theorem is to be selected to study the boundary value discrete fractional equations due to the properties exhibited by fractional difference operators. This article aims at investigating the stability results in the sense of Hyers and Ulam with application of Mittag–Leffler function hybrid fractional order difference equation of second type. The symmetric structure of the operators defined in this article is vital in establishing the existence results by using Krasnoselkii’s fixed point theorem. Banach contraction mapping principle and Krasnoselkii’s fixed point theorem are employed to establish the uniqueness and existence results for solution of fractional order discrete equation. A problem on heat transfer with fins is provided as an application to considered hybrid type fractional order difference equation and the stability results are demonstrated with simulations.

Caputo-Hadamard fractional differential equations on time scales: Numerical scheme, asymptotic stability, and chaos.

This study investigates Caputo-Hadamard fractional differential equations on time scales. The Hadamard fractional sum and difference are defined for the first time. A general logarithm function on time scales is used as a kernel function. New fractional difference equations and their equivalent fractional sum equations are presented by the use of fundamental theorems. Gronwall inequality, asymptotical stability conditions, and two discrete-time Mittag-Leffler functions of Hadamard type are obtained. Numerical schemes are provided and chaos in fractional discrete-time logistic equation and neural network equations are reported.

Modified Homotopy Perturbation Method and Approximate Solutions to a Class of Local Fractional Integrodifferential Equations

In this paper, the local fractional version of homotopy perturbation method (HPM) is established for a new class of local fractional integral-differential equation (IDE). With the embedded homotopy parameter monotonously changing from 0 to 1, the special easy-to-solve fractional problem continuously deforms to the class of local fractional IDE. As a concrete example, an explicit and exact Mittag–Leffler function solution of one special case of the local fractional IDE is obtained. In the process of solving, two initial solutions are selected for the iterative operation of local fractional HPM. One of the initial solutions has a critical condition of convergence and divergence related to the fractional order, and the other converges directly to the real solution. This paper reveals that whether the sequence of approximate solutions generated by the iteration of local fractional HPM can approach the real solution depends on the selection of the initial approximate solutions and sometimes also depends on the fractional order of the selected initial approximate solutions or the considered equations.

Application of the Laplace Transform to a New Form of Fractional Kinetic Equation Involving the Composition of the Galúe Struve Function and the Mittag–Leffler Function

The great importance of the fractional kinetic equations (FKEs) can be seen in the very recent research work of proposing such new equations and finding their solutions by different analytical and numerical methods. The paramount purpose of the present work is to proffer and study new FKEs involving the composition of the generalized Galúe Struve function of first kind and k-Mittag–Leffler function and apply a very influential effective analytical approach based on Laplace transform to find their analytical solutions. The obtained solutions have been presented with some real values and the simulation done via MATLAB. Furthermore, the numerical and graphical interpretations are also mentioned to illustrate the main results. Some special cases are to be taken under consideration to get the results in simpler form.

Local Stabilization of Delayed Fractional-Order Neural Networks Subject to Actuator Saturation

This paper investigates the local stabilization problem of delayed fractional-order neural networks (FNNs) under the influence of actuator saturation. First, the sector condition and dead-zone nonlinear function are specially introduced to characterize the features of the saturation phenomenon. Then, based on the fractional-order Lyapunov method and the estimation technique of the Mittag–Leffler function, an LMIs-based criterion is derived to guarantee the local stability of closed-loop delayed FNNs subject to actuator saturation. Furthermore, two corresponding convex optimization schemes are proposed to minimize the actuator costs and expand the region of admissible initial values, respectively. At last, two simulation examples are developed to demonstrate the feasibility and effectiveness of the derived results.

Editorial for Special Issue “Fractional Calculus Operators and the Mittag–Leffler Function”

Among the numerous applications of the theory of fractional calculus in almost all applied sciences, applications in numerical analysis and various fields of physics and engineering stand out [...]

(q1,q2)-Trapezium-Like Inequalities Involving Twice Differentiable Generalized m-Convex Functions and Applications

A new auxiliary result pertaining to twice (q1,q2)-differentiable functions is derived. Using this new auxiliary result, some new versions of Hermite–Hadamard’s inequality involving the class of generalized m-convex functions are obtained. Finally, to demonstrate the significance of the main outcomes, some applications are discussed for hypergeometric functions, Mittag–Leffler functions, and bounded functions.

Fractional simulations for thermal flow of hybrid nanofluid with aluminum oxide and titanium oxide nanoparticles with water and blood base fluids

Abstract The fractional model has been developed for the thermal flow of hybrid nanofluid due to the inclined surface. The thermal investigation of the hybrid nanomaterial is predicted by utilizing the molybdenum disulphide nanoparticles and graphene oxide nanomaterials. The flow computations for mixed convection flow of nanoparticles and base fluids are performed due to vertical oscillating plate. The simulations for the formulated model have been done ρ-Laplace transform technique for Caputo fractional simulations. Definitions of Mittage–Leffler function and ρ-Laplace transform are also presented for the governing model. The application of updated definitions of ρ-Laplace transform for the Caputo fractional model is quite interesting unlike traditional Laplace transforms. The comparative investigation for both types of nanoparticles is performed for heat and mass transfer rates. It is observed that the heat enhancement rate due to water-based nanoparticles is relatively impressive compared to the kerosene oil-based nanomaterials.

Trajectory controllability of nonlinear fractional Langevin systems

Abstract In this paper, we discuss the trajectory controllability of linear and nonlinear fractional Langevin dynamical systems represented by the Caputo fractional derivative by using the Mittag–Leffler function and Gronwall–Bellman inequality. For the nonlinear system, we assume Lipschitz-type conditions on the nonlinearity. Examples are given to illustrate the theoretical results.

An Inverse Problem for a Non-Homogeneous Time-Space Fractional Equation

We consider the inverse problem of finding the solution of a generalized time-space fractional equation and the source term knowing the spatial mean of the solution at any times t∈(0,T], as well as the initial and the boundary conditions. The existence and the continuity with respect to the data of the solution for the direct and the inverse problem are proven by Fourier’s method and the Schauder fixed-point theorem in an adequate convex bounded subset. In the published articles on this topic, the incorrect use of the estimates in the generalized Mittag–Leffler functions is commonly performed. This leads to false proofs of the Fourier series’ convergence to recover the equation satisfied by the solution, the initial data or the boundary conditions. In the present work, the correct framework to recover the decay of fractional Fourier coefficients is established; this allows one to recover correctly the initial data, the boundary conditions and the partial differential equations within the space-time domain.

Pathway fractional integral formula involving an extended Mittag–Leffler function

Abstract This work discovers the Laplace transform using a generalized pathway fractional integral formula involving an extended Mittag-Leffler function in the kernel for various parameters. Our findings are fairly broad in scope. Some well-known and novel results can also be obtained here.

Mittag–Leffler stability, control, and synchronization for chaotic generalized fractional-order systems

In this paper, we investigate the generalized fractional system (GFS) with order lying in (1, 2). We present stability analysis of GFS by two methods. First, the stability analysis of that system using the Gronwall–Bellman (G–B) Lemma, the Mittag–Leffler (M–L) function, and the Laplace transform is introduced. Secondly, by the Lyapunov direct method, we study the M–L stability of our system with order lying in (1, 2). Using the modified predictor–corrector method, the solutions of GFSs are calculated and they are more complicated than the classical fractional one. Based on linear feedback control, we investigate a theorem to control the chaotic GFSs with order lying in (1, 2). We present an example to verify the validity of control theorem. We state and prove a theorem to calculate the analytical formula of controllers that are used to achieve synchronization between two different chaotic GFSs. An example to study the synchronization for systems with orders lying in (1, 2) is given. We found an agreement between analytical results and numerical simulations.

On linear fractional differential equations with variable coefficients

We study and solve linear ordinary differential equations, with fractional order derivatives of either Riemann–Liouville or Caputo types, and with variable coefficients which are either integrable or continuous functions. In each case, the solution is given explicitly by a convergent infinite series involving compositions of fractional integrals, and its uniqueness is proved in suitable function spaces using the Banach fixed point theorem. As a special case, we consider the case of constant coefficients, whose solutions can be expressed by using the multivariate Mittag–Leffler function. Some illustrative examples with potential applications are provided.

Fractional Model of the Deformation Process

The article considers the fractional Poisson process as a mathematical model of deformation activity in a seismically active region. The dislocation approach is used to describe five modes of the deformation process. The change in modes is determined by the change in the intensity of the event stream, the regrouping of dislocations, and the change in and the appearance of stable connections between dislocations. Modeling of the change of deformation modes is carried out by changing three parameters of the proposed model. The background mode with independent events is described by a standard Poisson process. To describe variations from the background mode of seismic activity, when connections are formed between dislocations, the fractional Poisson process and the Mittag–Leffler function characterizing it are used. An approximation of the empirical cumulative distribution function of waiting time of the foreshocks obtained as a result of processing the seismic catalog data was carried out on the basis of the proposed model. It is shown that the model curves, with an appropriate choice of the Mittag–Leffler function’s parameters, gives results close to the experimental ones and can be allowed to characterize the deformation process in the seismically active region under consideration.

On the Finite-Time Boundedness and Finite-Time Stability of Caputo-Type Fractional Order Neural Networks with Time Delay and Uncertain Terms

This study investigates the problem of finite-time boundedness of a class of neural networks of Caputo fractional order with time delay and uncertain terms. New sufficient conditions are established by constructing suitable Lyapunov functionals to ensure that the addressed fractional-order uncertain neural networks are finite-time stable. Criteria for finite-time boundedness of the considered fractional-order uncertain models are also achieved. The obtained results are based on a newly developed property of Caputo fractional derivatives, properties of Mittag–Leffler functions and Laplace transforms. In addition, examples are developed to manifest the usefulness of our theoretical results.

A study on the dynamics of alkali–silica chemical reaction by using Caputo fractional derivative

In this paper, we propose a mathematical study to simulate the dynamics of alkali–silica reaction (ASR) by using the Caputo fractional derivative. We solve a non-linear fractional-order system containing six differential equations to understand the ASR. For proving the existence of a unique solution, we use some recent novel properties of Mittag–Leffler function along with the fixed point theory. The stability of the proposed system is also proved by using Ulam–Hyers technique. For deriving the fractional-order numerical solution, we use the well-known Adams–Bashforth–Moulton scheme along with its stability. Graphs are plotted to understand the given chemical reaction practically. The main reason to use the Caputo-type fractional model for solving the ASR system is to propose a novel mathematical formulation through which the ASR mechanism can be efficiently explored. This paper clearly shows the importance of fractional derivatives in the study of chemical reactions.

Quasi‐projective and finite‐time synchronization of delayed fractional‐order BAM neural networks via quantized control

This paper deals with the problems of quasi‐projective synchronization (QPS) and finite‐time synchronization (FTS) for a kind of delayed fractional‐order BAM neural networks (DFOBAMNNs). In order to reach the goals of synchronization and more accurately gauge of settling time and error level, several fresh quantized controllers are structured to make the utmost of confined communication resources. Then, based on the finite‐time theorem, quantized control strategy, Lyapunov function theory and properties of Mittag–Leffler function as well as inequality analysis techniques, some plentiful criteria are formed to set up a relation between control gains and quantization parameters. In addition, the corresponding error bound of QPS and guages of the settling time on FTS are also given. Finally, a few numerical examples are introduced to validate the effectiveness of the presented control protocols.