Terms Of Mittag-Leffler Function Research Articles

Remarks on the Solution of Fractional Ordinary Differential Equations Using Laplace Transform Method

In this work we used the Laplace transform method to solve linear fractional-order differential equation, fractional ordinary differential equations with constant and variable coefficients. The solutions were expressed in terms of Mittag-Leffler functions, and then written in a compact simplified form. As a special case for simplicity, the order of the derivative determined the order of the solution that was obtained. This paper presented several case studies involving the implementation of Fractional Order calculus-based models, whose results demonstrate the importance of Fractional Order Calculus.

Solution of Inhomogeneous Linear Fractional Differential Equations Involving Jumarie Fractional Derivative

This paper presents a method for solving inhomogeneous linear sequential fractional differential equations with constant coefficients (ILSFDE) involving Jumarie fractional derivatives in terms of Mittag-Leffler functions. For this purpose, the fundamental properties of the Jumarie derivative and Mittag-Leffler functions are given. After this, the successive jumarie fractional derivatives of Mittag-Leffler functions, fractional cosine, and sine functions are obtained. Further, we determined the particular integrals of these functions and then found the complete solutions of ILSFDE. in terms of Mittag-Leffler functions, fractional cosine, and sine functions. We have demonstrated this developed method with a few examples of ILSFDE. This method is similar to the method for finding the complete solutions of classical differential equations with constant coefficients.

New Results Involving the Generalized Krätzel Function with Application to the Fractional Kinetic Equations

Sun is a basic component of the natural environment and kinetic equations are important mathematical models to assess the rate of change of chemical composition of a star such as the sun. In this article, a new fractional kinetic equation is formulated and solved using generalized Krätzel integrals because the nuclear reaction rate in astrophysics is represented in terms of these integrals. Furthermore, new identities involving Fox–Wright function are discussed and used to simplify the results. We compute new fractional calculus formulae involving the Krätzel function by using Kiryakova’s fractional integral and derivative operators which led to several new identities for a variety of other classic fractional transforms. A number of new identities for the generalized Krätzel function are then analyzed in relation to the H-function. The closed form of such results is also expressible in terms of Mittag-Leffler function. Distributional representation of Krätzel function and its Laplace transform has been essential in achieving the goals of this work.

Analytical Investigation of the Heat Transfer Effects of Non-Newtonian Hybrid Nanofluid in MHD Flow Past an Upright Plate Using the Caputo Fractional Order Derivative

The objective of this paper is to examine the augmentation of the heat transfer rate utilizing graphene (Gr) and multi-walled carbon nanotubes (MWCNTs) as nanoparticles, and water as a host fluid in magnetohydrodynamics (MHD) flow through an upright plate using Caputo fractional derivatives with a Brinkman model on the convective Casson hybrid nanofluid flow. The performance of hybrid nanofluids is examined with various shapes of nanoparticles. The Caputo fractional derivative is utilized to describe the governing fractional partial differential equations with initial and boundary conditions on the flow model. Exact solutions are obtained for flow transport, temperature distribution besides that heat transfer rate and friction drag in terms of Mittag-Leffler function by using Fourier sine and Laplace techniques as hybrid methods. Further, we provided the limiting case solutions for classic partial differential equations on obtained governing fluid flow models. The influence of various physical parameters with different fractional orders are investigated on hybrid nanofluid’s fractional momentum and energy by plotting velocity and energy curves. Few of the findings suggest that fractional parameters have significant effect on flow parameters and that blade-shaped nanoparticles have a high heat transfer rate. The graphical results reveal that the Grashof number shows a symmetry effect in the case of cooling and heating the plate. Furthermore, the performance of hybrid nanofluid is considerably more effective with the Caputo-fractional derivatives rather than in the classic derivative approach.

Time fractional analysis of Casson fluid with application of novel hybrid fractional derivative operator

<abstract><p>A new approach is used to investigate the analytical solutions of the mathematical fractional Casson fluid model that is described by the Constant Proportional Caputo fractional operator having non-local and singular kernel near an infinitely vertical plate. The phenomenon has been expressed in terms of partial differential equations, and the governing equations were then transformed in non-dimensional form. For the sake of generalized memory effects, a new mathematical fractional model is formulated based on the newly introduced Constant Proportional Caputo fractional derivative operator. This fractional model has been solved analytically, and exact solutions for dimensionless velocity, concentration and energy equations are calculated in terms of Mittag-Leffler functions by employing the Laplace transformation method. For the physical significance of various system parameters such as $ \alpha $, $ \beta $, $ Pr $, $ Gr $, $ Gm $, $ Sc $ on velocity, temperature and concentration profiles, different graphs are demonstrated by Mathcad software. The Constant Proportional Caputo fractional parameter exhibited a retardation effect on momentum and energy profile, but it is visualized that for small values of Casson fluid parameter, the velocity profile is higher. Furthermore, to validated the acquired solutions, some limiting models such as the ordinary Newtonian model are recovered from the fractionalized model. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, from the literature, it is observed that to deriving analytical results from fractional fluid models developed by the various fractional operators is difficult, and this article contributes to answering the open problem of obtaining analytical solutions for the fractionalized fluid models.</p></abstract>

Transient analysis of M/M/1 fractional queue

Queuing models consisting of servers which may not work with full efficiency are modelled with the help of fractional differential equations. Such problems with partial activity of the server are designed with the help of differential-difference equations involving fractional derivatives whose solutions are expressed in terms of Mittag-Leffler function. In this paper, we propose an alternative approach which gives a transient solution of fractional M/M/1 queue in matrix form. The results obtained by this new approach are justified by comparing them with solutions of classical queue which are available in the literature. Efficacy of the model can be assessed by computing its state probabilities and also measures such as expected number of customers in the system etc. Also, the variations in these measures with respect to partial activity of the server have been presented graphically and numerically. Further, a mathematical procedure to find optimal efficiency of the server has been discussed.

An exact solution of heat and mass transfer analysis on hydrodynamic magneto nanofluid over an infinite inclined plate using Caputo fractional derivative model

<abstract> <p>This paper presents the problem modeled using Caputo fractional derivatives with an accurate study of the MHD unsteady flow of Nanofluid through an inclined plate with the mass diffusion effect in association with the energy equation. H<sub>2</sub>O is thought to be a base liquid with clay nanoparticles floating in it in a uniform way. Bousinessq's approach is used in the momentum equation for pressure gradient. The nondimensional fluid temperature, species concentration, and fluid transport are derived together with Jacob Fourier sine and Laplace transforms Techniques in terms of exponential decay function, whose inverse is computed further in terms of Mittag-Leffler function. The impact of various physical quantities interpreted with fractional order of the Caputo derivatives. The obtained temperature, transport, and species concentration profiles show behaviours for $0 < \mathtt{α} < 1$ where $\mathtt{α} $ is the fractional parameter. Numerical calculations have been carried out for the rate of heat transmission and the Sherwood number is swotted to be put in the form of tables. The parameters for the magnetic field and the angle of inclination slow down the boundary layer of momentum. The distributions of velocity, temperature, and concentration expand more rapidly for higher values of the fractional parameter. Additionally, it is revealed that for the volume fraction of nanofluids, the concentration profiles behave in the opposite manner. The limiting case solutions also presented on flow field of governing model.</p> </abstract>

Time fractional analysis of Casson fluid with Rabotnov exponential memory based on the generalized Fourier and Fick...s law

A new approach to investigates the analytical solutions of the mathematical fractional Casson fluid model that is described by the Yang-Abdel-Cattani fractional operator having Rabotnov exponential kernel near an infinitely vertical plate. The phenomenon has been expressed in terms of partial differential equations, then transformed the governing equations in non-dimentional form. For the sake of better rheology of Casson fluid, developed a fractional model by applying the new definition of Yang-Abdel-Cattani fractional derivative operator having Rabotnov exponential kernel that describe the generalized memory effects. For seeking exact solutions in terms of Mittag-Leffler functions, for fluid velocity, concentration and temperature, Laplace integral transformation method is used to solve the fractional model. For several physical significance of various fluid parameters such as α, β, Pr, Gr, Gm, Sc on velocity, concentration and temperature distributions are demonstrated through various graphs and it is noticed that the Yang-Abdel-Cattani fractional parameter portrayed the retardation effect on momentum and energy profile, but it is visualized that for small values of Casson fluid parameter the velocity profile is higher. Furthermore, for being validated the acquired solutions, some limiting models such as ordinary Newtonian model had recovered from Yang-Abdel-Cattani fractional Casson fluid. Moreover, the graphical representations of the analytical solutions illustrated the main results of the present work. Also, in the literature, it is observed that to derived analytical results from fractional fluid models developed by the various fractional operators, is difficult and this article contributing to answer the open problem of obtaining analytical solutions the fractionalized fluid models.

A NOVEL ANALYTICAL APPROACH FOR SOLVING TIME-FRACTIONAL DIFFUSION EQUATIONS

This article investigates the approximate analytical solutions of the time-fractional diffusion equations using a novel analytical approach, namely the Sumudu transform iterative method. The time-fractional derivatives are considered in the Caputo sense. The analytical solutions are found in closed form, in terms of Mittag-Leffler functions. Furthermore, the findings are shown graphically, and the solution graphs demonstrate a strong relationship between the approximate and exact solutions.

Mittag-Leffler form solutions of natural convection flow of second grade fluid with exponentially variable temperature and mass diffusion using Prabhakar fractional derivative

In this article, heat source impact on unsteady magnetohydrodynamic (MHD) flows of Prabhakar-like non integer second grade fluid near an exponentially accelerated vertical plate with exponentially variable velocity, temperature and mass diffusion through a porous medium. For the sake of generalized memory effects, a new mathematical fractional model is formulated based on newly introduced Prabhakar fractional operator with generalized Fourier's law and Fick's law. This fractional model has been solved analytically and exact solutions for dimensionless velocity, concentration and energy equations are calculated in terms of Mittag-Leffler functions by employing the Laplace transformation method. Physical impacts of different parameters such as α, Pr, β, Sc, Gr, γ, Gm are studied and demonstrated graphically by Mathcad software. Furthermore, to validate our current results, some limiting models such as classical second grade model, classical Newtonian model and fractional Newtonian model are recovered from Prabhakar fractional second grade fluid. Moreover, compare the results between second grade and Newtonian fluids for both fractional and classical which shows that the movement of the viscous fluid is faster than second grade fluid. Additionally, it is visualized that for both classical second grade and viscous fluid have relatively higher velocity as compared to fractional second grade and viscous fluid.

A Note on Natural Transformation & Meijer’s G-Function

The objective of this paper is to investigate natural transform of Meijer’s G-Function in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties in terms of Mittag-Leffler functions. Further, we show some applications of these natural transform in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

Generalized Mittag-Leffler Kernel Form Solutions of Free Convection Heat and Mass Transfer Flow of Maxwell Fluid with Newtonian Heating: Prabhakar Fractional Derivative Approach

In this article, the effects of Newtonian heating along with wall slip condition on temperature is critically examined on unsteady magnetohydrodynamic (MHD) flows of Prabhakar-like non integer Maxwell fluid near an infinitely vertical plate under constant concentration. For the sake of generalized memory effects, a new mathematical fractional model is formulated based on a newly introduced Prabhakar fractional operator with generalized Fourier’s law and Fick’s law. This fractional model has been solved analytically and exact solutions for dimensionless velocity, concentration, and energy equations are calculated in terms of Mittag-Leffler functions by employing the Laplace transformation method. Physical impacts of different parameters such as α, Pr, β, Sc, Gr, γ, and Gm are studied and demonstrated graphically by Mathcad software. Furthermore, to validate our current results, some limiting models such as classical Maxwell model, classical Newtonian model, and fractional Newtonian model are recovered from Prabhakar fractional Maxwell fluid. Moreover, we compare the results between Maxwell and Newtonian fluids for both fractional and classical cases with and without slip conditions, showing that the movement of the Maxwell fluid is faster than viscous fluid. Additionally, it is visualized that both classical Maxwell and viscous fluid have relatively higher velocity as compared to fractional Maxwell and viscous fluid.

Scaling Limits of Planar Symplectic Ensembles

We consider various asymptotic scaling limits $N\to\infty$ for the $2N$ complex eigenvalues of non-Hermitian random matrices in the symmetry class of the symplectic Ginibre ensemble. These are known to be integrable, forming Pfaffian point processes, and we obtain limiting expressions for the corresponding kernel for different potentials. The first part is devoted to the symplectic Ginibre ensemble with the Gaussian potential. We obtain the asymptotic at the edge of the spectrum in the vicinity of the real line. The unifying form of the kernel allows us to make contact with the bulk scaling along the real line and with the edge scaling away from the real line, where we recover the known determinantal process of the complex Ginibre ensemble. Part two covers ensembles of Mittag-Leffler type with a singularity at the origin. For potentials $Q(\zeta)=|\zeta|^{2\lambda}-(2c/N)\log|\zeta|$, with $\lambda>0$ and $c>-1$, the limiting kernel obeys a linear differential equation of fractional order $1/\lambda$ at the origin. For integer $m=1/\lambda$ it can be solved in terms of Mittag-Leffler functions. In the last part, we derive Ward's equation for planar symplectic ensembles for a general class of potentials. It serves as a tool to investigate the Gaussian and singular Mittag-Leffler universality class. This allows us to determine the functional form of all possible limiting kernels (if they exist) that are translation invariant, up to their integration domain.

Time fractional telegraph equation and its solution by Laplace transform method

In this paper, one-dimensional time fractional telegraph equation is deduced using fractional Ohm’s law in transmission lines. The nonlocal behavior, arising from the time fractality, is discussed via fractional calculus. Here, the time fractional differential equation related to electrical GLRC circuit is proposed in terms of the Liouville–Caputo fractional derivative. To keep the dimensionality of the physical parameters [Formula: see text] the new parameter [Formula: see text] is introduced. This parameter ([Formula: see text]) characterizes the existence of fractional structure in the system. The analytical solution of the proposed equation is obtained by using Laplace transform method in terms of Mittag-Leffler function which describes physical system with memory. The order of the time derivative being considered is [Formula: see text]. The result of an example shows that the behavior of voltage depends on the value of the fractional order of the proposed equation.

Study of S-Function in Relationship with Natural Transform

The objective of this paper is to investigate natural transform of the S -function defined and studied by Saxena and Daiya [4] in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties in terms of Mittag-Leffler functions. Further, we show some applications of these natural transform in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.

Robotnov function based operator for biological population model of biology

PurposeThe population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. This paper aims to propose a new Yang-Abdel-Aty-Cattani (YAC) fractional operator with a non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, reduced differential transform method (RDTM) and residual power series method (RPSM) taking the fractional derivative as YAC operator sense.Design/methodology/approachThis study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense.FindingsThis study has expressed the solutions in terms of Mittag-Leffler functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.Research limitations/implicationsThe population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this study, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.Practical implicationsThe population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation which is arised in biological population model. Here, this study has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.Social implicationsThe population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.Originality/valueThe population model has an important role in biology to interpret the spreading rate of viruses and parasites. This biological model is also used to identify fragile species. In this paper, the main aim is to propose a new YAC fractional operator with non-singular kernel to solve nonlinear partial differential equation, which is arised in biological population model. Here, this paper has explained the analytical methods, RDTM and RPSM taking the fractional derivative as YAC operator sense. This study has expressed the solutions in terms of Mittag-Leer functions. Also, this study has compared the solutions with the exact solutions. Three examples are described for the accuracy and efficiency of the results.

Delay-independent stability criteria for fractional order time delayed gene regulatory networks in terms of Mittag-Leffler function

This paper investigates the finite-time delay-free stability criteria for fractional-order gene regulatory networks (FOGRNs) with feedback regulation time delays. The existence of unique equilibrium points of considered systems is established based on the Banach fixed point theorem and Cauchy–Schwarz inequality. Further, when the order γ satisfies 0<γ<1 and 1<γ<2, some novel delay-free sufficient conditions are derived to ensure the finite-time stability for addressing fractional order systems by using the ideas of generalized Gronwall–Bellman inequality, equivalent norm techniques, and Laplace transform. In the end, the article comes up with two numerical cases to manifest the applicability and superiority of the obtained theoretical results.

Reflection and transmission of transient ultrasonic wave in fractal porous material: Application of fractional calculus

This paper provides a time domain model for the propagation of transient ultrasonic waves in a self-similar porous material having a rigid frame. This model is based on the formalism of Stillinger–Palmer–Stavrinou, which consists in modeling the fractal material as a porous medium with a non-integer dimensional space. This paper is devoted to the time-domain analytical calculus of the reflection and transmission operators that are expressed in terms of Mittag-Leffler functions. A sensitivity numerical study using ultrasonic reflected and transmitted waves is performed, highlighting the effect of the material’s physical parameters (fractal dimension, tortuosity, viscous characteristic length and porosity) on the waveforms.

Space-time fractional diffusion equation associated with Jacobi expansions

We consider the operator , where is the three term recurrence relation for the Jacobi polynomial normalized at the point x = 1 and I is the identity operator acting on . Knowing that for , these polynomials satisfy a product formula which provides a hypergroup structure on , we use the tools of harmonic analysis on this hypergroup to solve the space-time fractional diffusion equation Here, , is the Caputo time fractional derivative of order σ. The solution is given by , being the fundamental solution expressed in terms of Mittag-Leffler function and is the convolution on the hypergroup . We prove that, for , is nonnegative and we give some of its properties. Next, we study the one parameter operators associated with this fundamental solution.

Analytic and numerical solutions of discrete Bagley\u2013Torvik equation

In this research article, a discrete version of the fractional Bagley–Torvik equation is proposed: 1∇h2u(t)+AC∇hνu(t)+Bu(t)=f(t),t>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\nabla _{h}^{2} u(t)+A{}^{C} \\nabla _{h}^{\\nu }u(t)+Bu(t)=f(t),\\quad t>0, $$\\end{document} where 0<nu <1 or 1<nu <2, subject to u(0)=a and nabla _{h} u(0)=b, with a and b being real numbers. The solutions are obtained by employing the nabla discrete Laplace transform. These solutions are expressed in terms of Mittag-Leffler functions with three parameters. These solutions are handled numerically for some examples with specific values of some parameters.