Fractional-time Schrödinger equation (FTSE) describes the evolution of quantum processes endowed with memory effects. FTSE manifestly breaks all consistency requirements of quantum field theory (unitarity, locality and compliance with the clustering theorem), unless the order of fractional differentiation and integration ( ) falls close to the standard index . Working in the context of the minimal fractal manifold (where , ), we confirm here that FTSE approximates the attributes of gravitational metric and provides an unforeseen generation mechanism for massive fields.

In this paper, two types of the general fractional derivatives of distributed order as well as a corresponding fractional integral of distributed type are defined and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of the one-parametric Sonin kernels with a power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann-Liouville and the Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper.

In this paper we study the class of mixed-index time fractional differential equations in which different components of the problem have different time fractional derivatives on the left hand side. We prove a theorem on the solution of the linear system of equations, which collapses to the well-known Mittag-Leffler solution in the case the indices are the same, and also generalises the solution of the so-called linear sequential class of time fractional problems. We also investigate the asymptotic stability properties of this class of problems using Laplace transforms and show how Laplace transforms can be used to write solutions as linear combinations of generalisedMittag-Leffler functions in some cases. Finally we illustrate our results with some numerical simulations.

In the particular configuration of the scalar field K-essence in the Wheeler-DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is $\beta=\frac{2\alpha}{2\alpha - 1}$. This fractional equation belongs to different intervals, depending on the value of the barotropic parameter; when $\omega_{X} \in [0,1]$, the order belongs to the interval $1\leq \beta \leq 2$, and when $\omega_{X}\in[-1,0)$, the order belongs to the interval $0<\beta \leq 1$. In the quantum scheme, we introduce the factor ordering problem in the variables $(\Omega,\phi)$ and its corresponding momenta $(\Pi_\Omega, \Pi_\phi)$, obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time $T(\tau)$, however in the dust era we found a closed solution in the gauge time $\tau$.

This paper revisits the analytical theory of fractional vibrations with the highlights in five aspects. First, we address the cases of structures with frequency dependent mass or damping or stiffness in Sections 2-4. Second, we introduce the theory based on the general second-order vibration motion equation with frequency dependent elements (mass, damping, stiffness) in Sections 5-7. Third, we present the analytical theory of seven specific classes of second-order vibration systems with frequency dependent mass or damping or stiffness in Sections 8 and 9. Fourth, we bring forward the analytical theory of seven classes of fractional vibration systems in Sections 10-12. Finally, as an application, we give the closed form expression of the forced response to multi-fractional Euler-Bernoulli beam in Section 13. The explanation of the nonlinearity of fractional vibrations is given in Section 14.

In this paper, the 1st level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set of the kernels that satisfy this condition and posses an integrable singularity of power law type at the origin. The 1st level general fractional derivatives of arbitrary order are integro-differential operators of convolution type with the kernels from this set. They contain both the general fractional derivatives of arbitrary order of the Riemann-Liouville type and the regularized general fractional derivatives of arbitrary order considered in the literature so far. For the 1st level general fractional derivatives of arbitrary order, some important properties including the 1st and the 2nd fundamental theorems of Fractional Calculus are formulated and proved.

In this paper, we investigate an analytical solution of a class of nonlinear fractional integro-differential equation base on a generalized fractional power series expansion. The fractional derivatives are described in the conformable's type. The new approach is a modified form of the well-known Taylor series expansion. The illustrative examples are presented to demonstrate the accuracy and effectiveness of the proposed method

This paper proposes a definition of fractional line integral, generalising the concept of fractional definite integral. The proposal replicates the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It is based on the concept of fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integrals the Gr\"unwald-Letnikov and Liouville directional derivatives are introduced and their properties described. The integral is defined first for broken line paths and afterwards to any regular curve

In this research article we solve the problem of synchronization and anti-synchronization of chaotic systems described by discrete and time-delayed variable fractional order differential equations. To guarantee the synchronization and anti-synchronization of these systems, we use the well-known PID control theory and the Lyapunov-Krasovskii stability theory for discrete systems of variable fractional order.We illustrate the results obtained through simulation with examples, in which it can be seen that our results are satisfactory, thus achieving synchronization and anti-synchronization of chaotic systems of variable fractional order with discrete time delay.

In this study, we actually want to explore the time-fractional Phi-four equation via two methods, the exp _a function method and the hyperbolic function method. We transform a fractional order dierential equation into an ordinary differential equation using a wave transformation and the fractional derivative in conformable form. Then, the resulting equation has successfully been explored for new explicit exact solutions. The procured solutions are simply showed the effectiveness and plainness of the projected methods.

New exact solutions to conformable time fractional EW and modified EW equations are constructed by using Sine-Gordon expansion approach. The fractional traveling wave transform and homogeneous balance have significant roles in the solution procedure. The predicted solution is of the form of some finite series of multiplication of powers of cos and sin functions. The relation among trigonometric and hyperbolic functions in sense of Sine-Gordon expansion gives opportunity to construct the solutions in terms of hyperbolic functions.

Rough volatility models are popularized by \cite{gatheral2018volatility}, where they have shown that the empirical volatility in the financial market is extremely consistent with rough volatility. Fractional Riccati equation as a part of computation for the characteristic function of rough Heston model is not known in explicit form as of now and therefore, we must rely on numerical methods to obtain a solution. In this paper, we give a short introduction to option pricing theory and an overview of the current advancements on the rough Heston model.

Uncertain fractional differential equation (UFDE) is an excellent tool for describing complicated dynamic systems. This paper analyzes the valuation problems of currency option based on UFDE under optimistic value criterion. Firstly, an uncertain fractional currency model is formulated to describe the dynamics of foreign exchange rate. Then, the pricing formulae of European, American and Asian currency option are obtained under optimistic value criterion, respectively. Finally, numerical simulations are provided to illustrate the validity of our theoretical results.

The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and showed that it cannot be considered as a derivative.

This article deals with the conformable double Laplace transforms and their some properties with examples and also the existence Condition for the conformable double Laplace transform is studied. Finally, in order to obtain the solution of nonlinear fractional problems, we present a modified conformable double Laplace that we call conformable double Laplace decomposition methods (CDLDM). Then, we apply it to solve, Regular and singular conformable fractional coupled burgers equation illustrate the effectiveness of our method some examples are given.

The exact solutions of some conformable time fractional PDEs are presented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional Regularized Long Wave-Burgers (RLW-Burgers, potential Korteweg-de Vries (KdV) and clannish random walker's parabolic (CRWP) equations. Initially, the predicted solution in the finite series of a rational form of an exponential function is substituted to the ODE generated from the conformable time fractional PDE by using wave transformation. The coefficients used in the finite series are determined by solving the algebraic system derived from the coefficients of the powers of the predicted solution.

Recently, Gohar et al. introduced a novel, local, and well-behaved fractional calculus. It possesses all the classical properties, and Its locality imposes simplicity and accuracy in modeling fractional order systems. In this article, we further develop the definitions and extend the classical properties of Gohar fractional calculus to address some of the open problems in Calculus. The fractional Gronwall's integral inequality, Taylor power series expansion, and Laplace transform are defined and applied to overcome some of the limitations in the classical integer-order calculus. The fractional Laplace transform is applied to solve Bernoulli-type logistic and Bertalanffy nonlinear fractional differential equations, and the criteria under which it can be applied to solve linear differential equations are investigated.

DGJ (Daftardar-Gejii-Jafaris) method is used to obtain numerical solution of the third order fractional differential equation. Providing the DGJ method converges, the approximate solution is a good and effective numerical result which is close to the exact solution or the exact solution. For this,the examples of the explaning the method are presented. The proposed method is implemented for the approximation solution of the third order nonlinear fractional partial differential equations. The method was shown to be unsuitable and inconsistent for an example of a nonlinear fractional partial differential equation depend on initial-boundary value conditions. The fact that these numerical results are not consistent can be explained by the fact that the method is not convergent.

Based on the fractional order of nonlinear system for love model with a periodic function as an external force, analyzed the characteristics of the chaotic dynamic in this study. The relationship between the chaotic dynamic of the fractional-love model with the external force and the fractional-order system was analyzed when the parameters are fixed. Further, we also studied the relationship between the chaotic systemic dynamic and the parameters when the fractional-order system is fixed. The results show that when the parameters are fixed, the fractional-order system exhibited segmented chaotic states for the different fractional orders of the system. When fixed the fractional-order system, the system exhibited the periodic and chaotic states as parameter changes.

Power series expansions are useful in approximation theory and mathematical physics. The manuscript presents several types of fractional Taylor expansions of sufficiently smooth functions. This is achieved by employing an incremental regularization procedure to the computation of the derivative. The series are constructed in an algorithmic way, which allows for its implementation in computer algebra systems.

The fractional derivatives are a generalization the derivatives of integer order and find applications in studying memory processes in various scientific fields. Numerical methods are used to solve and analyze fractional models of real world problems. In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic formula, whose generating function is the polylogarithmic function. In the paper, we prove the convergence of the approximation and derive an estimate for the error and order. We consider an application of the approximation for the construction of finite difference schemes for numerical solution of the two-term ordinary fractional differential equation and the time-fractional Black-Scholes equation for option pricing. The properties of the approximation are used to prove the convergence of the methods used for numerical solution of the fractional differential equations. The theoretical results for the error and order of the methods are illustrated by the results of numerical experiments.

The paper's main goal is to introduce a novel type of interpolative convex contraction and build up some fresh findings for it using the interpolative convex contractions' progressive approach. We have established certain conclusions using orbitally S-complete and Suzuki type contractions in F-metric spaces. My research aims to examine the fixed point method's efficacy in solving fractional differential equation with boundary conditions.

We investigate the influence of a self-propelling, out-of-equilibrium active particle on generalized elastic systems, including flexible and semiflexible polymers, fluid membranes, and fluctuating interfaces, while accounting for long-ranged hydrodynamic effects. We derive the fractional Langevin equation governing the dynamics of the active particle, as well as that of any other passive particle (or probe) bound to the elastic system. Our study explores the diffusional behavior emerging for both the active particle and a distant probe.

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.

The space time fractional Korteweg-de Vries equation, modified Equal Width equation and Benney-Luke Equations are solved by using simple hyperbolic tangent Ansatz method. A simple compatible wave transformation in one dimension is employed to reduce the governing equations to integer ordered ODEs. Then, the ansatz approximation is used to derive exact solutions. Some illustrative examples are presented for some particular choices of parameters and derivative orders.

The emergence of the COVID-19 outbreak has caused a pandemic situation in over 210 countries. Controlling the spread of this disease has proven difficult despite several resources employed. Millions of hospitalization and deaths have been observed, and thousands of cases daily with many measures in place. Due to the complex nature of COVID-19, we proposed a system of time-fractional equations to understand the transmission of the disease better. Nonlocality involved in the model has made fractional differential equations appropriate for modeling the behavior. However, solving these types of models is computationally demanding. Our proposed generalized compartmental COVID-19 model incorporates effective contact rate, transition rate (from exposed quarantine and recovered to susceptible and infected quarantined individuals), quarantine rate, disease-induced death rate, natural death rate, natural recovery rate, recovery rate of quarantine infected for a holistic study of the coronavirus disease. A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, local and global stability analysis of the disease-free equilibrium analysis, and sensitivity analysis. Furthermore, numerical solutions of the proposed model are obtained with the generalized Adam-Bashforth-Moulton method developed for the fractional order model. Our analysis and solutions profile show that each of these incorporated parameters is very important in controlling the spread of COVID-19, especially quarantining exposed and infected individuals and the effective contact rate. Based on the results with different fractional order, we observe that there seems to be a third or even fourth wave of the spike in cases of COVID-19, which is what is happening right now in many countries.

The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave(RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The homogeneous balance procedure gives the order of the predicted polynomial-type solution that is inspired from well-known Sine-Gordon equation. The substitution of this solution follows the previous step. Equating the coefficients of the powers of predicted solution leads a system of algebraic equations. The solution of resultant system for coefficients gives the necessary relations among the parameters and the coefficients to be able construct the solutions. Some solutions are simulated for some particular choices of parameters.

In this research, a computation algorithm is established for a fractional order 2D fuzzy heat equation. In this study, Elzaki transform and HPM fusion is produced. Computing the desired outcome in series yields a fast convergence on an appropriate response. Examples are provided to support the conclusions, which are then compared with a particular approach to show the effectiveness and potential of the suggested approach. Two crisp equations—one for the lower bound solution and one for the upper bound solution are constructed from the input fuzzy fractional heat equation. The contour and surface representations of the approximate and exact results are offered for the lower and upper-bound solutions. The l∞-error norm is used in this study to validate the numerical convergence aspect. Together with the absolute inaccuracy, the approximate and exact solutions are matched. It has been demonstrated that the proposed regime will make it feasible to work with fuzzy fractional partial differential equations in a wide range of dimensions.

This experimental research presents an easier model for fitting experimental results that can be simulate the dynamical behaviour of a single degree freedom system, SDFS, subjected in a forced vibration in a shaking table. The fractional damping using a Kelvin-Voigt fractional derivative rheological model, KVFRdm, to represent a fractional oscillator and compare with the experimental model.

Here are given tight probabilistic inequalities that provide nearly best estimates for the Csiszar's f-divergence. These use the right and left psi -Hilfer fractional derivatives of the directing function f. Csiszar's f- divergence or the so called Csiszar's discrimination is used as a measure of dependence between two random variables which is a very essential aspect of stochastics, we apply our results there. The Csiszar's discrimination is the most important and general measure for the comparison between two probability measures. We give also other applications.

In this paper the D 3 dihedral logistic map of fractional order is introduced. The map 1 presents a dihedral symmetry D 3 . It is numerically shown that the construction and interpretation 2 of the bifurcation diagram versus the fractional order require special attention. The system stability 3 is determined and the problem of hidden attractors is analyzed. Also, analytical and numerical 4 results show that the chaotic attractor of integer order, with D 3 symmetries, looses its symmetry 5 in the fractional-order variant.

Exact solutions to conformable time fractional (3+1)- dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE. The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo-Miwa, the Zakharov-Kuznetsov and the modified Zakharov-Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.

A stochastic dominance (SD) relation can be defined by two different perspectives: One from the view of distributions, and the other one from the view of expected utilities. In early days, Fishburn investigated SD from the view of distributions and we refer this perspective as Fishburn’s SD. One of his many results was the development of fractional-order SD for continuous distributions. However, discrete fractional-order SD may not be generalized directly since some properties of fractional calculus do not have a discrete counterpart. In this paper, we develop a discrete analogue of fractional-order SD from the view of distributions. We generalize the order of SD by Lizama’s fractional delta operator, show the preservation of SD hierarchy, and formulate the utility classes that are congruent with our SD relations. This work brings a message that some results of discrete SD cannot be generalized directly from continuous SD. We characterize the difference between discrete and continuous fractional-order SD, as well as the way to handle them.

An expansion method based on time fractional Sine-Gordon equation is implemented to construct some real and complex valued exact solutions to the Korteweg-de Vries and modified Korteweg-de Vries equation in time fractional forms. Compatible fractional traveling wave transform plays a key role to be able to apply homogeneous balance technique to set the predicted solution. The relation between trigonometric and hyperbolic functions based on fractional Sine-Gordon equation allows to form the exact solutions with multiplication of powers of hyperbolic functions.

In this present paper, we deal with the generalized (k, s)-fractional integral and differential operators recently defined by Nisar et al. and obtain some generalized (k, s)-fractional integral and differential formulas involving the class of a function as its kernels. Also, we investigate a certain number of their consequences containing the said function in their kernels.

The main objective of this present paper is to introduce further extension of extended Caputo fractional derivative operator and establish the extension of an extended fractional derivative of some known elementary functions. Also, we investigate the extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and generating relations for extension of extended hypergeometric functions.

In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.

The gradient descent algorithm has become the standard algorithm for computing the extreme values of functions, but for multivariate functions this algorithm is mostly ineffective. This is because the convergence rate of each element is inconsistent in most cases. However, in this paper, we found that the gradient sign is a self-optimizing operator, ensuring that the convergence rate is consistent across all elements. This also explains, from an optimization perspective, the success of the Fast Gradient Sign Method (FGSM) in generating adversarial samples that are indistinguishable from the normal input, but can easily fool neural networks. We also found that the fractional order gradient is also self-optimizing, and that the convergence speed of this algorithm can be controlled by adjusting the order of the gradient. Experiments suggest that this algorithm not only generates adversarial samples faster than other algorithms, but that a single source image can generate many such samples. This algorithm is also more effective than others at generating adversarial samples from simple images.

Fractional calculus gained a lot of attention in the last couple of years. Researchers discovered that processes in various fields follow rather fractional dynamics than ordinary integer-ordered dynamics, meaning the corresponding differential equations feature non-integer valued derivatives. There are several arguments for why this is the case, one of them being that fractional derivatives’ inherit spatiotemporal memory and/or the ability to express complex naturally occurring phenomena. Another popular topic nowadays is machine learning, i.e., learning behavior and patterns from historical data. In our ever-changing world with ever-increasing amounts of data, machine learning is a powerful tool for data analysis, problem-solving, modeling, and prediction. It further provides many insights and discoveries in various scientific disciplines. As these two modern-day topics provide a lot of potential for combined approaches to describe complex dynamics, this article reviews combined approaches of fractional derivatives and machine learning from the past, puts them into context, and thus provides a list of possible combined approaches and the corresponding techniques. Note, however, that this article does not deal with neural networks, as there already is profound literature on neural networks and fractional calculus. We sorted past combined approaches from the literature into three categories, i.e., preprocessing, machine learning &amp; fractional dynamics, and optimization. The contributions of fractional derivatives to machine learning are manifold as they provide powerful preprocessing and feature augmentation techniques, can improve physically informed machine learning, and are capable of improving hyperparameter optimization. Thus, this article serves to motivate researchers dealing with data-based problems, to be specific machine learning practitioners, to adopt new tools and enhance their existing approaches.

The role of fractional calculus in circuit systems has received increased attention in recent years. In order to evaluate the effectiveness of time-domain calculation methods in the analysis of fractional-order piecewise smooth circuit systems, an experimental prototype is developed and the effects of three typical calculation methods in different test scenarios are compared and studied in this paper. It is proved that Oustaloup’s rational approximation method usually overestimates the peak-to-peak current and brings in pulse-voltage phenomenon in piecewise smooth test scenarios, while the results of two iterative recurrence-form numerical methods are in good agreement with the experimental results. The study results are dedicated to provide a reference for efficiently deploying calculation methods in fractional-order piecewise smooth circuit systems. Some quantitative analysis results are concluded in this paper.

This study presents finite-time synchronization of chaotic fractional order systems with disturbance uncertainty and unknown time delay. First, a PID sliding surface is presented. Then, for the finite-time synchronization of the master and slave systems, a robust sliding-adaptive control method is provided. Update rules have been retrieved to estimate the system parameters by using the Lyapunov function while establishing the stability of the suggested mechanism and assuring the convergence of errors to zero. The proposed approach was used to solve the fractional order system of Gensou-Tsei with time-varying parameters, and the simulation results show that the presented approach performs well.

In this paper, we propose to explore a nonlinear fractional mathematical model of HFMD- Hand Foot Mouth Disease with vaccinated compartment. Firstly, we prove the non negativity and boundedness for the fractional order dynamical model. The system’s existence and uniqueness are studied using the formulation of the Caputo derivative operator. The fixed-point approach was used to get results indicating the presence of at least one solution. In numerical simulation, we obtained the approximate solution by using fractional Euler method.

In this paper, we establish the generalized Qi-type inequality involving conformable fractional integrals. The results presented here would provide extensions of those given in earlier works.

This work aims to introduce a novel concept: the Riesz-Dunkl fractional derivatives, within the context of Dunkl type operators. A particularly noteworthy revelation is that when a specific parameter $\kappa$ equals zero, the Riesz-Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform.

The exact solutions in the wave form are derived for the time fractional KdV and the time fractional Burgers' equations in conformable fractional derivative sense. The fractional variable change using the fundamental properties of the conformable derivative reduces both equations to some nonlinear ODEs. The predicted solution is assumed to be a finite series form of a function satisfying a particular first-order ODE whose solution contains an exponential function in the denominator. The solutions are represented in the explicit forms and illustrated by some choices of the parameters for various fractional orders of the equations.

In view of the problem of cracking easily and partial distortion of images after encryption or decryption, a novel image encryption and decryption algorithm based on Arnod Transform and fractional chaotic is proposed. To begin with, the Arnold transform is used to encrypt. So that the spatial confidence of the original image has been comprehensively disturbed. Secondly, the XOR involving the fractional order chaotic sequence is used to encrypt. The key sequence is dynamically generated to ensure the randomness and difference of key generation. When decryption is required, the first decryption is performed using the key and XOR. Then the second decryption is carried out by using the inverse Arnold transform, and finally the decrypted image is obtained. Experimental results show that the improved algorithm has achieved better performance in encryption and decryption.

This paper studies the finite-time synchronization problem of fractional-order stochastic memristive bidirectional associative memory neural networks (MBAMNNs) with discontinuous jumps. A novel criterion for finite-time synchronization is obtained by utilizing the properties of quadratic fractional-order Gronwall inequality with time delay and the comparison principle. This criterion provides a new approach to analyze the finite-time synchronization problem of neural networks with stochasticity. Finally, numerical simulations are provided to demonstrate the effectiveness and superiority of the obtained results.

In this paper we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, we analyze the analytical solution for a class of fractional models associated with Logistic model, Von Foerster model and Bertalanffy model is presented graphically for various fractional orders and solution of corresponding classical model is recovered as a particular case.

Fractional differential systems have received much attention in recent decades likely due to its powerful ability in modeling memory processes, which are mostly observed in real world. Fractional models have been widely investigated and applied in many fields such as physics, biochemistry, electrical engineering, continuum and statistical mechanics. It is shown by many researchers that fractional derivatives can provide more accurate models than integer order derivatives do. Obviously, bang-bang property is of a significant importance in optimal control theory. In this paper a Bang-Bang property and time-optimal control problem for time-Fractional Differential System (FDS) is under consideration. First we formulate our problem and we prove the existence theorem. We state and prove the bang-bang theorem. Finally we give the optimality conditions that characterized the optimal control. Some illustrate applications are given to clarify our results.

The theory of fractional calculus extends the order of classical integer calculus from integer to non-integer. As a new engineering application tool, it has made many important research achievements in many fields, including image processing. This paper mainly studies the application of fractional calculus theory in image enhancement and denoising, including the basic theory of fractional calculus and its amplitude frequency characteristics, the application of fractional differential operator in image enhancement, and the application of fractional integral operator in image denoising. The experimental results show that the fractional calculus theory has more special advantages in image enhancement and denoising. Compared with the existing integer order image enhancement operators, the fractional differential operator can more effectively enhance the "weak edge" and "strong texture" details of the image. The fractional order integral image denoising operator can not only improve the signal-to-noise ratio of the image compared to traditional denoising methods, but also better preserve detailed information such as edges and textures of the image.

In this paper, we compare the performance between systems of ordinary and (Caputo) fractional differential equations depicting the susceptible-exposed-infectious-recovered (SEIR) models of diseases. In order to understand the origins of both approaches as mean-field approximations of integer and fractional stochastic processes, we introduce the fractional differential equations as approximations of some type of fractional nonlinear birth--death processes. Then, we examine validity of the two approaches against empirical courses of epidemics; we fit both of them to case counts of three measles epidemics that occurred during the pre-vaccination era in three different locations. While FDEs appear more flexible in fitting empirical data, our ODEs offered better fits to two out of three data sets. Important differences in transient dynamics between these modeling approaches are discussed.

Recently, different extensions of the fractional derivative operator are found in many research papers. The main aim of this paper is to establish an extension of the extended Caputo fractional derivative operator. The extension of an extended fractional derivative of some elementary functions derives by considering an extension of beta function which includes the Mittag-Leffler function in the kernel. Further, an extended fractional derivative of some familiar special functions, the Mellin transforms of newly defined Caputo fractional derivative operator and the generating relations for extension of extended hypergeometric functions also presented in this study.

Our aim in this paper is to establish some new Hermite-Hadamard- Mercer type integral inequalities by utilizing the fractional proportional-integral operators.For this purpose, Hermite-Hadamard-Mercer inequalities for di¤er- antiable mappings whose derivatives in absolute value are convex.

Here we present Hilfer-Polya, psi-Hilfer Ostrowski and psi-Hilfer-Hilbert- Pachpatte types fractional inequalities. They are univariate inequalities involving left and right Hilfer and -Hilfer fractional derivatives. All estimates are with respect to norms ||.||p, 1 <_ p <_infinite. At the end we provide applications.

This paper presents Chebyshev Type inequalities for the Riemann-Liouville variable-order fractional integral operator using two synchronous functions on the set of real numbers. It is the first result of its kind in the current literature using variable-order Riemann-Liouville fractional integral operator. Some special cases for the result obtained in the paper are discussed.

Fractional calculus is defined by expanded integer order integration and differentiation. In this paper, multiple mathematical models of a nonlinear dual tank device are precisely formulated by fractional calculus. Using the accurate model, multivariable Fractional-Order Controller is designed for the nonlinear device. The merits of the fractional-order design include: 1) Control of multivariable nonlinearities, 2) Compensation of uncertainties, 3) Elimination of coupling effects. Simulations and experiments are conducted to verify the precision of the fractional order models and the effectiveness of the multivariable fractional-order control design.

The three dimensional conformable time fractional Kadomtsev-Petviashvili and the conformable time fractional modified Kawahara equations are solved by implementing the Kudryashov's procedure. The corresponding wave transformation reduces both equations to some ODEs. Balancing the nonlinear and the highest order derivative terms gives the structure of the solutions in the finite series form. The useful symbolic tools are used to solve the resultant algebraic systems. The solutions are expressed in explicit forms.

In this paper, we succeed to discover the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of concerned model. Finally, this technique is simple, fruitful and reliable to handle the nonlinear FPDEs.

The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its extensive use in optics, engineering, and signal processing. In the present work, we aimed to expand the fractional Hilbert transform to a space of generalized functions known as Boehmians. We introduce a new fractional convolution operator for the fractional Hilbert transform to prove a convolution theorem similar to the classical Hilbert transform and also to extend the fractional Hilbert transform to Boehmians. We also construct a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform.

The Fourier Continuous Derivative ($D_C$) offers a unique perspective on fractional differentiation grounded in the theory of Fourier series. This approach has the potential to address problems across various disciplines, including physics, engineering, and mathematics. The primary insight underpinning this approach is that a convex function defined on $\mathbb{Z}$ retains its convexity on $\mathbb{R}$. This paper delves into the Fourier Continuous Derivative, compares it with traditional fractional derivatives, and outlines its possible real-world applications, such as modeling viscoelastic materials, solving wave equations, and financial data analysis.

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm's accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of problem. Under Caputo meaning the obtained results are illustrated numerically, and graphically. Geometrically, the behavior of the solution declares that the changing of the fractional derivative parameter values' in their domain alters the style of the attained solution in a symmetrical meaning and fully coinciding to the ordinary derivative value'. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for Integral, and integro-differental equations of fractional order.

In this paper, we pursue and investigate the solutions for fractional kinetic equations, involving Bessel-Struve function by means of their Sumudu transforms. In the process, one Important special case is then revealed, and analyzed. The results obtained in terms of Bessel-Struve function are rather general in nature and can easily construct various known and new fractional kinetic equations.

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.

The main objective of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. We establish some results related to the newly defined fractional operator such as Mellin transform and relations to k-hypergeometric and k-Appell's functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and Wright hypergeometric functions.

Recently, representation formulae and monotonicity properties of generalized k-Bessel functions, Wk v,c., were established and studied by SR Mondal [24]. In this paper, we pursue and investigate some of their image formulae. We then extract solutions for fractional kinetic equations, involving Wk v,c, by means of their Sumudu transforms. In the process, Important special cases are then revealed, and analyzed.

This article introduces a new fractional approach to the concept of information dimension of complex networks, based on a (q,q′)-entropy proposed in the literature. The q parameter measures how far is the number of subsystems (for a given size ε) from the mean number of overall sizes. The q′ (interaction index) measures when the interactions between subsystems are greater (q′&gt;1), lesser (q′&lt;1) or equal to the interactions into these subsystems. The computations of the proposed information dimension are carried out on several real-world and synthetic complex networks. The results from the proposed information dimension are compared with those from the information dimension based on the Shannon entropy.

The use of salt rock for underground radioactive waste disposal facilities requires a comprehensive analysis of creep-damage process in salt rock. A computer-controlled creep setup is employed to carry out a creep test of salt rock lasted as long as 359 days under a constant uniaxial stress. The AE space-time evolution and energy releasing characteristics during creep test are studied in the meantime. A new creep-damage model is proposed on the basis of fractional derivative by combining the AE statistical regularity. It indicates that the AE data in non-decay creep process of salt rock can be divided into three stages. Furthermore, the parameters of new creep-damage model are determined by Quasi-Newton method. The fitting analysis suggests that the creep-damage model based on fractional derivative in this paper provides a precise description of full creep regions in salt rock.

Suitable measures of grazing impacts on ground cover, that enable separation of the effects of climatic variations, are needed to inform land managers and policy makers across the arid rangelands of the Northern Territory of Australia. This work developed and tested a time-series, change-point detection method for application to time series of vegetation fractional cover derived from Landsat data to identify irregular and episodic ground-cover growth cycles. These cycles were classified to distinguish grazing impacts from that of climate variability. A measure of grazing impact was developed using a multivariate technique to quantify the rate and degree of ground cover change. The method was successful in detecting both long term (> 3 years) and short term (< 3 years) growth cycles. Growth cycle detection was assessed against rainfall surplus measures indicating a relationship with high rainfall periods. Ground cover change associated with grazing impacts was also assessed against field measurements of ground cover indicating a relationship between both field and remotely sensed ground cover. Cause and effects between grazing practices and ground cover resilience can now be explored in isolation to climatic drivers. This is important to the long term balance between ground cover utilisation and overall landscape function and resilience.

In this study, we first discover some new concept coordinated UD-convex mappings with fuzzy-number values. After that, we look into Hermite-Hadamard type inequalities via fuzzy-number-valued coordinated UD-convex fuzzy-number-valued mapping (coordinated UD-convex FNVM). In the case of coordinated UD-convex FNVM, novel conclusions are derived by making particular decisions in recently proven inequalities. Additionally, it is demonstrated that the recently discovered inequalities are expansions of comparable findings in the literature. It is important to note that the main outcomes are validated by nontrivial examples.

This paper discusses the concept of fractional derivative with complex order from the application point of view. It is shown that a fractional derivative is hermitian, if and only if the order is real. The hermitian part of complex order derivatives behave like allpass filters with almost logarithmic phase.

In this article, we consider the three dimensional $\alpha$-fractional nonlinear delay differential system of the form \begin{align*} D^{\alpha}\left(u(t)\right)&=p(t)g\left(v(\sigma(t))\right),\\D^{\alpha}\left(v(t)\right)&=-q(t)h\left(w(t))\right),\\D^{\alpha}\left(w(t)\right)&=r(t)f\left(u(\tau(t))\right),~ t \geq t_0, \end{align*} where $0 < \alpha \leq 1$, $D^{\alpha}$ denotes the Katugampola fractional derivative of order $\alpha$. We have established some new oscillation criteria of solutions of differential system by using generalized Riccati transformation and inequality technique. The obtained results are illustrated with suitable examples.

Local polynomial smoother (LPS) is a weighted local least-squares nonparametric method. It provides a local Taylor series fit of the data at any location and can be directly used in a differential equation to provide a numerical scheme. In this article, we introduce this new nonparametric idea based on local polynomial smoother, for acquiring the numerical solution of the Bagley-Torvik fractional-order differential equations. Furthermore, this paper will present a numerical comparison with some methods, such as legendre operational matrix and pseudo-spectral method. The efficiency and accuracy of the LPS method are demonstrated by two numerical examples.

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approxi- mation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

In the present study, nonlinear vibration of a nanobeam resting on fractional order viscoelastic Winkler-Pasternak foundaion is studied using nonlocal elasticity theory. D'Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. Detailled parametric study is conducted, the effects of variation in different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations

In the paper, we develop some numerical schemes including finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulted numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by a integral operator with zero boundary condition. Second, because of the solution of fractional diffusion equation is usually singular near the boundary, we apply finite difference scheme to discrete fractional Laplacian operator and fractional diffusion equation by fractional interpolation function. Moreover, it is find that the differential matrix of the above scheme is symmetric matrix, and strictly row-wise diagonally dominant in special fractional interpolation function. Third, we show finite volume scheme to discrete fractional diffusion equation by fractional interpolation function, and analyze the properties of differential matrix. Finally, the numerical experiments are given, and verify the correctness of theoretical results and the efficiency of the schemes.

Fractional differential equations have become a fundamental modelling approach for understanding and simulating the many aspects of nonlocality and spatial heterogeneity of complex materials and systems. Yet, while real-order fractional operators are nowadays widely adopted, little progress has been made in extending such operators to complex-order counterparts. In this work, we introduce new definitions for the complex-order fractional Laplacian, fully consistent with the theory of averaging of smooth functions over fractal sets, and present tailored spectral methods for their numerical treatment. The proposed complex-order operators exhibit a dual particle-wave behaviour, with solutions manifesting wave-like features depending on the magnitude of the imaginary part of the fractional order. Reaction-diffusion systems driven by the complex-order fractional Laplacian exhibit unique spatio-temporal dynamics, such as equilibrium of diffusion in random materials by interference of scattered waves, conduction block and highly fractionated propagation, or the generation of completely novel Turing patterns. Taken together, our results support that the proposed complex-order operators hold unparalleled capabilities to advance the description of multiscale transport phenomena in physical and biological processes highly influenced by the heterogeneity of complex media.

The differential transformation, an approach based on Taylor's theorem, is proposed as convenient for finding exact or approximate solution to the initial value problem with multiple Caputo fractional derivatives of generally non-commensurate orders. The multi-term differential equation is first transformed into a multi-order system and then into a system of recurrence relations for coefficients of formal fractional power series. The order of the fractional power series is discussed in relation to orders of derivatives appearing in the original equation. Application of the algorithm to an initial value problem results in a reliable and expected outcome.

Recently, Li et al [1] investigated the nonlinear Langevin equations existence including two fractional orders with infinite-point boundary conditions. It has shown that the outcomes given it does count on the solution form for which it has boundary values. It means that their solution needs more to be in the final step. In current work, we will have discussion about the existence and uniqueness for the same boundary conditions by investigating the closely explicit solution form that has no boundary values. Numerical example is supported to have more vision to compare the between new and previous results. It turns out that our results are superior.

This study develops a new definition of fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. Such developed definition encompasses many types of fractional derivatives, such as the Riemann-Liouville and Caputo fractional derivatives for singular kernel type as well as the Caputo-Fabrizio, the Atangana-Baleanu and the generalized Hattaf fractional derivatives for non-singular kernel type. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, newly numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application to computational biology is presented.

In this article, we establish the existence of and then compare smallest eigenvalues for nabla fractional boundary value problems involving a fractional difference boundary condition, using the theory of u0-positive operators with respect to a cone in a Banach space.

The main objective of this present paper is to establish the extension of an extended fractional derivative operator by using an extended beta function recently defined by Parmar et al. by considering the Bessel functions in its kernel. Also, we give some results related to the newly defined fractional operator such as Mellin transform and relations to extended hypergeometric and Appell's function via generating functions.

Due to the nonlinear stiffness and fractional-order damping characteristics of some real suspension systems, coupled with the presence of viscoelastic materials, fractional-order nonlinear suspension system modeling has become a focus of research in recent years. In this study, an electric oil and gas actuator based on fractional-order PID position feedback control is proposed, through which the damping coefficient of the suspension system is adjusted to realize the active control of the suspension. A fractional order PID algorithm is used to control the motor's rotational angle to realize the damping adjustment of the suspension system. In this process, the road roughness is collected by the sensors as the criterion of damping adjustment, and the particle swarm algorithm is utilized to find the optimal objective function under different road surface slopes, to obtain the optimal cornering value. According to the mathematical and physical model of the suspension system, the simulation model and the corresponding test platform of this type of suspension system are built. The simulation and experimental results show that the simulation results of the fractional-order nonlinear suspension model are closer to the actual experimental values than those of the traditional linear suspension model, and the accuracy of each performance index is improved by more than 18.5%. The designed active suspension system optimizes the body acceleration, suspension dynamic deflection, and tire dynamic load to 89.8%, 56.7%, and 73.4% of the passive suspension, respectively. Notably, the designed fractional order PID control circuit for the motor shows better control results compared to the conventional PID control circuit. This study provides an effective theoretical and empirical basis for the control and optimization of fractional-order nonlinear suspension systems.

The operation status of manufacturing equipment is directly related to the reliability of the operation of manufacturing equipment and the continuity of operation of the production system. Based on the analysis of the operation status of manufacturing equipment and its characteristics, it is proposed that the concept of assessing the operation status of manufacturing equipment can be realized by applying real-time acquisition of accurate inspection data of important parts of weak motion units and comparing them with their motion status evaluation criteria. Through the study of the application characteristics of fractional-order calculus theory, a differential data fusion model based on the fractional-order differential operator is established. The advantages of Internet of Things (IoT）technology and fractional order differential fusion algorithm are integrated to obtain real-time high-precision data of the operating parameters of manufacturing equipment, and the research objective of operating condition assessment of manufacturing equipment is realized. The feasibility and effectiveness of the method are verified by the application of the method in machining center operation status assessment.

In this paper, by means of the Banach fixed point theorem and the Krasnoselskii's fixed point theorem, we investigate the existence of solutions for some fractional neutral functional integro-differential equations involving infinite delay. This paper deals with the fractional equations in the sense of Caputo fractional derivative and in the Banach spaces. Our results generalize the previous works on this issue. Also, an analytical example is presented to illustrate our results.

In this work, we establish some new Hermite-Hadamard type inequalities for convex functions via conformable fractional integral. Moreover, we show that through the conformable fractional integral we can find some new Hermite-Hadamard type inequalities for convex functions via the classical integrals.

This paper is related to the fractional view analysis of Helmholtz equations, using innovative analytical techniques. The fractional analysis of the proposed problems has been done in terms of Caputo-operator sense. In the current methodology, first, we applied the r-Laplace transform to the targeted problem. The iterative method is then implemented to obtain the series form solution. After using the inverse transform of the r-Laplace, the desire analytical solution is achieved. The suggested procedure is verified through specific examples of the fractional Helmholtz equations. The present method is found to be an effective technique having a closed resemblance with the actual solutions. The proposed technique has less computational cost and a higher rate of convergence. The suggested methods are therefore very useful to solve other systems of fractional order problems.

This paper focus on the modeling and time-domain analyzing method of piecewise-smooth fractional-order circuit systems. Firstly, an analytical framework is constructed based on a generalized piecewise-smooth two-port impedance network model, which takes the topological characteristics of Dickson type switched-capacitor converters (SCCs) as examples, and considers the fractional-order characteristics of class-2 X7R ceramic capacitors, which are widely used in SCCs. Secondly, the time-domain analyzing method for such a piecewise-smooth fractional-order circuit system is developed, numerical solutions are obtained, and the influence of the fractional-order capacitor on the power conversion efficiency of the system are revealed. Finally, in order to validate the analysis, both numerical simulation and experiments are provided in the work, the results of which confirm that, the proposed fractional-order analytical framework can capture the power loss mechanism of piecewise-smooth fractional-order circuit systems more effectively than the traditional integer-order based approach, thus benefiting the design of circuit systems.

In the present work we study the existence of solutions for hybrid nonlinear fractional integrodifferential equations. We developed an algorithm by using the operator theoretical techniques in order to obtain the approximate solutions. The main results depend on the Dhage iteration method that were incorporated with the modern hybrid fixed point theorems. The approximate solutions were obtained by using Lipschitz conditions and weaker form of mixed partial continuity. Further, we provide some examples to explain the hypotheses and the related results.

In this paper, we have established Hermite-Hadamard inequalities for r-preinvex functions via fractional integrals.

This study develops a predictor-corrector algorithm for the numerical simulation of IVPs involving singular generalized fractional derivatives with Mittag-Leffler kernels. The proposed algorithm converts the considered IVP into a Volterra-type integral equation and then uses Trapezoidal rule to obtain approximate solutions. Numerical approximate solutions of some singular generalized fractional derivative with Mittag-Leffler kernels models have been presented to demonstrate the efficiency and accuracy of the proposed algorithm. The algorithm describes the influence of the fractional derivative parameters on the dynamics of the studied models. The suggested method is expected to be effectively employed in the field of simulating generalized fractional derivative models

In this paper, we established the existence of PC-mild solutions for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained by using the techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we have used the distributed characteristic operators to define the mild solution of the system. Results obtained here improve and extend some known results.

In this paper, the new optical wave solutions of truncated M-fractional perturbed Kundu-Eckhaus model with full non-linearity are obtained by utilizing the expa function technique and modified extended tanh expansion function technique. The solutions are in the form of dark soliton, bright soliton, singular solitons and other form of solutions. The gained solutions are helpful for the further development of concerned model. The obtained results have also been presented graphically in both two-dimensional and three-dimensional formats to discuss the dynamical features as well as the parametric dependence of the constructed solutions. The study offers a highly spectacular and acceptable techniques to combine various intriguing wave demonstrations for more sophisticated models of the modern day.

In this paper, we prove the existence of three solutions for a $p$-Kirchhoff problem with $\psi$-Hilfer fractional derivative. To be more precise, we use the variational method and we prove that the associated functional energy admits a critical point in each of the three constructed sets, these critical points are weak solutions for a studied problem. Moreover, by definition of these sets, one of these solutions is positive, the second is negative, and the third one change sign. At the end of this work, we present an example to validate our main results.

The paper focuses on the admissibility problem of descriptor fractional-order systems (DFOSs). The alternate admissibility criteria are addressed for DFOSs with order in (0,2) which involve a non-strict linear matrix inequality (LMI) method and a strict LMI method, respectively. The forms of non-strict and strict LMIs are brand-new and distinguished with the existing literature, which fill the gaps of studies for admissibility. These approaches are available to the order in (0,2) without separating the order ranges into (0,1) and [1,2). In addition, a method involved least real decision variables in terms of strict LMIs is derived which is more convenient to process the practical solution. Three numerical examples are given to illustrate the validity of proposed results.

The article is devoted to the research of economic cycles and crises, which are studied within the framework of the theory of long waves by N. D. Kondratiev (K-waves) using the mathematical apparatus of fractional calculus. The object of the study is the mathematical model of S.V. Dubovsky 3 and some of its modifications, which describe the dynamics of the economic system, characterizing the interaction between the efficiency of new technologies and the efficiency of capital productivity. Modifications in the model are determined by the following points: the mathematical model takes into account the dependence of the rate of accumulation on capital productivity, there is an external harmonic influence - the influx of investments and new technological solutions, the economic system has heredity, including time-dependent, which leads to a delayed effect - the system’s response to impact. The property of heredity in the mathematical model is taken into account using fractional derivatives of constant and variable orders. Mathematical model S.V. Dubovsky and its modifications are studied numerically using the Adams-Bashforth-Moulton algorithm. Oscillograms and phase trajectories are constructed, and an interpretation of the simulation results is given.

This paper first discusses a fractional-order Liu system of order as low as 2.7 and shows its chaotic characteristics by carrying out numerical simulations such as Lyapunov exponents, bifurcation diagrams and phase portraits. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of chaos in the system is verified theoretically. Finally, the fractional-order system is implemented on a Field Programmable Gate Array (FPGA) and the results obtained show that the fractional-order Liu system is indeed chaotic.

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

In this study, we first present a time-fractional L$\hat{e}$vy diffusion equation of the exponential option pricing models of European option pricing and the risk-neutral parameter. Then, we modify a particular L$\hat{e}$vy-time fractional diffusion equation of European-style options. Introduce a more general model from the models based on the L$\hat{e}$vy-time fractional diffusion equation and review some recent findings regarding of the Europe option pricing of risk-neutral free.

The main goal of the paper is to use a generalized proportional Riemann-Liouville fractional derivative (GPRLFD) to model BAM neural networks and to study some stability properties of the equilibrium. Initially, several properties of the GPRLFD are proved such as the fractional derivative of a squared function. Also some comparison results for GPRLFD are provided. Two types of equilibrium of the BAM model with GPRLFD are defined. In connection with the applied fractional derivative and its singularity at the initial time the Mittag-Leffler exponential stability in time of the equilibrium is introduced and studied. An example is given illustrating the meaning of the equilibrium as well as its stability properties.

Background. The treatment of coronary stenosis relies on invasive high risk surgical assessment to generate the fractional flow reserve diagnostics index, a ratio of distal to proximal pressures in respect of coronary atherosclerotic plaque causing stenosis. Non-invasive methods are therefore a need of the times. This study proposes an extensible mathematical description of the coronary vasculature that permits rapid estimation of the coronary fractional flow reserve. Methods. By adapting an existing closed loop model of human coronary blood flow, the effects of large vessel stenosis and microvascular disease on fractional flow reserve were quantified. Several simula-tions generated flow and pressure information which was used to compute fractional flow re-serve under a spectrum of conditions including focal stenosis, diffuse stenosis, and microvascular disease. Sensitivity analysis stratified the influence of model parameters on the index. The model was simulated as coupled non-linear ordinary differential equations and numerically solved us-ing an implicit higher order method. Results. Large vessel stenosis affected fractional flow re-serve. The model predicts that the presence, rather than severity, of microvascular disease affect coronary flow deleteriously. Sensitivity analysis revealed that heart rate may not affect the index. Conclusions. The model provides a computationally inexpensive instrument for future in silico coronary blood flow investigations as well as clinical-imaging decision making. A combination of focal and diffuse stenosis appears to be essential in reducing the index. In addition to pressure measurements in the large epicardial vessels, diagnosis of microvascular disease is essential. The independence of the index with respect to heart rate suggests that computationally inexpensive steady state simulations may provide sufficient information to reliably compute the index.