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In this paper, we present a novel pseudospectral (PS) method for solving a new class of initial-value problems (IVPs) of time-dependent one-dimensional fractional partial differential equations (FPDEs) with variable coefficients and periodic solutions. A main ingredient of our work is the use of the recently developed periodic RL/Caputo fractional derivative (FD) operators with sliding positive fixed memory length of Bourafaet al. (2021) or their reduced forms obtained by Elgindy (2023) as the natural FD operators to accurately model FPDEs with periodic solutions. The proposed method converts the IVP into a well-conditioned linear system of equations using the PS method based on Fourier collocations and Gegenbauer quadratures. The reduced linear system has a simple special structure and can be solved accurately and rapidly by using standard linear system solvers. A rigorous study of the computational storage requirements as well as the error and convergence of the proposed method is presented. The idea and results presented in this paper are expected to be useful in the future to address more general problems involving FPDEs with periodic solutions. [ABSTRACT FROM AUTHOR]

Some errors exist in the above paper. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to focus on the accurate analysis of the fractional heat transfer in a two-dimensional (2D) rectangular monolayer tissue with three different kinds of lateral boundary conditions and the quantitative evaluation of the degree of thermal damage and burn depth. Design/methodology/approach: A symplectic method is used to analytically solve the fractional heat transfer dual equation in the frequency domain (s-domain). Explicit expressions of the dual vector can be constructed by superposing the symplectic eigensolutions. The solution procedure is rigorously rational without any trial functions. And the accurate predictions of temperature and heat flux in the time domain (t-domain) are derived through numerical inverse Laplace transform. Findings: Comparison study shows that the maximum relative error is less than 0.16%, which verifies the accuracy and effectiveness of the proposed method. The results indicate that the model and heat source parameters have a significant effect on temperature and thermal damage. The pulse duration (Δt) of the laser heat source can effectively control the time to reach the peak temperature and the peak slope of the thermal damage curve. The burn depth is closely correlated with exposure temperature and duration. And there exists the delayed effect of fractional order on burn depth. Originality/value: A symplectic approach is presented for the thermal analysis of 2D fractional heat transfer. A unified time-fractional heat transfer model is proposed to describe the anomalous thermal behavior of biological tissue. New findings might provide guidance for temperature prediction and thermal damage assessment of biological tissues during hyperthermia. [ABSTRACT FROM AUTHOR]

The construction of fractional derivatives with the right properties for use in field theory is reputed to be a difficult task, essentially because of the absence of a unique definition and uniform properties. The conformable fractional derivative introduced in 2014 by Khalil et al. in their seminal paper is a novel and well-behaved definition of fractional derivative for a function that is derivable in the usual sense. In this paper, we investigate the consistency of the Euler–Lagrange formalism for a field theory defined on such a fractional space–time. We especially focus on the relation between symmetries and conservation laws (Noether's currents), about the symmetry group introduced to construct the Lagrangian of the field. In particular, we show that the use of the conformable derivative induces additional terms in the calculation of the action variation. We also investigate the conservation of the Noether current and show that this property only takes place on condition that the equations of motion are verified with a new definition of the conserved law. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the fractional Sturm–Liouville operators and present the polar fractional Laplacian as a particular case of these operators. We also consider the distributed order space–time fractional diffusion equation involving the polar fractional Laplacian and propose three approaches for studying this problem on the circle and ring domains. First, we apply the integral transforms as the analytical methods to get the solution in terms of the Laplace-type integrals using the Titchmarsh theorem. Second, we use the finite difference method for discretization of the space variable and employ the matrix transfer technique to obtain a system of the distributed order fractional equations. For this system, we modify the Putzer's algorithm and get the solution in terms of the eigenvalues of discretization matrix. Third, we employ the backward Euler numerical method for discretization of the time variable and find the corresponding error bound. Moreover, we compare and verify the approximate solutions with the exact solutions in analytical form. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to compare the suitability of fractional derivatives in the modelling of practical capacitors. Such suitability refers to ability to provide the analytical capacitance function that matches the experimental ones of each fractional derivative. Design/methodology/approach: The analytical capacitance functions based on various fractional derivatives of both local and nonlocal types including the author's have been derived. The derived capacitance functions have been simulated and compared with the experimental ones of aluminium electrolytic and electrical double layer capacitors (EDLCs). Findings: This paper has found that any local fractional derivative with fractional power law-based relationship with the conventional one is suitable for modelling the aluminium electrolytic capacitor (AEC) by incorporating with the conventional capacitance definition. On the other hand, the author's nonlocal fractional derivatives have been found to be more suitable than the others for modelling the EDLC by incorporating with the revisited definition of capacitance. Originality/value: The proposed comparative analysis has been originally presented in this work. The criterion for local fractional derivative, to be suitable for modelling the AEC, has been found. The nonlocal fractional operators which are most suitable for modelling the EDLC have been derived where the unsuitable one has been pointed out. [ABSTRACT FROM AUTHOR]

This paper presents an efficient stabilized meshless technique with a hybrid kernel to simulate the fractional Rayleigh–Stokes problem for an edge in a viscoelastic fluid. The proposed method approximates the unknown solution through two phases. Firstly, the temporal discretization of the govern model is performed by integrating both sides of it, and then the space discretization is obtained by making use of the local hybrid Gaussian-cubic kernel meshless. The localized approach considers merely neighboring collocation nodes, while avoiding the ill-conditioning occurring in other techniques with dense and large matrix systems. The semi-discretized approach in terms of the convergence and stability properties is discussed theoretically in the weighted H 1 -norm and confirmed numerically. Two illustrative examples are given to show the efficiency and applicability of the proposed strategy. [ABSTRACT FROM AUTHOR]

This paper introduces a mixed meshless method for the numerical solution of time-fractional fourth-order reaction–diffusion model in multi-dimensional case. By additional variable σ = Δ ψ , the fourth-order derivative in the equation can be converted to a system of paired two second-order equations. The current method converts the problem into a system of linear algebraic equations. To demonstrate the capability and flexibility of the proposed method, some examples with different geometric domains are given in one, two and three dimensional cases. [ABSTRACT FROM AUTHOR]

In this paper, the Lie symmetry analysis is proposed for a space–time convection–diffusion fractional differential equations with the Riemann–Liouville derivative by (2+1) independent variables and one dependent variable. We find a reduction form of our governed fractional differential equation using the similarity solution of our Lie symmetry. One-dimensional optimal system of Lie symmetry algebras is found. We present a computational method via the spectral method based on Bernstein's operational matrices to solve the two-dimensional fractional heat equation with some initial conditions. [ABSTRACT FROM AUTHOR]

This paper explores the dynamics of vector-borne diseases transmitted by mosquitoes through the lens of systems incorporating fractional-order derivatives. Specifically, we extend the classical SISUV and SIRUV models by introducing fractional-order systems with the Caputo sense derivative. This augmentation introduces memory effects into the modelling process. Our study delves into the local and global stability analyses of both models, providing valuable insights into the dynamics of these diseases. To validate our theoretical results, we conduct numerical simulations confirming the robustness and applicability of the proposed models. • We analyse models describing the dynamics of mosquito-borne diseases. • We generalize the classical SISUV epidemic model to its fractional-order version. • We generalize the classical SIRUV epidemic model to its fractional-order version. • We use fractional derivative in the Caputo type. • We investigate local and global stability of the SISUV and SIRUV models. • We verify our theoretical results by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a neural network approach is suggested to the product design analysis. Namely, fractional-order neural network models are proposed as more flexible mechanism to study product form design. Since control and stability methods are fundamental in the construction and practical significance of a neural network model, appropriate controllers are designed and practical stability criteria are proposed for the fractional-order model under consideration. The stability and control analysis are based on the Lyapunov function method. Examples are elaborated to demonstrate the established results. The proposed modeling approach and the stability results are also applicable to numerous industrial design tasks. • Fractional-order neural network modeling approach is introduced to the product form design. • Appropriate controllers are utilized. • The practical stability notion is adopted to the introduced model. • Practical stability criteria are established using the Lyapunov function technique. • Examples and discussion are also offered to verify and justify the proposed results. [ABSTRACT FROM AUTHOR]

In this paper, the application of complex fractional moment(CFM) method in the stochastic dynamic systems with Caputo-type fractional derivative under the excitation of Gaussian white noise are investigated. By the approximation method, the FPK equation governing the amplitude is derived, and the semi-analytic solution of FPK equation is obtained by CFM method. The results are verified by numerical simulation. Meanwhile, the probability evolution of transient Probability Density Function(PDF) is analyzed when the stochastic bifurcation induced by fractional coefficient for the first time. In addition, the relation of equivalence between first passage time(FPT) and CFM is established by a novel method, the accuracy of this method is verified, the influences of system parameters change on reliability function and the FPT are discussed by the novel method. • The applicability and advantage of the CFM method in stochastic systems with Caputo fractional derivative are investigated. • A novel method based on CFM to obtain the probability distribution of first passage time is proposed in this paper. • The probability evolution of transient PDF is analyzed when the stochastic bifurcation occurs. [ABSTRACT FROM AUTHOR]

The effect of resource usage on economic growth has been investigated in many papers and the generic effect of resource usage efficacy by improving technical skills has been discussed. However, the impact of environmental pollution and the effect of memory property of the economic variables on economic growth have not been studied in detail. In this paper, a new resource depletion model, in which fractional derivative describes the memory property of economic variables, is built. Firstly, the influence of the environmental pollution on economic growth is analyzed. The results show the model is extreme sensitive to some parameters with considering the impact of environmental pollution. The small change of the parameters can lead a collapse of the economy in a short time. Meanwhile, new equilibrium region of economic growth can be obtained with considering the effect of the memory property of economic variables. Secondly, the effect of externally stochastic perturbation on the average of capital is studied, and the result shows random excitation can not only impede but also promote the development of the economy. [ABSTRACT FROM AUTHOR]

In this work, we introduce ABC-Caputo operator with ML kernel and its main characteristics are discussed. Viral diseases models for AIDS and Zika are considered, and finally, as third model, the macroeconomic model involving ABC fractional derivatives is investigated, respectively. It is presented that the AB Caputo derivatives satisfy the Lipschitz condition along with superposition property. The numerical methods for solving the fractional models are presented by means of ABC fractional derivative in a detailed manner. Finally the simulation results obtained in this paper according to the suitable values of parameters are also manifested. [ABSTRACT FROM AUTHOR]

• The fractional Chen system is analyzed. • Stability properties with respect to free entry parameters are described. • A fractional Hopf bifurcation is investigated rigorously. • Other bifurcation phenomena are observed numerically. • The occurrence of a chaotic attractor for low derivative orders is discussed. The paper provides a theoretical analysis of some local bifurcations in the fractional Chen system. Contrary to the integer-order case, basic bifurcation properties of the fractional Chen system are shown to be qualitatively different from those described previously for the fractional Lorenz system. Further, the fractional Hopf bifurcation in the Chen system is expressed rigorously with respect to general entry parameters. Based on these observations, some particularities of the fractional dynamics of the Chen system are documented and its chaotic behavior for low derivative orders is discussed. [ABSTRACT FROM AUTHOR]

This paper investigates the stochastic dynamics in the nonlinear vibroimpact system with time delay and fractional derivative under Gaussian white noise excitation. Firstly, based on the definition of Caputo-type fractional derivative and the method of non-smooth transformation, the original system is transformed into an equivalent delayed stochastic vibroimpact system without fractional derivative. Then, by using the stochastic averaging method, the stationary density function of the stochastic I t o ˆ equation is obtained. The effectiveness of the proposed method was validated by comparing the consistency between the original system and the optimized system without fractional derivative or the term of time delay. At last, we also explore the stochastic P-bifurcation induced by the power spectral density of two uncorrelated noises, time delay, fractional order, and restitution coefficient of the system. • The stochastic vibroimpact system with time delay and fractional derivative is investigated. • The period-like motion of the mass in vibroimpact system is used to make an effective approximation of the system. • Considering the effect of time delay on the system, the accuracy of the vibration frequency approximation is relatively high. • The behavior of stochastic P-bifurcation induced by different parameters has been explored. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the stochastic bifurcations in the nonlinear vibroimpact system with fractional derivative under random excitation. Firstly, the original stochastic vibroimpact system with fractional derivative is transformed into equivalent stochastic vibroimpact system without fractional derivative. Then, the non-smooth transformation and stochastic averaging method are used to obtain the analytical solutions of the equivalent stochastic system. At last, in order to verify the effectiveness of the above mentioned approach, the van der Pol vibroimpact system with fractional derivative is worked out in detail. A very satisfactory agreement can be found between the analytical results and the numerical results. An interesting phenomenon we found in this paper is that the fractional order and fractional coefficient of the stochastic van der Pol vibroimpact system can induce the occurrence of stochastic P-bifurcation. To the best of authors’ knowledge, the stochastic P-bifurcation phenomena induced by fractional order and fractional coefficient have not been found in the present available literature which studies the dynamical behaviors of stochastic system with fractional derivative under Gaussian white noise excitation. [ABSTRACT FROM AUTHOR]

The main concern of this paper is to address the synchronization problem of chaotic fractional-order fuzzy cellular neural networks (FOFCNNs) through designing the novel adaptive control scheme. The objective of the study is to explore the importance of considering fractional order derivatives (FODs) and time-varying delays. Even though numerous works have been reported in the literature regarding the derivation of sufficient conditions, there has been a lack of research on involving the dynamical analysis of FOFCNNs. Hence, this study focuses on the dynamical analysis of FOFCNNs. Particularly, both asymptotical and exponential synchronization of drive-response FOFCNN model is guaranteed via sufficient conditions that are derived by constructing the fractional Lyapunov functional candidate and solvable linear matrix inequalities (LMIs). Besides that, numerical simulations are performed to reveal the significance of the FODs. Also, an image encryption algorithm is designed based on the chaotic FOFCNNs solutions that result in better security measures. In summary, the overall contribution of the study is categorized into two: (1) sufficient conditions which ensure the global asymptotic and exponential stability are derived in a novel manner; (2) an image encryption algorithm is proposed by considering the FOFCNN as pseudo-random number generator (PRNG), which outperforms the existing encryption algorithms. [ABSTRACT FROM AUTHOR]

Highlights • Distinctive features of self-organization with memory are described. • Appearance of memory in process can generate a self-organization. • Principles of self-organization my power-law memory are suggested. • Change of memory fading can lead to significant change of characteristic times. • Generalization of method of adiabatic elimination of fast variables is proposed. Abstract In this paper we propose a generalization of method of adiabatic elimination of fast relaxing variables, which is a special form of the slaving principle, to the case of processes with power-law memory. Simple models to describe the distinctive features of the self-organized processes with power-law memory are considered. We have proved that when the memory fading parameters change, the processes with memory can significantly change its rates (by many orders of magnitude) and change the characteristic times both upward and downward. Therefore the parameters of memory fading can be considered as control parameters that change of characteristic time of processes. Some principles to describe specific features of self-organization with memory are proposed. [ABSTRACT FROM AUTHOR]

Highlights • It presents a revision of previously proposed variable order fractional derivatives. • It introduces a new approach based on the inverse Laplace transform. • Variable order fractional linear systems are defined. • A variable order Mittag-Leffler function is introduced. • Variable order two-sided fractional derivatives. Abstract Fractional Calculus had a remarkable evolution during recent decades, and paved the way towards the definition of variable order derivatives. In the literature we find several different alternative definitions of such operators. This paper presents an overview of the fundamentals of this topic and addresses the questions of finding out which of them are reasonable according to simple criteria used for constant order fractional derivatives. This approach leads to the definitions of variable order fractional derivative based on the Grünwald–Letnikov and the Liouville formulations defined on R , as well as to the definition of a Mittag-Leffler function for variable orders, and to the application of these definitions to dynamical systems. [ABSTRACT FROM AUTHOR]

Highlights • A two-step family of Adams–Bashforth method is derived for the approximation of the Atangana–Baleanu fractional derivative. • Mathematical analysis and numerical simulation of time-fractional multicomponent systems • Application of the Atangana–Baleanu derivative based on nonlocal and nonsingular kernels. • Simulation results showing chaotic behaviors for different values of alpha. Abstract In this paper, we consider the mathematical analysis and numerical simulation of time-fractional multicomponent systems. Here, the classical time derivatives in such systems are replace with the Atangana–Baleanu fractional derivative in the sense of Caputo. This derivative is found useful in the sense that it combines both the non-local and nonsingular kernels in its formulation. A two-step family of Adams–Bashforth method is derived for the approximation of the Atangana–Baleanu derivative. Numerical experiments presented for different instances of α , 0 < α ≤ 1 correspond to our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper proposes a new strategy to study some useful properties of growth rates of functions in C α ∈ R spaces in order to analyze the Leibniz rule for fractional derivatives. The differential operators are taken in the Riemann–Liouville sense. Moreover, stability analysis of the proposed strategy is investigated. The results demonstrate that the proposed theoretical analysis is accurate. [ABSTRACT FROM AUTHOR]

The paper discusses the characteristic properties of fractional derivatives of non-integer order. It is known that derivatives of integer orders are determined by properties of differentiable functions only in an infinitely small neighborhood of the considered point. Therefore differential equation, which is considered for this point and contains a finite number of integer-order derivatives, cannot describe nonlocality in space and time. This allows us to propose a principle of nonlocality for fractional derivatives. We state that if the differential equation with fractional derivative can be presented as a differential equation with a finite number of integer-order derivatives, then this fractional derivative cannot be considered as a derivative of non-integer order. This means that all results obtained for this type of fractional derivatives can be derived by using differential operators with integer orders. To illustrate the application of the nonlocality principle, we prove that the conformable fractional derivative, the M-fractional derivative, the alternative fractional derivative, the local fractional derivative and the Caputo–Fabrizio fractional derivatives with exponential kernels cannot be considered as fractional derivatives of non-integer orders. [ABSTRACT FROM AUTHOR]

Mathematical modeling of biological systems is an interesting research topic that attracted the attention of many researchers. One of the main goals in this area is the design of mathematical models that more accurately illustrate the characteristics of the real-world phenomena. Among the existing research projects, modeling of immune systems has given a growing attention due to its natural capabilities in identifying and destroying abnormal cells. The main objective of this paper is to investigate the pathological behavior of HIV-infection using a new model in fractional calculus. The proposed model is examined through three different operators of fractional derivatives. An efficient numerical method is also presented to solve these fractional models effectively. In fact, we believe that the new models presented on the basis of these three operators show various asymptomatic behaviors that do not appear during the modeling with the integer-order derivatives. Therefore, the fractional calculus provides more precise models of biological systems that help us to make more realistic judgments about their complex dynamics. Finally, simulations results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper aimed to investigate the stationary response of non-linear system with fractional derivative damping term under Gaussian white noise excitation. The corresponding Fokker–Plank–Kolmogorov (FPK) equation can be deduced by utilizing the stochastic averaging method and Stratonovich–Khasminskii theorem in the first place. And then we can solve the FPK equation to obtain the stationary probability densities (SPDs) of amplitude, which in fact can be used to describe the response of system. Furthermore, the analytical results coincide with the Monte Carlo results. Finally, one found that reducing fractional derivative order is able to enhance the response of system and increasing fractional coefficient can weaken the response of system. So the fractional derivative damping term has a great effect on the response of Duffing-Van der Pol oscillator. In addition, the response can also be influenced by other system parameters. [ABSTRACT FROM AUTHOR]

In this paper, we use stochastic averaging method and data-driven artificial neural network to study the response and reliability of a fractional stochastic dynamical system. Firstly, a generalized Harmonic transformation is introduced to get an approximate expression for the fractional derivative. Based on this procedure, stochastic averaging method is utilized to get Fokker–Planck Kolmogorov equation for the system response and Backward Kolmogorov equation for the system reliability. An exact stationary solution of amplitude response is obtained by solving Fokker–Planck Kolmogorov equation, in the meanwhile, numerical results regarding to the reliability by Crank–Nicolson difference method are derived too. We propose an algorithm of data-driven artificial neural network and apply it for solving these two equations simulatively, in order to verify the correction and accuracy of the analytical methods. All results illustrate that data-driven artificial neural network is highly effective to achieve the system response and reliability. The advantage of this deep learning algorithm mainly includes the meshless structure, boundary sample data collection, small amount of training data and unconstrained optimization. In addition, the increase of the fractional order can enhance the system response and leads to stochastic bifurcation. • Response and reliability are analyzed by stochastic averaging and DDANN. • Proposed DDANN is mesh free and high effectiveness. • All results obtained by analytical methods and DDANN are agreement. • Higher order of fractional derivative is helpful for improving the reliability. [ABSTRACT FROM AUTHOR]

Our attempt is an axiomatic approach to find all classes of possible definitions for fractional derivatives with three axioms. In this paper, we consider a special case of linear integro-differential operators with power functions' convolution kernel a (α) (t − s) b (α) of order α ∈ (0 , 1). We determine analytic functions a (α) and b (α) such that when α → 0 + , the corresponding operator becomes identity operator, and when α → 1 − the corresponding operator becomes derivative operator. Then, a sequential operator is used to extend the fractional operator to a higher order. Some properties of the sequential operator in this regard also are studied. The singularity properties, Laplace transform and inverse of the new class of fractional derivatives are investigated. Several examples are provided to confirm theoretical achievements. Finally, the solution of the relaxation equation with diverse fractional derivatives is obtained and compared. • A class of linear fractional operators that behave like an integer order derivative is obtained. • Inverse and Laplace transforms of introduced fractional derivatives are investigated. • The relaxation equation with diverse fractional operators is solved analytically. [ABSTRACT FROM AUTHOR]

The recent outbreaks of Ebola encourage researchers to develop mathematical models for simulating the dynamics of Ebola transmission. We continue the study of the models focusing on those with a variable population. Hence, this paper presents a compartmental model consisting of 8-dimensional nonlinear differential equations with a dynamic population and investigates its basic reproduction number. Moreover, a dimensionless model is introduced for numerical analysis, thus proving the disease-free equilibrium is locally asymptotically stable whenever the threshold condition, known as a basic reproduction number, is less than one. Finally, we use a fractional differential form of the model to sufficiently fit long time-series data of Guinea, Liberia, and Sierra Leone retrieved from the World Health Organization, and the numerical results demonstrate the performance of the model. • A variable population model for the dynamics transmission of Ebola disease is introduced. • The analysis of the is carried out via an equivalent dimensionless model. • Fractional version of the model is used to sufficiently fit long time-series data of Guinea, Liberia, and Sierra Leone obtained from WHO. [ABSTRACT FROM AUTHOR]

The paper is concerned with the strong approximation of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time; namely, a Caputo fractional derivative of order α ∈ (0 , 1) , and a Riemann–Liouville fractional integral operator of order γ ∈ [ 0 , 1 ] applied to a Gaussian noise. We approximate the model by a standard piecewise linear finite element method (FEM) in space and the classical Grünwald–Letnikov method in time (for both time-fractional operators), and the noise by the L 2 -projection. Spatially semidiscrete and fully discrete schemes are analyzed and strong convergence rates are obtained by exploiting the temporal Hölder continuity property of the solution. Numerical experiments are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Purpose – Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus. Design/methodology/approach – This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given. Findings – Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations. Originality/value – Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared. [ABSTRACT FROM AUTHOR]

In this paper we discuss a concept of dynamic memory and an application of fractional calculus to describe the dynamic memory. The concept of memory is considered from the standpoint of economic models in the framework of continuous time approach based on fractional calculus. We also describe some general restrictions that can be imposed on the structure and properties of dynamic memory. These restrictions include the following three principles: (a) the principle of fading memory; (b) the principle of memory homogeneity on time (the principle of non-aging memory); (c) the principle of memory reversibility (the principle of memory recovery). Examples of different memory functions are suggested by using the fractional calculus. To illustrate an application of the concept of dynamic memory in economics we consider a generalization of the Harrod–Domar model, where the power-law memory is taken into account. [ABSTRACT FROM AUTHOR]

In this paper, two numerical techniques are introduced to study numerically the general fractional advertising model. This system describes the flux of the consumers from unaware individuals group to aware or purchased group. The first technique is an asymptotically stable difference scheme, which was structured depending on the nonstandard finite difference method. This scheme preserves the properties of the solutions of the model problem as the positivity and the boundedness. The second technique is the Jacobi–Gauss–Lobatto spectral collocation method which is exponentially accurate. By means of this approach, such problem is reduced to solve a system of nonlinear algebraic equations and are greatly simplified the problem. Numerical comparisons to test the behavior of the used techniques are run out. We conclude from the computational work that: the Jacobi–Gauss–Lobatto spectral collocation method is more accurate whereas the nonstandard finite difference method requires less computational time. [ABSTRACT FROM AUTHOR]

In this paper, a novel fractional equivalent linearization (EL) approach is developed by incorporating a fractional derivative term into the classical linearization equation. Due to the introduction of the fractional derivative term, the accuracy of the new linearization is improved, illustrated by a Duffing oscillator that is subjected to a harmonic excitation. Furthermore, a new method for solving stochastic response of nonlinear SDOF system is developed by combining Karhunen–Loève (K-L) expansion and fractional EL. The method firstly decomposes the stochastic excitation in terms of a set of random variables and deterministic sub-excitations using K-L expansion, and then construct sub-fractional equivalent linear system according to each sub-excitation by fractional EL, the response of the original nonlinear system is finally approximated as the weighed summation of the deterministic response of each sub-system multiplied by the corresponding random variable. The random nature of the final response comes from the set of random variables that is obtained in K-L expansion. In this way, the stochastic response computation is converted to a set of deterministic response analysis problems. The effectiveness of the developed method is demonstrated by a Duffing oscillator that is subjected to stochastic excitation modeled by Winner process. The results are compared with the numerical method and Monte Carlo simulation (MCS). [ABSTRACT FROM AUTHOR]

This paper presents a study on amplitude death in nonlinear coupled oscillators containing fractional derivatives. Analytical criterion for amplitude death region is obtained by eigenvalue analysis and verified by numerical results. It is found that amplitude death regions can be enlarged to a large extent by fractional derivatives. For this reason, amplitude death can be detected in fractional Stuart–Landau systems with weak coupling strength and low frequency, whereas it never appears in integer-order systems. Interestingly, the widening of amplitude death region induced by fractional derivative is shared by a variety of oscillators with different types of coupling mechanisms. An interpretation for the underlying mechanism of this phenomenon is briefly addressed, based on which we further suggest a coupling organization leading to amplitude death only in fractional oscillators. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on the analysis of a hyperchaotic financial system as well as its chaos control and synchronization. The phase diagrams of the above system are plotted and its dynamical behaviours like equilibrium points, stability, hyperchaotic attractors and Lyapunov exponents are investigated. In order to control the hyperchaos, an efficient optimal controller based on the Pontryagin’s maximum principle is designed and an adaptive controller established by the Lyapunov stability theory is also implemented. Furthermore, two identical financial models are globally synchronized by using an interesting adaptive control scheme. Finally, a fractional economic model is introduced which can also generate hyperchaotic attractors. In this case, a linear state feedback controller together with an active control technique are used in order to control the hyperchaos and realize the synchronization, respectively. Numerical simulations verifying the theoretical analysis are included. [ABSTRACT FROM AUTHOR]

This paper proposes a continuous time irrational filter structure via a set of the fractional order Gammatone components instead of via a set of integer order Gammatone components. The filter design problem is formulated as a nonsmooth and nonconvex infinite constrained optimization problem. The nonsmooth function is approximated by a smooth operator. The domain of the constraint functions is sampled into a set of finite discrete points so the infinite constrained optimization problem is approximated by a finite constrained optimization problem. To find a near globally optimal solution, the norm relaxed sequential quadratic programming approach is applied to find the locally optimal solutions of this nonconvex optimization problem. The current or the previous locally optimal solutions are kicked out by adding the random vectors to them. The locally optimal solutions with the lower objective functional values are retained and the locally optimal solutions with the higher objective functional values are discarded. By iterating the above procedures, a near globally optimal solution is found. The designed filter is applied to perform the denoising. It is found that the signal to noise ratio of the designed filter is higher than those of the filters designed by the conventional gradient descent approach and the genetic algorithm method, while the required computational power of our proposed method is lower than those of the conventional gradient descent approach and the genetic algorithm method. Also, the signal to noise ratio of the filter with the fractional order Gammatone components is higher than those of the filter with the integer order Gammatone components and the conventional rational infinite impulse response filters. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Caputo-type fractional model to study the Nipah virus transmission dynamics. The model describes the impact of unsafe contact with an infectious corpse as a possible way to transmit this virus. The corresponding area to the system properties, including the positivity and boundedness of the solution, is explored by using the generalized fractional mean value theorem. Also, we investigate sufficient conditions for the local and global stability of the disease-free and the endemic steady-states based on the basic reproduction number R 0. To show these important stability features, we employ fractional Routh–Hurwitz criterion and LaSalle's invariability principle. For the implementation of this epidemic model, we also use the Adams–Bashforth–Moulton numerical method in a fractional sense. Finally, in addition to compare the fractional and classical results, as one of the main goals of this research, we demonstrate the usefulness of minimal unsafe touch with the infectious corpse. Simulation and comparative results verify the theoretical discussions. • A Caputo-type fractional SIRD model is established for the Nipah virus transmission. • The new model describes the impact of unsafe contact with an infected corpse. • The corresponding area to the system feasibility and equilibrium points is explored. • For the aim of implementation, the Adams–Bashforth–Moulton scheme is employed. • Finally, we demonstrate the usefulness of minimal unsafe touch with an infectious corpse. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient computational approach named as Sumudu residual power series method (SRPSM) to solve fractional Bloch equations appearing in an NMR flow. This method is a copulation of the residual power series method (RPSM) and the Sumudu transform to construct approximate solution in shapes of sharp convergent series by adopting the notion of limit with a view of smooth calculations and time saving strategy as compared to the classical RPSM in which computations of fractional derivatives are required. This work also verifies and compares the solution obtained by the proposed technique with the classical RPSM. To assure the applicability, performance, and reliability of the introduced method, a system of fractional Bloch equations are examined along with numerical simulations. The aspect of application supported with computer simulations demonstrates that the suggested approach is precise, and appropriate to explore the solutions of linear & nonlinear initial value problems with fractional-order derivatives. • Fractional Bloch equations appearing in an NMR flow are studied. • A hybrid computational technique is used to slove the fractional order problem. • The method is a combination of residual power series method and the sumudu transform. • The results for fractional Bloch equations are presented graphically. [ABSTRACT FROM AUTHOR]

In this paper, the design of fractional order digital differentiator is investigated. First, the radial basis function interpolation method is described. Then, the non-integer delay sample estimation of discrete-time sequence is derived by using the radial basis function interpolation approach. Next, the Grünwald-Letnikov derivative and non-integer delay sample estimation are applied to obtain the transfer function of fractional order digital differentiator. Finally, the applications in digital image sharpening and parameter estimation of fractional noise process are studied to demonstrate the usefulness of this new design approach. [ABSTRACT FROM AUTHOR]

• The basic motivation of our article is the approximate controllability issues of delay evolution equations for Sobolev type with order 1

This paper analyzes the dynamics of a business cycle model with fractional derivative of order   α  (0 < α < 1) subject to narrow-band random excitation, in which fractional derivative describes the memory property of the economic variables. Stochastic dynamical system concepts are integrated into the business cycle model for understanding the economic fluctuation. Firstly, the method of multiple scales is applied to derive the model to obtain the approximate analytical solution. Secondly, the effect of economic policy with fractional derivative on the amplitude of the economic fluctuation and the effect on stationary probability density are studied. The results show macroeconomic regulation and control can lower the stable amplitude of economic fluctuation. While in the process of equilibrium state, the amplitude is magnified. Also, the macroeconomic regulation and control improves the stability of the equilibrium state. Thirdly, how externally stochastic perturbation affects the dynamics of the economy system is investigated. [ABSTRACT FROM AUTHOR]

In this paper the generalized differential transform method is applied to obtain an approximate solution of linear and nonlinear differential equation of fractional order with boundary conditions. Several numerical examples are considered and comparisons with the existing solution techniques are reported. Results show that the method is effective, easier to implement and very accurate when applied for the solution of fractional boundary values problems. [ABSTRACT FROM AUTHOR]

In this paper, a classical type of two-dimensional time-fractional telegraph equation defined by Caputo sense for ( 1 < α < 2 ) is analyzed by an approach based on the Galerkin weak form and moving least squares (MLS) approximation subject to given appropriate initial and Dirichlet boundary conditions. In the proposed method, which is a kind of the Meshless local Petrov–Galerkin (MLPG) method, meshless Galerkin weak form is applied to the interior nodes while the meshless collocation method is used for the nodes on the boundary, so the Dirichlet boundary condition is imposed directly. In MLPG method, it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes, such as circles or squares. The moving least squares approximation is proposed to construct shape functions. Two numerical examples are presented and satisfactory agreements are achieved. [ABSTRACT FROM AUTHOR]

In this paper a new model for the liquid motion within a Tuned Liquid Column Damper (TLCD) device is developed, based on the mathematical tool of fractional calculus. Although the increasing use of these devices for structural vibration control, it is shown that existing model does not always lead to accurate prediction of the liquid motion. A better model is then needed for accurate simulation of the behavior of TLCD systems. As regards, it has been demonstrated how correctly including the first linear liquid sloshing mode, through the equivalent mechanical analogy well established in literature, produces numerical results that highly match the corresponding experimental ones. Since the apparent effect of sloshing is the deviation of the natural frequency from the theoretical one, the authors propose a fractional differential equation of motion. The latter choice is supported by the fact that the introduction a fractional derivative of order α alters simultaneously both the resonant frequency and the degree of damping of the system. It will be shown, through an extensive experimental analysis, how the proposed model accurately describes liquid surface displacements. [ABSTRACT FROM AUTHOR]

Purpose – The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the analytical solutions of the models and show the performance of several parameters. Design/methodology/approach – To describe the heat conduction, the Caputo type time fractional heat conduction equation is used and a convective term is included since the changes in density give rise to motion of the liquid phase. The similarity variables are used to simplify the models. Findings – The analytical solutions describing the changes of temperature in both liquid and solid phases are obtained. For the solid phase, the solution is given in the Wright function form. While for the liquid phase, since the appearance of the advection term, an approximate solution in series form is given. Based on the solutions, the performance of the parameters is discussed in detail. Originality/value – From the point of view of mathematics, the moving boundary problems are nonlinear, so barely any analytical solutions for these problems can be obtained. Furthermore, there are many applications in which a material undergoes phase change, such as in melting, freezing, casting and cryosurgery. [ABSTRACT FROM AUTHOR]

• In this paper, we present an existence and uniqueness result for positive solution of a kind of fractional boundary value problem with Riemann-Liouville fractional derivative. Before this, we provide two new fixed point results for mappings satisfying a certain conrtractive condition that obtained by simulation and Q-functions on quasi metric spaces. This study is important because of the first time that a fixed point theorem obtained in quasi metric spaces is applied to such an equation. In this paper, we present two fixed point results on quasi metric spaces by taking into account a contractive condition that obtained via simulation and Q -functions. Then one of these fixed point results has been applied to guarantee the existence and uniqueness of positive solution of a kind of fractional boundary value problem with Riemann-Liouville fractional derivative. [ABSTRACT FROM AUTHOR]

• Birhythmic zone expand with decreasing Gaussian white noise intensity. • Increasing fractional-orders brings about multimodal structures of probability distribution. • Energy imbalance takes place with increasing the fractional-orders. This paper addresses the occurrence of P-bifurcation in a fractional-delay modified birhythmic van der Pol (BVDP) oscillator, for enzyme-substrate reactions in brain waves, under Gaussian white noise excitation. The minimum mean-square error is used to reduce the system to its equivalent integer-order nonlinear stochastic equation. An averaged Itô equation is obtained via the stochastic averaging method, with the amplitude being the solution of the Fokker-Planck-Kolmogorov equation. From the latter, the stationary density functions are found analytically. This helps to predict the appearance of birhythmicity theoretically and shows itself to respond to parameter changes, namely, fractional-orders, fractional coefficient, and noise intensity. There is an agreement between the theoretical solutions and the numerical solution, which confirms the accuracy of our predictions. In general, bifurcation scenarios are dominated by the changes in fractional orders, as strongly supported by the behaviors of the calculated potential barriers. [ABSTRACT FROM AUTHOR]

Abstract: The stochastic solution to diffusion equations with polynomial coefficients is called a Pearson diffusion. If the time derivative is replaced by a distributed order fractional derivative, the stochastic solution is called a distributed order fractional Pearson diffusion. This paper develops a formula for the covariance function of distributed order fractional Pearson diffusion in the steady state, in terms of generalized Mittag-Leffler functions. The correlation function decays like a power law. That formula shows that distributed order fractional Pearson diffusions exhibits long range dependence. [Copyright &y& Elsevier]

Abstract: In this paper, we implement the radial basis functions for solving a classical type of time-fractional telegraph equation defined by Caputo sense for . The presented method which is coupled of the radial basis functions and finite difference scheme achieves the semi-discrete solution. We investigate the stability, convergence and theoretical analysis of the scheme which verify the validity of the proposed method. Numerical results show the simplicity and accuracy of the presented method. [Copyright &y& Elsevier]

Abstract: In this paper, the two-dimensional Legendre wavelets are applied for numerical solution of the fractional Poisson equation with Dirichlet boundary conditions. In this way, a new operational matrix of fractional derivative for the Legendre wavelets is derived and then this operational matrix has been employed to obtain the numerical solution of the above-mentioned problem. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence of the two-dimensional Legendre wavelets expansion is investigated. Also the power of this manageable method is illustrated. [Copyright &y& Elsevier]