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The objective of this paper is to recommend an adjusted Susceptible-Exposed-Infectious-Removed (SEIR) model that characterizes the temporal patterns of various individuals affected by the omicron variant in an epidemic. This model considers factors such as vaccination, asymptomatic cases, indoor and outdoor air, hospitalizations, and deaths, as well as the impact of immunization and memory trace. While many recent studies overlook the complexities of multiple strains, including their diverse transmission rates and reaction to vaccines, this study introduces a new fractional derivative model to examine the spread of the omicron variant of COVID-19 and the implementation of a vaccination campaign. A thorough theoretical analysis is conducted, and the critical factor (R c r n z) is calculated using the model equations. It is demonstrated that when R c r n z is less than 1, the disease-free state is globally asymptotically stable, meaning that the epidemic diminishes. Moreover, the stability of both global and local equilibrium points is examined. Numerical simulations are employed to demonstrate the alignment between the numerical findings and theoretical characteristics. The model is adjusted to experimental data that reflect the progression of the omicron variant of COVID-19 in Guangzhou, exhibiting a satisfactory performance in predicting the number of infected individuals, thereby suggesting its capability to accurately estimate asymptomatic cases. Furthermore, to emphasize the benefits of employing fractional differential equations, the paper provides examples related to memory trace and hereditary characteristics. Moreover, the examined models are expected to be applied and expanded upon in order to contribute to the formulation of policies for disease control during times of limited vaccine availability. To summarize, the model appears to be a sufficient tool for researching and managing infectious diseases. It is projected that as the Omicron variant's prevalence declines, there will be a reduced need for respiratory-focused precautions. • An SVEIAOPHRD model is put forward to mathematically analyze the spread of the omicron variation of COVID-19. • Mathematical criteria for determining the stability of the equilibria are established. • An efficient nonstandard approach for resolving the continuous problem is suggested and examined. • The numerical simulations validate the analytical and numerical outcomes depicted in this study. [ABSTRACT FROM AUTHOR]

The article highlights the growing importance of computational mathematics in tackling intricate real-world phenomena governed by distributed order fractional dynamics. These dynamics, characterized by memory effects and non-local interactions, are pervasive in various fields such as physics, biology, finance, and engineering. Despite their prevalence, analytical solutions for distributed order fractional differential equations remain difficult to find, necessitating the development of robust numerical methods. The paper explores the effectiveness of Romanovski–Jacobi spectral collocation techniques in this context and provides a thorough analysis of how to use them to solve distributed order fractional differential equations. By leveraging these schemes, researchers gain valuable insights into efficiently analyzing and simulating fractional systems, thereby advancing our understanding of complex dynamics across diverse domains. It provides precise numerical findings by using finite expansion to evaluate residuals. The precision of the approach is illustrated by numerical simulations, especially in distributed order fractional differential equations. Additionally, we provide a few numerical tests to demonstrate the method's ability to preserve the underlying problem's non-smooth solution. [ABSTRACT FROM AUTHOR]

This work proposes an iterative RBF-based method for solving nonlinear time-fractional Benjamin-Bona-Mahony-Burgers (BBMB) equations in arbitrary domains. The method relies on a newly developed approach for solving fractional ordinary differential equations (FODEs). To tackle the nonlinearity of the problem, we employ a simple quasi-linearization procedure. Subsequently, the solution is constructed using modified radial basis functions (RBFs), where the FODEs emerge automatically. The temporal domain is discretized using the Müntz polynomial basis, while the spatial domain discretization is achieved through the meshless backward substitution method. By conducting a series of problems in one, two, and three spatial dimensions, we verify the accuracy, efficiency, and robustness of the proposed scheme, and compare the computed results with references from the literature. The goal of this paper is to demonstrate the promising accuracy and efficiency of the proposed technique for solving time-fractional BBMB equations. [ABSTRACT FROM AUTHOR]

This paper proposes a method to solve a 2D time-fractional parabolic equation on arbitrary domains using an RBF partition of unity procedure based on a non-uniform time-stepping framework. The proposed method involves partitioning the main domain into overlapping subdomains, computing local approximations on each subdomain, and joining them together by PU weights to obtain a global approximation. This approach enhances stability, reduces computational costs, and replaces an ill-conditioned linear system with several small well-conditioned linear systems. To handle the weak singularity near the initial time, a time-graded mesh is used to obtain a non-uniform time-stepping scheme, which increases the temporal convergence rate. The convergence analysis of the method is examined to obtain an error bound for the local approximation. Numerical tests are conducted to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper develops a singularity-separation Legendre collocation method for solving two-term linear Volterra integral equation with algebraic and logarithmic singular kernels. The psi-series solution at the origin is obtained by an efficient Picard iteration, which is very accurate near the origin. By splitting the interval into a singular subinterval and a regular one, the equation is converted to one with a sufficiently smooth solution on the regular interval, which is discretized by the Legendre collocation method. The efficient computational process is discussed in detail, and the convergence under the square norm is proved. Finally, three numerical examples, including a Caputo fractional differential equation and a Basset equation with initial conditions, are provided to illustrate the efficiency and accuracy of the proposed singularity-separation method. • The VIE includes two integral terms possessing algebraic-logarithmic singularities. • The series solution about the origin is obtained for the two-term VIE. • A singularity-separation Legendre collocation method is developed. • A recurrence formula is derived to evaluate the singular weight integrals efficiently. • Convergence analysis shows that the method has an optimal convergence rate. [ABSTRACT FROM AUTHOR]

The main objective of the present paper is to establish the existence and uniqueness (EU) results for nonlinear fractional Langevin equation involving Liouville-Caputo generalized fractional derivative (GFD) of different order with non-local boundary conditions. The existence solution is obtained by using Krasnoselskii's fixed point theorem, and the uniqueness result is obtained by using the Banach contraction mapping principle. An example is introduced to validate the effectiveness of the results. The results are novel and provide an extension to some of the findings known in the literature. • We have studied the existence and uniqueness (EU) results for nonlinear fractional Langevin equation involving Liouville-Caputo generalized fractional derivative (GFD) of different order with non-local boundary conditions. • We have established the existence of Solutions of nonlinear fractional Langevin equation involving Liouville-Caputo generalized fractional derivative (GFD) by using Krasnoselskii's fixed point theorem. • We have established the Uniqueness of Solutions of nonlinear fractional Langevin equation involving Liouville-Caputo generalized fractional derivative (GFD) by using Banach contraction mapping principle. • An illustrative example of our main results is also presented. [ABSTRACT FROM AUTHOR]

In this paper, a collocation approach for the fractional Newell-Whitehead-Segel equation is described. The finite difference method is used to discretized the time-fractional derivative, and Chebyshev functions are used for interpolating the spatial derivatives. An a priori error estimate is derived. The stability of the suggested approach is presented. Several numerical examples are provided to show the viability and effectiveness of the suggested approach. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear space-time fractional extended Korteweg-de Vries (F-eKdV) equation involving the conformable fractional derivative (CFD) is considered. Bernoulli and Riccati equations as a simplest equation method together with applying modified traveling wave transformation are used to obtain the exact solution of F-eKdV equation. Also, by exp ⁡ (− Φ (ζ)) -expansion method, different kind of exact traveling wave solutions are gained, including periodic-singular, dark-singular and plane-wave solutions. As a result, several soliton solutions of the proposed equation are revealed. These novel soliton waves are constructed using the software package Maple with some given parametric values and displayed graphically in 2-dimensional (2D) and 3-dimensional (3D) plots by using MATLAB. To illustrate how the fractional operator affects the results, some of the acquired solutions are presented for various values of the fractional order α and also compared to their exact solutions in classical case (i.e. α = 1). In the given graphical representations, the physical meaning of the reported solutions is illustrated. From the obtained results, it is observed that the methods are effective and dependable to solve nonlinear fractional partial differential equations (NFPDEs). [ABSTRACT FROM AUTHOR]

The goal of the current paper is to create a novel model for the two-dimensional flow (TDF) of a second-grade fluid and the transfer of heat due to free convection. The second-grade fluid travels through the porous media and is electrically conducted. The governing equation for fluid velocity and temperature is formulated as a set of partial differential equations jointly. The model is developed in fractional form by applying the Caputo-Fabrizio (CF) fractional derivative approach. The two-dimensional flow problem is modeled in terms of one-sided coupled fractional partial differential equations with corresponding physical conditions imposed on the fluid flow and temperature. The problem is initially converted to a dimensionless form and then solved for the exact solutions by applying joint Fourier and Laplace transforms. Exact solutions are examined for fluid velocity and temperature and then in limiting cases are reduced to some well-known fluid motions. Some special cases are also obtained from the general solutions. For the numerical results, Mathcad-15 software is used and the results are plotted in various graphs. The temperature, and velocity profiles are discussed and highlighted for embedded flow parameters. The model obtained in this work is new and can be extended for other fluid models hence this article provides a base for future research. [ABSTRACT FROM AUTHOR]

In this paper, we study small-noise asymptotic behaviors for a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H ∈ (1 / 2 , 1) and magnitude ϵ H. By building up a variational framework and two weak convergence criteria in the factional Brownian motion setting, we establish the large and moderate deviation principles for these types of equations. Besides, we also obtain the central limit theorem, in which the limit process solves a linear equation involving the Lions derivative of the drift coefficient. [ABSTRACT FROM AUTHOR]

[Display omitted] • The modified time-fractional diffusion model plays an important role in heat transfer and fluid flow problems. • A meshless method based on hybridization of Gaussian and cubic kernels is developed to solve the model. • The proposed method has good accuracy and applicable over complex domains with various node distributions. • The time-discrete algorithm is examined in terms of the convergence and unconditional stability properties using the energy method. During the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arises in porous media. The modified time-fractional diffusion equation provides a deeper understanding of several dynamic phenomena. The purpose of the paper is to develop an efficient meshless technique for approximating the modified time-fractional diffusion problem formulated in the Riemann–Liouville sense. The temporal discretization is performed by integrating both sides of the modified time-fractional diffusion model. The unconditional stability of the time discretization scheme and the optimal convergence rate are obtained. Then, the spatial derivatives are discretized through a local hybridization of the cubic and Gaussian radial basis function. This hybrid kernel improves the condition of the system matrix. Therefore, the solution of the linear system can be obtained using direct solvers that reduce significantly computational cost. The main idea of the method is to consider the distribution of data points over the local support domain where the number of points is almost constant. Three examples show that the numerical procedure has good accuracy and applicable over complex domains with various node distributions. Numerical results on regular and irregular domains illustrate the accuracy, efficiency and validity of the technique. This paper adopts a local hybrid kernel meshless approach to solve the modified time-fractional diffusion problem. The main results of the research is the numerical technique with non-uniform distribution in irregular grids. [ABSTRACT FROM AUTHOR]

In this paper, we are going to deal with fractional operators (FOs) with non-singular kernels which is not an easy task because of its restriction at the origin. In this work, we first show the boundedness of the extended form of the modified Atangana-Baleanu (A-B) Caputo fractional derivative operator. The generalized Laplace transform is evaluated for the introduced operator. By using the generalized Laplace transform, we solve some fractional differential equations. The corresponding form of the Atangana-Baleanu Caputo fractional integral operator is also established. This integral operator is proved bounded and obtained its Laplace transform. The existence and Hyers-Ulam stability is explored. In the last results, we studied the relation between our defined operators. The operators in the literature are obtained as special cases for these newly explored FOs. [ABSTRACT FROM AUTHOR]

In this work, four problems for stochastic fractional pseudo-parabolic containing bounded and unbounded delays are investigated. The fractional derivative and the stochastic noise we consider here are the Caputo operator and the fractional Brownian motion. For the two problems involving bounded delays, we aim at establishing global existence, uniqueness, and regularity results under integral Lipschitz conditions for the non-linear source terms. Such behaviors of mild solutions are also analyzed in the unbounded delay cases but under globally and locally Lipschitz assumptions. We emphasize that our results are investigated in the novel spaces C ([ − r , T ] ; L p (Ω , W l , q (D))) , C μ ((− ∞ , T ] ; L p (Ω , W l , q (D))) , and the weighted space F μ ɛ ((− ∞ , T ] ; L p (Ω , W l , q (D))) , instead of usual ones C ([ − r , T ] ; L 2 (Ω , H)) , C μ ((− ∞ , T ] ; L 2 (Ω , H)). The main technique allowing us to overcome the rising difficulties lies on some useful Sobolev embeddings between the Hilbert space H = L 2 (D) and W l , q (D) , and some well-known fractional tools. In addition, we also study the Hölder continuity for the mild solutions, which can be considered as one of the main novelties of this paper. Finally, we consider an additional result connecting delay stochastic fractional pseudo-parabolic equations and delay stochastic fractional parabolic equations. We show that the mild solution of the first model converges to the mild solution of the second one, in some sense, as the diffusion parameter β → 0 + . [ABSTRACT FROM AUTHOR]

In this paper, the splitting-up scheme for rough partial differential equations (RPDE) is introduced. The convergence rate of the splitting-up scheme for weak solutions to semilinear partial differential equations with linear rough noises is given. An application to stochastic partial differential equations with fractional Browian motions is considered. [ABSTRACT FROM AUTHOR]

• Nonlinear vibrations of viscoelastic functionally graded micro-beam is considered. • To consider the effects of micro scale, the modified couple stress theory is utilized. • Viscoelastic part is modeled using the fractional Kelvin-Voigt. • In the case of resonance, choosing the correct fractional order is vital. • Galerkin method and finite difference method were used to solve the fractional partial differential equations. In this paper, nonlinear vibrations of fractional viscoelastic functionally graded micro-beam are studied based on the fractional calculus. The Modified Couple Stress Theory (MCST) is utilized for considering the effect of the micro-scale. The micro-beam is modeled based on the Euler-Bernoulli theory and the Von Karman's nonlinear strain relations. The viscoelastic part of the micro-beam is considered using the fractional Kelvin-Voigt viscoelastic model. Functionally graded properties vary along the thickness based on a power-law function. The small-scale effects of the micro-beam are modeled using the MCST. Hamiltons principle is used to derive the governing equation of motion. For the solution, the Finite Difference Method (FDM) and the Finite Element Method (FEM) are employed. The FDM is used for discretizing the time domain, and the FEM is used for discretizing the space domain. Effects of the fractional-order, microstructure parameters, and Functionally Graded Material (FGM) properties on the time response of the viscoelastic micro-beam are analyzed numerically. Numerical results show that the fractional-order modeling of the viscoelastic micro-beam can change the behavior of the structure, and increasing the fractional-order, can increase the damping of the system. Numerical simulations also suggest that the effect of the fractional derivative order on the responses of free and forced vibrations is different because there is an important correlation between the fractional-order derivative and the excitation frequency. The results of this paper can be used for modeling the damping of the viscoelastic structures. [ABSTRACT FROM AUTHOR]

Physics informed neural networks (PINNs) are a type of deep learning framework which can use the knowledge of physical laws to facilitate the learning of various forward and inverse problems. PINNs have displayed significant efficiency in numerical computation, but there are still some limitations such as their inability in tackling differential equations with oscillations or singular perturbation. To alleviate this difficulty, a new deep learning framework called improved fractional physics informed neural networks (IFPINNs) is proposed in this paper. Compared to PINNs, the calculation in the weight propagation between layers is replaced with nonlinear functions rather than linear ones. This new architecture allows for a better usage of physical information of differential equations, thus leading to an improved fitting quality for the cases with high oscillations or singular perturbation. Under certain conditions, IFPINNs can be degenerated into PINNs. Moreover, this paper investigates the performance of IFPINNs in dealing with fractional differential equations with time-fractional orders ranging from 1 to 2. The numerical results demonstrate the proposed architecture can achieve a higher accuracy for various integral and fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper discusses controllability results for active types with infinite-time delay of non-instantaneous impulsive fractional differential equations. The model is constructed based on the generalized Caputo (Caputo-Katugampola) fractional derivative and the control function with non-local Katugampola fractional integral as a boundary condition. Our principal results are established by giving some sufficient hypotheses, utilizing well-known fractional calculus truths and using Krasnoselskii's fixed point theorem. The infinite time delay has been treated with the abstract phase space techniques and fulfilling the ensuing axioms due to Hale and Kato. It turns out that under some sufficient conditions, the problem has at least one controllable solution. An implementation of our theoretical results is demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]

In this research paper, the third-order fractional partial differential equation (FPDE) in the sense of the Caputo fractional derivative and the Atangana-Baleanu Caputo (ABC) fractional derivative is investigated for the first time. The importance of the proposed problem is more general than the third-order linear time-varying systems model. This fractional partial differential equation is converted to the abstract form of the ordinary differential equation at β = 1. A first-order finite difference scheme is created for the converted problem. The stability inequality of the (FDS) is demonstrated for this abstract form problem using Cardano's relation method in Hilbert space. The explicit finite difference method (EFDM) is used to obtain an approximate solution of the third-order fractional partial differential equation (FPDE) based on the Caputo fractional derivative and the (ABC) fractional derivative of order β∈(0,1]. Concerning these equations, the obtained approximate solutions using the proposed method are contrasted with the exact solution. Finally, the simulation for the third-order (FPDE) based on the fractional derivatives of Caputo and (ABC) for both the exact and approximate solutions is given. [ABSTRACT FROM AUTHOR]

This paper analyzes the system describing the absorption solar collector as an application of nanoparticles for the storage of solar energy. The system involves two fractional partial differential equations (FPDEs) utilizing the Caputo fractional definition (CFD). Explicit solutions are determined for the temperature and velocity in terms of the Wright function (WrFn) via Laplace transform (LT). In addition, the solutions are expressed in terms of some well–known special functions at selected values of the fractional–order. Moreover, the behaviors of the temperature and velocity are investigated graphically using the thermo–physical data of the Cu/Al 2 O 3 –nanoparticles. Furthermore, some numerical results are conducted about the performance of the Cu and Al 2 O 3. The results reveal the higher efficiency/performance of the Cu–nanoparticles over the Al 2 O 3 and thus copper enjoys higher capability than alumina for the purpose of solar energy storage. In addition, it is found that 1.41% in the enhancement of heat transfer is achieved by adding 1% of the Cu–nanoparticles while the corresponding enhancement rate of the other three nanoparticles was less. Besides, the enhancement rate reaches 7.1% by increasing the volume fraction of the Cu–nanoparticles to 5%. Furthermore, the enhancement rates for the TiO 2 , Ag, and Al 2 O 3 are 5.41%, 6.47%, and 6.53%, respectively when using 5% of these nanoparticles. Finally, the advantages/effectiveness of the current approach over a previous work in the literature are discussed in detail. [ABSTRACT FROM AUTHOR]

Local fractional calculus has gained wide attention in the field of circuit design. In this paper, we propose the zero-input response(ZIR) of fractal RC circuit modeled by local fractional derivative(LFD) for the first time. With help of the law of switch and the Kirchhoff Voltage Laws, the transient local fractional ordinary differential equation is established, and the corresponding exact solution behavior defined on Cantor sets is presented. What we found especially interesting was that the fractal RC becomes the ordinary one in the particular case κ = 1. The results obtained in this paper reveal that the local fractional calculus is a powerful tool to analyze the fractal circuit systems. [ABSTRACT FROM AUTHOR]

This paper presents an approximate solution of time variable order fractional differential equations with sub-diffusion and super-diffusion. The aim of paper is to solve and analyze this problem by a fully discrete local discontinuous Galerkin scheme. The method is based on local discontinuous Galerkin method in space and a finite difference technique in time. The numerical stability and convergence of the proposed method are investigated then the convergence rate O (h k + 1 + △ t 2 − α (t n)) in the case of sub-diffusion and O (h k + 1 + △ t) in the case of super-diffusion are proven for the presented scheme. Finally, provided numerical examples illustrate efficiency of the method and accuracy of the theory. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce Caputo fractional LC derivative defined in the linear correlated fuzzy-valued number space R F (A) and some of its applications, specifically the feedback control problem in the space R F (A) in the sense of Caputo fractional LC derivative. Firstly, we give definition of the Riemann-Liouville-LC integral and the Caputo fractional LC derivative of a function that takes value in R F (A). Besides, some Caputo factional LC derivative properties of the sum and difference of two functions have also been proved. Moreover, the problem of dynamic systems in space R F (A) is given in both cases where A is a symmetric or non-symmetric fuzzy number. Along with that, the stability theorems of the equilibrium point are also taken into consideration. Finally, we are interested in building a state feedback control function for the fractional differential equation system in space R F (A) to ensure the equilibrium point of the system is asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new finite class of non-polynomial basis functions for the spectral tau approximation of time-fractional partial differential equations on a semi-infinite interval. Different from many other spectral tau approaches, the singular basis functions are based on a finite class of Romanovski–Jacobi polynomials through a fractional coordinate transform. As such, the singularity of the new basis can be tailored to suit that of the singular solutions to a class of time-fractional partial differential equations, leading to spectrally accurate approximations. We introduce the fractional Romanovski-Jacobi-Gauss-type quadrature formulae along with basic approximation results for some weighted projection operators defined in suitable weighted Sobolev spaces. We discuss the relationship between such kind of finite orthogonal functions and other classes of infinite orthogonal functions such as fractional Laguerre functions and fractional modified generalized Laguerre functions. Moreover, we derive a new formula expressing explicitly the fractional integral of the fractional Romanovski-Jacobi functions in terms of the same functions themselves. This family of orthogonal systems offers great flexibility to match a wide range of time-fractional differential equations with Robin boundary conditions and with nonsmooth solutions. [ABSTRACT FROM AUTHOR]

In this paper, we apply some special functions (Mittag-Leffler, Gauss Hypergeometric, Bessel-Maitland and Fox H functions) to investigate a class of matrix-valued random control functions. Using this kind of control function helps us to introduce the concept of multi-stability and get an approximation of non homogenous vector-valued fractional differential equations via the fixed point method. Further, some Ulam-Hyers stability results for the mentioned fractional differential equations in different cases are obtained. Also, we determine some sufficient conditions for the stability of non homogenous vector-valued fractional differential equations. [ABSTRACT FROM AUTHOR]

In the present paper, we analyze a coupled system of nonlinear three point boundary value problems (BVPs) consisting of a coupled system of higher order hybrid sequential differential equations formulated by fractional operators. We study several new conditions in the direction of existence theory for solutions of the given hybrid system under the weaker hypotheses based on the method of measure of non-compactness in degree theory. The required conditions confirming the uniqueness and stability are derived in relation to the coupled system of hybrid sequential BVPs. Lastly, we extend our hybrid sequential system to a real mathematical model of typhoid treatment with four compartments and by deriving numerical schemes for this generalized hybrid sequential system, we analyze the variations of different fractional orders and their impact on dynamical behaviors of the mentioned hybrid system of typhoid model. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the accuracy and the stability of the CN-WSGD when it is applied to the initial–boundary value problem for a space fractional diffusion equation. In the CN-WSGD scheme, the weighted and shifted Grünwald difference (WSGD) scheme is used to discretize the space fractional derivative and the Crank–Nicolson (CN) scheme is used to discretize the temporal derivative. There is a free parameter in a CN-WSGD scheme that affects the stability and the accuracy of the scheme. In general, a CN-WSGD scheme has temporally and spatially second order accuracy, and there exists a scheme that has spatially third order accuracy. However, the third order accuracy scheme is not always stable. In this paper, we present and prove some theoretical results (old and new) on the stability of several CN-WSGD schemes, and consider choosing a value for the parameter such that the corresponding CN-WSGD scheme is unconditioned stable and has optimal upper bound for the accuracy. Numerical results are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u (t) = f (t , u (t)) , α ∈ (n , n + 1) , 0 ≤ t < + ∞ , u (0) = 0 , u ′ ′ (0) = 0 , ... , u (n) (0) = 0 , lim t → + ∞ c D α − 1 u (t) = β u (ξ). The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ (R +) satisfying the Hölder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy. [ABSTRACT FROM AUTHOR]

In this paper, the mean-square synchronization problem is considered for the fractional-order stochastic complex network (FOSCN) via pinning control. Firstly, suppose that the dynamics of the node are modeled by continuous-time nonlinear fractional stochastic differential equations (FSDEs). Next, the pinning nodes are selected according to the low-dimensional pinning criteria. Furthermore, the controller is designed for each pinning node, and a set of sufficient conditions are given to guarantee the mean-square synchronization. Then, the convergence analysis of the closed-loop system is finished by directly utilizing the properties of the integral solution. It proved that the mean-square synchronization of FOSCN can be achieved with the designed controller under some conditions. In addition, the synchronization problem of general fractional-order complex networks (FOCN) without stochastic noise is solved by using a similar method. Finally, compared simulation examples are performed to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper inspires us to propose an adaptive meshless numerical method of solving a 2D variable order time fractional mobile-immobile advection-diffusion equation on arbitrary domain. The mentioned method has three main contributions. (i) The adaptive meshless method could decrease amounts of computations effectively. (ii) On arbitrary domain, we successfully apply Legendre multiwaves in reproducing kernel Hilbert space (RKHS) defined on rectangle domain to approximate the exact solution. (iii) We skillfully build a special spline space, convergence order is obtained in view of convergence theories of spline space. Numerical experiments further demonstrate the validity of the method. • The adaptive meshless method could decrease amounts of computations effectively. • On arbitrary domain, we apply Legendre multiwaves to approximate the exact solution using some approximation theorems. • We successfully give the error estimate in the case of low regularity of the function. [ABSTRACT FROM AUTHOR]

This paper is devoted to identifying a time-dependent source term for a multi-term time-fractional diffusion equation with a nonlocal dynamic boundary and integral type over-determination condition. The time-fractional derivatives are considered in Caputo's sense. By applying Fourier's method we obtained multi-term ordinary fractional order differential equation which has been reduced to an algebraic equation by using Laplace transform. Inverse Laplace transform is used to obtain the solution of multi-term ordinary fractional order differential equation which involves multinomial Mittag-Leffler functions. Under some regularity and consistency conditions on the data, unique existence and stability of the regular solution of the inverse problem is proved. [ABSTRACT FROM AUTHOR]

• Variable thickness viscoelastic circular plate is modeled by variable fractional model at the first time. • Shifted Chebyshev polynomials algorithm solves ternary government equation of the plate accurately in the time domain. • Convergence analysis proves the applicability of proposed algorithm. • Numerical examples prove the correctness and sensitivity of proposed model and algorithm. • Vibration of the plate can be reduced by increasing thickness, reducing area or strengthening restraint. In this paper, a variable fractional derivative model (VFDM) is used to model the viscoelastic circular plate whose thickness varies exponentially. Based on the VFDM, a ternary variable fractional differential equation of viscoelastic circular plate is established. The shifted Chebyshev polynomials (SCPs) algorithm is introduced to numerically solve the complex equation directly in time domain. The effectiveness of proposed SCPs algorithm is verified by convergence analysis. Then the dynamic responses of viscoelastic circular plate are researched under different loads, radius, thickness parameters, boundary conditions and viscoelastic materials by SCPs algorithm. The displacement of circular plate increases with the increasing load. The vibration can be reduced by increasing the thickness of plate, reducing plate area or strengthening the restraint on plate. The bending resistance of polyethylene terephthalate (PET) plate is better than polyurea. These results are consistent with the results of literatures. Therefore, the proposed model and algorithm are correct and sensitive in the numerical simulation of viscoelastic circular plate with different conditions. [ABSTRACT FROM AUTHOR]

In this paper we study a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H ∈ (0 , 1 / 2) ∪ (1 / 2 , 1). We prove the well-posedness of this type equations, and then establish a general result on the Bismut formula for the Lions derivative by using Malliavin calculus. As applications, we provide the Bismut formulas of this kind for both non-degenerate and degenerate cases, and obtain the estimates of the Lions derivative and the total variation distance between the laws of two solutions. [ABSTRACT FROM AUTHOR]

This paper considers a new bi-directional cascaded system of a fractional ordinary differential equation (FODE) and a fractional partial differential equation (FPDE) which interacts at an intermediate point. The space-dependent coefficients, interaction between the FODE and FPDE at an intermediate point and the presence of fractional calculus makes the FODE–FPDE cascaded system, representative. In this note, we first apply an invertible integral transformation to convert the system into a FODE–FPDE coupled system, as the target system, which is Mittag–Leffler stable. Using the backstepping method and under some assumptions of the space-dependent coefficients, we work out the kernel functions in the transformation and we design a boundary controller. Also, by the invertibility of the transformation, we show the Mittag–Leffler stability of the closed-loop system via the Lyapunov approach. Second, we propose an observer for which we prove that it can well estimate the original cascaded system. Then, we design an output feedback boundary control law and show that the closed-loop system is Mittag–Leffler stable under the designed output feedback control law. Finally, we present some numerical illustrations to show the correctness of the theoretical results. • A new FODE–FPDE coupled system interacting at an intermediate point is considered. • Using backstepping method, observer and observer-based output feedback are presented. • The state and the output feedback controllers for the coupled system are designed. • The Mittag–Leffler stability of the closed-loop system is proved. [ABSTRACT FROM AUTHOR]

In this paper, we study the L 1 − 2 method for solving numerically a class of time fractional parabolic differential equations on a graded mesh. Based on the nonsmooth regularity assumptions, the stability and convergence of the proposed numerical method are investigated. Using rigorous and sharp theoretical analysis, we obtain the optimal graded parameter r = (3 − α) / (1 + α) of the graded mesh. Numerical experiments are given to demonstrate that even if the α is small, the graded mesh based on the optimal graded parameter is still able to recover the optimal convergence order of the L 1 − 2 method. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to develop a technique for approximating the numerical solution of certain class of tempered fractional differential equations. The proposed technique is based on the Newton–Cotes quadrature formula, and it is established by using generalized Lagrange interpolation polynomials. The product integration technique is used to develop a formula for the left and right-sided tempered fractional integrals. Next, by using these formulas, a method is developed for solving a general class of tempered fractional differential equations. Furthermore, three distinct classes of initial value problems and one of the terminal value problems for tempered fractional differential equations are taken into consideration and each class is illustrated by a numerical experiment to demonstrate the validity and effectiveness of the proposed methodology. Moreover, upper bound of error in the numerical approximations have been provided. [ABSTRACT FROM AUTHOR]

The paper explores numerical approaches to a class of fractional boundary value problems that can be converted into Volterra integral equations with multiple singular kernels. We propose a collocation boundary value method with a graded mesh for the resulting weakly singular integral equations. Moreover, we demonstrate the local convergence property of the presented method using the Gronwall's inequality. In addition, we assess the stability of the algorithm by examining the L 2 -norm of the approximate solution. It is found that the proposed collocation method exhibits both fast convergence order and good stability. The numerical experiments provide rigorous validation of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the existence results of the solution and the finite-time stability (FTS) are focused for fractional fuzzy differential equations (FFDEs) involving non-instantaneous impulsive effects and perturbance parameters. In view of the definitions of short-memory fractional derivatives and a short-memory fractional modeling approach, the existence of a unique solution is proved through utilizing the fixed point theorem of a weakly contractive mapping on the partially ordered fuzzy space, and the sufficient condition of FTS is obtained by using the generalized fractional Gronwall inequality. The fuzzy fractional derivatives used in this study include the concept of derivatives based on the generalized Hukuhara difference (gH-difference) and the granular difference (gr-difference). It is demonstrated that the obtained results based on the gr-difference can overcome restrictions associated with the previous approaches defined via the gH-difference concept. Finally, some numerical examples are provided to verify our main results. [ABSTRACT FROM AUTHOR]