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This manuscript investigates a fractional piecewise dengue transmission model using singular and non-singular kernels. The existence results and uniqueness of the solution are established by using the approach of fixed point and in the framework of piecewise derivative and integral. To obtain the approximate solution of the considered models we apply a piecewise numerical iteration scheme which is based on Newton interpolation polynomials. Furthermore, the numerical scheme for piecewise derivatives encompasses singular and non-singular kernels. This study aims to enhance our understanding of dengue internal transmission dynamics by using a novel piecewise derivative approach that considers both singular and non-singular kernels. This work contributes to clarifying the concept of piecewise derivatives and their significance in understanding crossover dynamics. Moreover, a deep neural network approach is employed with high accuracy in training, testing, and validation of data to investigate the specified disease problem. This methodology is employed to thoroughly investigate the intricacies of the specified disease problem. [ABSTRACT FROM AUTHOR]

AbstractMeasles, a member of the Paramyxoviridae family and the Morbillivirus genus, is an infectious disease caused by the measles virus that is extremely contagious and can be prevented through vaccination. When a person with the measles coughs or sneezes, the virus is disseminated by respiratory droplets. Normally, the appearance of measles symptoms takes 10–14 d following viral exposure. Conjunctivitis, a high temperature, a cough, a runny nose, and a distinctive rash are some of the symptoms. Despite the measles vaccination being available, it is still widespread worldwide. To eradicate measles, the Reproduction Number (i.e. R0<1) must remain less than unity. This study examines a SEIVR compartmental model in the caputo sense using a double dose of vaccine to simulate the measles outbreak. The reproduction number R0 and model properties are both thoroughly examined. Both the local and global stabilities of the proposed model are determined for R0 less and greater than 1. To achieve the model’s global stability, the Lyapunov function is used while the existence and uniqueness of the proposed model are demonstrated In addition to the calculated and fitted biological parameters, the forward sensitivity indices for R0 are also obtained. Simulations of the proposed fractional order (FO) caputo model are performed in order to analyse their graphical representations and the significance of FO derivatives to illustrate how our theoretical findings have an impact. The graphical results show that the measles outbreak is reduced by increasing vaccine dosage rates. [ABSTRACT FROM AUTHOR]

The type of symmetry exhibited by a travelling wave can have important implications for its behaviour and properties, such as its polarization, dispersion, and interactions with other waves or boundaries. The fractional differential Duffing problem refers to the mathematical modelling of nonlinear, damped oscillations of a system with fractional derivatives. It is a generalization of the classical Duffing equation, which describes the behaviour of a nonlinear, damped oscillator (the equation becomes symmetric under time-reversal). The fractional derivatives allow for a more accurate description of the system's memory and hereditary properties. The solution of the fractional Duffing equation can provide insight into the complex dynamic behaviour of various physical, biological, and engineering systems. We are concerned with studying a new differential Duffing fractional problem. It involves some sequential Caputo derivatives with an infinite series of Riemann-Liouville integrals and some other functions. We begin by proving a first existence and uniqueness result, then we discuss two types of stability for the obtained uniqueness result. An illustrative examples is given to show the applicability of the result. We are also concerned with applying the Tanh method to obtain new classes of travelling wave solutions for three important classes of (Khalil) fractional conformable problems; the generalized equation of Duffing, the Landau-Ginzburg-Higgs equation and the Sine-Gordon one. Some numerical simulations are plotted and a conclusion is given at the end. [ABSTRACT FROM AUTHOR]

Awareness of virus spreading is an important issue for building various defence strategies and protecting personal computers, smart devices, network devices, etc. In this research work, we develop epidemiological models to address this problem and introduce certain modified epidemiological fractional SAIR model, where we consider the fractional derivative in Caputo sense. We utilize the residual power series method to construct approximate solutions for the governing system. To show the efficiency and suitability of the proposed technique, we introduce a comparative between the obtained solutions and those solutions that are constructed using the fourth-order Runge-Kutta method. We derive some numerical results by considering specific values for the parameters in the governing model, and then we depict some of these results into two and three dimensions. [ABSTRACT FROM AUTHOR]

In this work, we analyze and develop an efficient numerical scheme for the Lotka–Volterra competitive population dynamics model involving fractional derivative of order α ∈ ( 0 , 1 ) . The fractional derivative is defined in the Caputo sense. The solution exhibits a weak singularity near t = 0. Using the L1 technique, the fractional operator is discretized. The differential equations are reduced to a system of nonlinear algebraic equations. To solve the corresponding nonlinear system, we employed the generalized Newton–Raphson method. The presence of singularities creates a layer at the origin, and as a result, the proposed scheme fails to achieve its optimal convergence on a uniform mesh. To accelerate the rate of convergence, we used a graded mesh with a suitably chosen grading parameter. The stability analysis and error estimates are derived on a maximum norm. Finally, numerical experiments are conducted to show the validity and applicability of the proposed scheme. [ABSTRACT FROM AUTHOR]

The pivotal aim of the present study is to find the solution for a nonlinear system describing the aggregation process of cellular slime mold by using q -homotopy analysis transform method. The coupled system is considered within the frame of the Caputo fractional operator. We examine three different cases with distinct values of sensitivity function χ ρ = 1 , ρ and ρ 2 to exemplify the efficiency and applicability of the considered scheme. We capture the nature of the obtained results with respect to the fractional order, with distinct initial conditions, and illustrate them using 2D and 3D plots for particular values of the parameters. The considered scheme offers parameters, which help to adjust the convergence region, and we plotted the ℏ-curves to dissipate the effect in the present framework. Moreover, some simulating and important behavior of the considered model using attained results shows the prominence of the hired operator while analyzing the coupled equations and confirms the competence of the projected scheme. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to construct an efficient recursive algorithm to solve a time-space fractional Poisson's equation which can be treated as a time-space fractional diffusion equation in two dimensions. The fractional derivatives in both time and space are defined in the Caputo sense. A homotopy perturbation method is introduced to approximate the solution, and a comparison is made between the exact and the approximate solutions. In addition, we present a procedure for solving higher-order fractional Poisson's equations. In this case, the equation is converted to a system of fractional differential equations in which the order of the time derivatives is less than or equal to one. The convergence analysis is carried out, and an apriori bound of the solution is obtained for the present problem. Numerical examples are provided and the experimental evidence proves the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

A numerical scheme is developed to solve the time-fractional linear Kuramoto-Sivahinsky equation in this work. The time-fractional derivative (of order γ) is taken in the Caputo sense. The scheme comprises the backward Euler formula in the temporal direction and the quintic B-spline collocation approach in the spatial direction. Through rigorous analysis, the proposed method is shown to be unconditionally stable and convergent of order 2 − γ and two in the temporal and spatial directions, respectively. Two test problems are solved numerically to demonstrate the convergence and accuracy of the method. [ABSTRACT FROM AUTHOR]

The main aim of this work is to construct an efficient recursive numerical technique for solving multi-term time fractional nonlinear KdV equation. The fractional derivatives are defined in Caputo sense. A modified Laplace decomposition method is introduced to approximate the solution. The Adomian polynomials play an important role to execute such a recursive process. In addition, the mathematical importance and some applications of KdV equation are discussed. The approximate solution obtained by the proposed method can be expressed in the form of an infinite convergent series. The experimental evidences demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, efficient methods are derived for seeking a numerical solution to the time-fractional convection–diffusion equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time-fractional derivative in the Caputo sense with order in (0 , 1) is discretized by the L 2 - 1 σ methods with uniform and non-uniform meshes and the spatial derivative is approximated by the local discontinuous Galerkin (LDG) finite element methods. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm. [ABSTRACT FROM AUTHOR]

An algorithm based on a class of the Chebyshev polynomials family called the fifth-kind Chebyshev polynomials (FCPs) is introduced to solve the multi-term variable-order time-fractional diffusion-wave equation (MVTFD-WE). Appeared fractional derivative operators in these equations are of the Caputo type. Coupling FCPs and the collocation method leads to reduce the MVTFD-WE to a system of algebraic equations. The convergence of the proposed scheme is investigated in a weighted Sobolev space via obtaining error bounds for approximate solutions which shows the method error tends to zero if the number of terms of the series solution is selected sufficiently large. The applicability and efficiency of the suggested method are examined through several illustrative examples. [ABSTRACT FROM AUTHOR]

This paper considers the problem of the semi-global asymptotic stabilisation of fractional-order (FO) linear systems subject to actuator saturation by output feedback. To solve the problem, a family of observer-based linear output feedback laws, parameterised in a positive scalar, is proposed by means of the low gain feedback design technique. The design applies to FO linear systems that are stabilisable and detectable, but not exponentially unstable. For such an FO system under the proposed observer-based linear low gain output feedback, the peak value of the control input for a given initial condition can be made arbitrarily small to avoid actuator saturation by decreasing the value of the parameter towards zero and thus semi-global asymptotic stabilisation is achieved. To obtain these results, we establish the properties of low gain feedback, derive asymptotic expansions and the bounds of high-order derivatives of the Mittag-Leffler (ML) functions to estimate the state responses of FO linear systems, and explicitly construct a Hermitian matrix to satisfy a linear matrix inequality (LMI) stability condition for FO linear systems. The results in this paper extend the corresponding results for integer-order (IO) linear systems. [ABSTRACT FROM AUTHOR]

Due to the shortcomings of power law distributions in the heat transfer laws of fractional calculus, some other forms of derivatives with few other kernel functions have been proposed. This literature survey focuses on the mathematical model of thermo-viscoelasticity which investigates the transient phenomena in a three-dimensional thermoelastic medium in the context of two-temperature Kelvin–Voigt three-phase-lag model of generalized thermoelasticity, defined in integral form on a slipping interval incorporating the memory-dependent heat transport law. The bounding plane is subjected to a time-dependent thermal loading and is free of tractions. Incorporating normal mode as a tool, the problem has been solved analytically in terms of normal modes and the physical fields have been depicted graphically for a copper-like material. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the presence of memory-dependent derivative, effect of delay time and viscosity also. [ABSTRACT FROM AUTHOR]

In this paper, we show Lyapunov and Hartman-Wintner-type inequalities for a fractional partial differential equations with Dirichlet conditions and we give some applications of these inequalities for the eigenvalue problem. Also, we give de La Vallée Poussin-type inequality for the fractional elliptic boundary value problem and Lyapunov-type inequalities for the fractional elliptic systems with Dirichlet conditions. [ABSTRACT FROM AUTHOR]

In this manuscript, a numerical method based on the conjunction of Paraskevopoulos's algorithm and operational matrices is developed to solve numerically the multi-order linear and nonlinear fractional differential equations. By means of this conjunction, the multi-order problem is decomposed into a system of differential equations of fractional order which are then solved by employing operational matrices approach. The accuracy and efficiency of the method is examined by taking some examples. In addition, the numerical results presented in Pak et al. [Analytical solutions of linear inhomogeneous fractional differential equation with continuous variable coefficients. Adv Differ Equ. 2019;2019(256):1–22] are improved in our work. [ABSTRACT FROM AUTHOR]

We obtain the distribution of the sum of independent and non-identically distributed generalized Mittag–Leffler random variables. We then apply this result to study some related fractional point processes. We present their explicit probability mass functions as well as their connections with the fractional integral/differential equations. In the case of a point process with Mittag–Leffler distributed waiting times which alternate two indexes ν 1 , ν 2 ∈ (0 , 1 ] and two rates λ 1 , λ 2 > 0 , we also study the conditional arrival times and we show an application to the telegraph process. [ABSTRACT FROM AUTHOR]

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial [ABSTRACT FROM AUTHOR]

Fractional derivative is a widely accepted theory to describe the physical phenomena and the processes with memory effects which is defined in the form of convolution having kernels as power functions. Due to the shortcomings of power-law distributions, some other forms of derivatives with few other kernel functions are proposed. This present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena in a fiber-reinforced functionally graded unbounded medium due to the presence of periodically varying heat source in the context of three-phase-lag model of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. Employing Laplace and Fourier transforms as tools, the problem has been solved analytically in the transformed domain. The inversion of the Fourier transform is carried out using residual calculus, and the numerical inversion of the Laplace transform has been executed using a method on Fourier series expansion. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the presence of memory dependent derivative, reinforcement, and non-homogeneity also. Communicated by S. Velinsky [ABSTRACT FROM AUTHOR]

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of elasto-thermodiffusion to investigate the transient phenomena for a spherical shell in the context of two temperature theory based on Lord–Shulman model, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. The inner and outer boundaries of the spherical shell are free of traction and are subjected to time-dependent thermal and chemical shocks. Employing Laplace transform as a tool, the problem has been solved analytically in the transformed domain. Numerical inversion of the Laplace transform is carried out incorporating the method of Bellman et al. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the presence of memory-dependent derivative, effect of delay time and thermodiffusion also. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to find an approximate solution of time-fractional coupled nonlinear Schrödinger equations (TFCNLS) through one of the meshless approach based on radial basis functions (RBFs) collocation. The time-fractional derivative is described in terms of the Caputo derivative. Discretizing the time-fractional derivative of the mentioned equation, we first use a scheme of order O (Δ t 2 − α) , 0 < α ≤ 1 , and then the average value of the function in a consecutive time step is used for other terms. Also, we use the RBFs collocation method to approximate the spatial derivative. On the other hand, the stability analysis of the suggested scheme is investigated in a similar way to the classic von Neumann technique for TFCNLS equations. This present paper is to indicate that the meshfree methods are appropriate and reliable to obtain a numerical solution of fractional partial differential equations. This efficiency and accuracy of the present method are verified by solving two examples. We obtain the numerical results from solving this problem on the rectangular domain. All obtained numerical experiments are presented in tables and figures. Finally, it can be said that the main advantage of the mentioned scheme is that the algorithm is very simple and easy to apply. [ABSTRACT FROM AUTHOR]

A time-space fractional reaction-diffusion equation in a bounded domain is considered. Under some conditions on the initial data, we show that solutions may experience blow-up in a finite time. However, for realistic initial conditions, solutions are global in time. Moreover, the asymptotic behavior of bounded solutions is analysed. [ABSTRACT FROM AUTHOR]

In this article, efficient methods are derived for seeking numerical solution to the time fractional convection-diffusion-reaction equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time fractional derivative in the Caputo sense with order in (0, 1) is discretized by the finite difference methods with uniform and non-uniform meshes and the spatial derivative by the local discontinuous Galerkin finite element method. The derived numerical schemes are stable and convergent. Finally, some numerical experiments are presented to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this study, a fractional Bernstein series solution method has been submitted to solve the fractional-order biological population model with one carrying capacity. The numerical method has been implemented by an effective algorithm written on the computer algebraic system Maple 15. An error-bound analysis is performed by using a process similar to the RK45 method. An error estimation technique relating to residual function is presented. In the numerical application, the variations in the population of prey and predator with respect to time and situations of these two species relative to each other are plotted. The outputs obtained from our method are compared with the homotopy perturbation Sumudu transform method and reproducing kernel Hilbert space method. The approximate solutions gained from the Bernstein series method are consistent with those of other methods. The advantage of our method is that it requires less computational cost compared with methods involving more complex operations. [ABSTRACT FROM AUTHOR]

We show that any given function can be approximated with arbitrary precision by solutions of linear, time-fractional equations of any prescribed order. This extends a recent result by Claudia Bucur, which was obtained for time-fractional derivatives of order less than one, to the case of any fractional order of differentiation. In addition, our result applies also to the ψ-Caputo-stationary case, and it will provide one of the building blocks of a forthcoming paper in which we will establish general approximation results by operators of any order involving anisotropic superpositions of classical, space-fractional and time-fractional diffusions. [ABSTRACT FROM AUTHOR]

In this paper, an efficient and robust numerical technique is suggested to solve fractional Volterra integro-differential equations (FVIDEs). The proposed method is mainly based on the generalized fractional-order Legendre wavelets (GFLWs), their operational matrices and the Collocation method. The main advantage of the proposed method is that, by using the GFLWs basis, it can provide more efficient and accurate solution for FVIDEs in compare to integer-order wavelet basis. A comparison between the achieved results confirms accuracy and superiority of the proposed GFLWs method for solving FVIDEs. Error analysis and convergence of the GFLWs basis is provided. [ABSTRACT FROM AUTHOR]

This paper presents an approximate method to solve a class of two-dimensional fractional optimal control problems with nonlinear dynamical system. To implement the new method, by considering the initial-boundary conditions, the unknown state and control functions are approximated by the Bernstein polynomials (B-polynomials) basis using spectral Ritz method, then the problem is reduced to an unconstrained nonlinear optimisation problem. Meanwhile, to reduce computational complexity, a new fractional operational Bernstein matrix generalised on an arbitrary interval is constructed and applied. The choice of polynomial basis functions along with the Ritz method provides good flexibility in which all the given initial and boundary conditions are imposed. At last, we extensively argue the convergence of the new method and several illustrative test problems are added to demonstrate the applicability and effectiveness of the new procedure. Moreover, our achievements are compared with the previous results to show the superiority of the proposed method. [ABSTRACT FROM AUTHOR]

Symmetric stress distribution in an infinite isotropic plane with a line crack, in which surfaces are exposed to the heat flux loading is considered in the framework of fractional thermoelasticity. The heat conduction is described by the time-fractional heat conduction equations with the Caputo derivative of fractional order α. The solution is obtained using the integral transform technique and is expressed in terms of the Mittag-Leffler function. The stress intensity factor is calculated for different values of the order of fractional derivative. The numerical results are illustrated graphically. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a class of eigenvalue problems of fractional differential equations of order with variable coefficients. The method of solution is based on utilizing the fractional series solution to find theoretical eigenfunctions. Then, the eigenvalues are determined by applying the associated boundary conditions. A notable result, for certain cases, is that the eigenfunctions are characterized in terms of the Mittag-Leffler or semi Mittag-Leffler functions. The present findings demonstrate, for certain cases, the existence of a critical value at which the problem has no eigenvalue (for ), only one eigenvalue (at ), a finite or infinitely many eigenvalues (for ). The efficiency and accuracy of the present algorithm are demonstrated through several numerical examples. [ABSTRACT FROM AUTHOR]

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena for a fiber-reinforced thick plate due to the gravitational effects having a heat source, in the context of three-phase-lag model of generalized thermoelasticity, which is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. The upper surface of the plate is free of traction having a prescribed surface temperature while the lower surface rests in a rigid foundation and is thermally insulated. Employing Laplace and Fourier transforms as tools, the problem has been solved analytically in the transformed domain. The inversion of the Fourier transform is carried out using suitable numerical techniques while the numerical inversion of Laplace transform is done incorporating a method on Fourier series expansion technique. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the presence of memory dependent derivative, reinforcement and gravitational effect. [ABSTRACT FROM AUTHOR]

The paper is mainly concerned with a class of neutral stochastic fractional integro-differential equation with Poisson jumps. First, the existence and uniqueness for mild solution of an impulsive stochastic system driven by Poisson jumps is established by using the Banach fixed point theorem and resolvent operator. The exponential stability in the pth moment for mild solution to neutral stochastic fractional integro-differential equations with Poisson jump is obtained by establishing an integral inequality. [ABSTRACT FROM AUTHOR]

This paper has presented a new semi-analytic numerical method for solving multi-point problems for nonlinear singular ordinary differential equations (ODEs) of a high order. The method consists of replacing the original equation by an approximate equation which has an exact analytic solution with a set of free parameters. These free parameters are determined by the use of the collocation procedure. Some examples are given to demonstrate the validity and applicability of the new method and a comparison is made with the existing results. In particular, the singular multi-point boundary value problems (BVPs) with equations of the fourth, fifth and sixth orders are considered. Numerical results show that the proposed method is of high accuracy and is efficient for solving a wide class of the nonlinear singular ODEs. The algorithm of the method is very simple and does not require any specific technique in handling the singularity at the endpoints of the solution domain. [ABSTRACT FROM AUTHOR]

In this paper, the high-dimensional Caputo-type parabolic equation with fractional Laplacian is studied by using the finite difference method. The convergence and error estimate of the established finite difference scheme are shown. And the illustrative examples are also displayed which support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, using the piecewise linear and quadratic Lagrange interpolation functions, we propose a novel numerical approximate method for the Caputo fractional derivative. For the obtained explicit recursion formula, the truncation error is investigated, which shows the involved convergence order isO(τ3−β) withβ∈(0,1). As an application, we use this proposed numerical approximation to solve the time fractional diffusion equations by the barycentric rational interpolations in space. The resultant systems of algebraic equations, truncation error, convergence, and stability are analyzed. Theoretical analysis and numerical examples show this constructed method enjoys accuracy of, wheredis the degree of the rational polynomial. [ABSTRACT FROM AUTHOR]

In this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the Riesz fractional derivative in space. A trapezoidal formula is used to solve a system of Volterra integral equations transformed from spatial discretization. Stability and convergence of the proposed scheme is discussed which shows second-order accuracy both in temporal and spatial directions. Finally, examples are presented to show the accuracy and effectiveness of the schemes. [ABSTRACT FROM AUTHOR]

In this paper, we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operatorwhereand the fractional derivatives,,are in the Caputo sense. Applying integral transform methods, we describe a complete family of eigenfunctions and fundamental solutions of the operatorin classes of functions admitting a summable fractional derivative. The solutions are expressed using the Mittag–Leffler function. From the family of fundamental solutions obtained, we deduce a family of fundamental solutions of the corresponding fractional Dirac operator, which factorizes the fractional Laplace operator introduced in this paper. [ABSTRACT FROM AUTHOR]

This paper presents two numerical solutions of time fractional Fokker– Planck equations (TFFPE) based on the local discontinuous Galerkin method (LDGM). Two time-discretization schemes for the fractional order part of TFFPE are investigated. The first discretization utilizes the fractional finite difference scheme (FFDS) and in the second scheme the fractional derivative is replaced by the Volterra integral equation which it computed by the trapezoidal quadrature scheme (TQS). Then the LDGM has been applied for space-discretization in both schemes. Additionally, the stability and convergence analysis of the proposed methods have been discussed. Finally some test problems have been investigated to confirm the validity and convergence of two proposed methods. The results show that FFDS and TQS areand second-order accurate in time variable, respectively. [ABSTRACT FROM AUTHOR]

In this paper, a series of new high-order numerical approximations toα-th Caputo derivatives (0<α<2) is derived based on a compound of shift operators and high-order approximations to Riemann–Liouville derivatives. The convergence order is independent of the derivative orderα, rather than the previous error estimates. Several numerical examples including the Caputo-type advection–diffusion equation are displayed, which support the derived numerical schemes. [ABSTRACT FROM PUBLISHER]

Time-nonlocal generalization of the classical Fourier law with the “long-tail” power kernel can be interpreted in terms of fractional calculus (theory of integrals and derivatives of noninteger order) and leads to the time-fractional heat conduction equation with the Caputo derivative. Fractional heat conduction equation with the harmonic source term under zero initial conditions is studied. Different formulations of the problem for the standard parabolic heat conduction equation and for the hyperbolic wave equation appearing in thermoelasticity without energy dissipation are discussed. The integral transform technique is used. The corresponding thermal stresses are found using the displacement potential. [ABSTRACT FROM PUBLISHER]

In this paper, a Galerkin finite element scheme to approximate the time–space fractional diffusion equation is studied. Firstly, the fractional diffusion equation is transformed into a fractional Volterra integro-differential equation. And a second-order fractional trapezoidal formula is used to approach the time fractional integral. Then a Galerkin finite element method is introduced in space direction, where the semi-discretization scheme and fully discrete scheme are given separately. The stability analysis of semi-discretization scheme is discussed in detail. Furthermore, convergence analysis of semi-discretization scheme and fully discrete scheme are given in details. Finally, two numerical examples are displayed to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We present the fractional wave equation in a conducting material. We used a Maxwell’s equations with the assumptions that the charge densityand current densityJwere zero, and that the permeabilityand permittivitywere constants. The fractional wave equation will be examined separately; with fractional spatial derivative and fractional temporal derivative, finally, consider a Dirichlet conditions, the Fourier method was used to find the full solution of the fractional equation in analytic way. Two auxiliary parametersandare introduced; these parameters characterize consistently the existence of the fractional space-time derivatives into the fractional wave equation. A physical relation between these parameters is reported. The fractional derivative of Caputo type is considered and the corresponding solutions are given in terms of the Mittag-Leffler function show fractal space-time geometry different from the classical integer-order model. [ABSTRACT FROM AUTHOR]

A quintic B-spline collocation technique is employed for the numerical solution of time-fractional fourth-order partial differential equations. These equations occur in many applications in real-life problems such as modelling of plates and thin beams, strain gradient elasticity and phase separation in binary mixtures, which are basic elements in engineering structures and are of great practical significance to civil, mechanical and aerospace engineering. The time-fractional derivative is described in the Caputo sense. Backward Euler scheme is used for time discretization and the quintic B-spline-based numerical method is used for space discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. The given problem is solved with three different boundary conditions, including clamped-type condition, simply supported-type condition, and a transversely supported-type condition. Numerical results are considered to investigate the accuracy and efficiency of the proposed method. [ABSTRACT FROM PUBLISHER]

In this paper, an implicit finite difference scheme for the nonlinear time-space-fractional Schrödinger equation is presented. It is shown that the implicit scheme is unconditionally stable with experimental convergence order ofO(τ2−α+h2), where τ andhare time and space stepsizes, respectively, and α (0<α<1) is the fractional-order in time. In order to reduce the computational cost, the explicit–implicit scheme is proposed such that the nonlinear term is easily treated. Meanwhile, the implicit finite difference scheme for the coupled time-space-fractional Schrödinger system is also presented, which is unconditionally stable too. Numerical examples are given to support the theoretical analysis. [ABSTRACT FROM PUBLISHER]

This work presents an alternative solution for the mathematical analysis of the fractional waves in dielectric media. For the fractional wave equation, the Caputo fractional derivative was considered, the order of the spatial and temporal fractional derivatives are, respectively. In this analysis, we introduce the appropriate fractional dimensional parameters which characterize consistently the existence of the fractional space and time derivatives into the fractional wave equation. We will consider source free Maxwell equations in isotropic and homogeneous dielectric medium. The general solutions obtained in our research have been expressed in terms of the multivariate Mittag–Leffler functions, these functions depend only on the parametersandpreserving the appropriated physical units according to the system studied. [ABSTRACT FROM AUTHOR]

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with orderO(τ2−α+h4) inL2norm, where τ, α, andhare the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme. [ABSTRACT FROM AUTHOR]

In the present paper we construct the cubic B-spline operational matrix of fractional derivative in the Caputo sense, and use it to solve fractional differential equation. The main characteristic of the approach is that it overcomes the computational difficulty induced by the memory effect. There is no need to save and call all historic information, which can save memory space and reduce computational complexity. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation. The results from this method are good in terms of accuracy. [ABSTRACT FROM PUBLISHER]

It is well known that, under standard assumptions, initial value problems for fractional ordinary differential equations involving Caputo-type derivatives are well posed in the sense that a unique solution exists and that this solution continuously depends on the given function, the initial value, and the order of the derivative. Here, we extend this well-posedness concept to the extent that we also allow the location of the starting point of the differential operator to be changed, and we prove that the solution depends on this parameter in a continuous way too if the usual assumptions are satisfied. Similarly, the solution to the corresponding terminal value problems depends on the location of the starting point and of the terminal point in a continuous way too. [ABSTRACT FROM AUTHOR]

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis. [ABSTRACT FROM PUBLISHER]

In this paper, we present a framework to obtain analytical solutions to a fractional oscillator by the homotopy perturbation method. The equation of motion of a driven fractional oscillator is obtained from the corresponding equation of motion of a driven harmonic oscillator by replacing the second-order time derivative by a fractional derivative of order α with 0<α≤2. The fractional derivative is described in the Caputo sense. Some examples are tested, and the results reveal that the technique introduced here is very effective and convenient for solving a fractional oscillator. The response characteristics of the fractional oscillator are studied for several cases. [ABSTRACT FROM AUTHOR]