Fractional Ordinary Differential Equations Research Articles

A pseudo-spectral based method for time-fractional advection-diffusion equation

In this paper, a pseudo-spectral method with the Lagrange polynomial basis is proposed to solve the time-fractional advection-diffusion equation. A semi-discrete approximation scheme is used for conversion of this equation to a system of ordinary fractional differential equations. Also, to protect the high accuracy of the spectral approximation, the Mittag-Leffler function is used for the integration along the time variable. Some examples are performed to illustrate the accuracy and efficiency of the proposed method.

Bandwidth optimization of information application system under fine integral method of fuzzy fractional order ordinary differential equations

In order to solve the problems that the network bandwidth of previous information application systems can’t guarantee the quality of big data transmission, resulting in low transmission efficiency and slow data processing, etc., an improved information application system is proposed by optimizing the fine integration method of fuzzy fractional ordinary differential equations and combining with software-defined networking (SDN). Firstly, the fractional-order fuzzy differential equation fine integration method of Pade approximation is derived. Then, based on the error analysis theory, the optimization formula for the adaptive selection of weighted parameter N and expanded item number q is obtained. Combined with SDN, an improved information application system is designed, and the improved algorithm and system are detected by example simulation and performance test. The results show that the numerical accuracy and the computational efficiency of the improved algorithm are higher than that of the improved algorithm. The improved port data merge rate and task completion efficiency under different bandwidth are also significantly higher than the original system. It shows that the information application system proposed in this research can better solve the problems of insufficient bandwidth and low communication speed of traditional systems, and provide a new perspective for bandwidth optimization of big data.

Exact Solutions and Conservation Laws of Time-Fractional Levi Equation

Exact solutions were derived for a time-fractional Levi equation with Riemann–Liouville fractional derivative. The methods involve, first, the reduction of the time-fractional Levi equation to fractional ordinary differential equations with Erdélyi-Kober fractional differential operator with respect to point symmetry groups, and second, use of the invariant subspace to reduce the time-fractional Levi equation into a system of fractional ordinary differential equations, which were solved by the symmetry group method. The obtained explicit solutions have interesting analytic behaviors connected with blow-up and dispersion. The conservation laws generated by the point symmetries of the time-fractional Levi equation are shown via nonlinear self-adjointness method.

The optimization effect of fuzzy factional-order ordinary differential equation in block chain financial cross-border E-commerce payment mode

To solve the problems of low execution efficiency and insufficient security of traditional electronic payment mode, in this study, an improved mode of cross-border e-commerce payment based on fuzzy fractional-order ordinary differential equation and secure electronic transaction protocol is proposed. Firstly, the competition model of cross-border e-commerce payment platform is constructed using fuzzy fractional order ordinary differential equation. Based on the theory of Hopf bifurcation, the stability of the model is analyzed. Then, the cross-border e-commerce improved payment mode is built based on the secure electronic transaction protocol. Finally, the efficiency and security performance of the improved payment mode are compared with that of the traditional payment mode with face through simulation experiments. The results show that the improved mode takes less time in encryption and decryption than the traditional mode; the time spent on the security payment of the improved mode remains around 1.2 s, while that of the traditional mode is around 2.7 s; the improved mode security payment takes significantly less time than the traditional mode, which indicates that the payment mode proposed in this research based on fuzzy fractional order ordinary differential equation is not only more secure than the traditional payment mode, but also can effectively improve the payment speed of users and greatly enhance their shopping experience.

Well-posedness of fractional differential equations with variable-order Caputo-Fabrizio derivative

We propose a nonlinear fractional ordinary differential equation (FODE) with variable-order Caputo-Fabrizio derivative, denoted by VO-CF-FODE, and prove its well-posedness. In particular, we prove that when the variable order is an integer at the initial time, the well-posedness of the proposed model does not require additional conditions imposed on the coefficient and the source term that is common in the context of constant-order CF-FODEs. The proposed methods are further extended to prove some well-posedness results of the corresponding linear partial differential equations.

Automatic cyber security risk assessment based on fuzzy fractional ordinary differential equations

Real-time network security risk detection is of great significance to the research of dynamic network security, and it is one of the current research hotspots of network security. Based on the idea of artificial immune, a dynamic network intrusion detection and prediction model based on fuzzy fractional ordinary differential equations is proposed. The uniqueness of the solution is studied in the square integrable equation space using the principle of compression operator, and then the second type of solution is proposed. The differential-type piecewise Taylor series expansion method of the linear Fredholm integral equation is used to obtain the approximate solution expressions and perform convergence and error estimation. The process of network attack detection, antibody concentration calculation method and risk prediction process based on time series autoregressive moving average model are given. The experimental results show that the model can quantitatively analyze the current security situation of the network in real time and make predictions about the risks faced by the network. The prediction effect of catastrophe network risk is better than the GM (1, 1) model, and it is close to the actual risk situation, with high prediction accuracy.

Approximate solutions of one-dimensional systems with fractional derivative

The fractional calculus is useful to model nonlocal phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution of ordinary fractional differential equations. Due to the nonlocality of the fractional derivative, we may establish an equivalence between fractional oscillators and ordinary oscillators with a dissipative term.

Adoption of fuzzy mixed fractional-order ordinary differential equations in spatial information oriented data grid

To improve the efficiency of resource sharing among spatial information, firstly, the mathematical characteristics of fractional order ordinary differential equations are analyzed. By the numerical method, the fractional operator is discretized and approximated to construct the linear differential equation between fractions. Secondly, the fractal spatial model and data grid mining algorithm are designed based on fuzzy mathematical control rules. Finally, the correlation performance of data mining algorithm is tested based on the massive spatial data of network teaching, and the self-tuning of fuzzy fractional order system parameters is simulated. The results show that the algorithm in this research can effectively mine and utilize spatial information when confronted with a large amount of network teaching data. After a long time running, it shows good robustness and avoids the fearless waste of time and space resources. The data mining algorithm based on grid design in this research has a better mining effect on information resources, provides a new solution and method for data management and collaborative transmission in many fields, and plays a positive role in promoting the development of spatial information related fields.

Symmetries, similarity invariant solution, conservation laws and exact solutions of the time-fractional Harmonic Oscillator equation

In this paper, group analysis of the time-fractional Harmonic Oscillator equation with Riemann–Liouville derivative is performed and its reduced fractional ordinary differential equations are determined. With the aid of the concept of nonlinear self-adjoint and the fractional generalization of the Noether operators are obtained conserved vector for this equation. Furthermore, by using Laplace transform and invariant subspace method, exact solutions of the Riemann–Liouville and Caputo fractional partial differential equations are also constructed.

A superlinear numerical scheme for multi-term fractional nonlinear ordinary differential equations

In this paper, we present a superlinear numerical method for multi-term fractional nonlinear ordinary differential equations (MTFNODEs). First, the presented problem is equivalently transformed into its integral form with multi-term Riemann–Liouville integrals. Second, the compound product trapezoidal rule is used to approximate the fractional integrals. Then, the unconditional stability and convergence with the order [Formula: see text] of the proposed scheme are strictly established, where [Formula: see text] and [Formula: see text] are the maximum and the second maximum fractional indexes in the considered MTFNODEs, respectively. Finally, two numerical examples are provided to support the theoretical results.

A new approach to solving multiorder time‐fractional advection–diffusion–reaction equations using BEM and Chebyshev matrix

In this paper, the boundary element method is combined with Chebyshev operational matrix technique to solve two‐dimensional multiorder time‐fractional partial differential equations: nonlinear and linear in respect to spatial and temporal variables, respectively. Fractional derivatives are estimated by Caputo sense. Boundary element method is used to convert the main problem into a system of a multiorder fractional ordinary differential equation. Then, the produced system is approximated by Chebyshev operational matrix technique, and its condition number is analyzed. Accuracy and efficiency of the proposed hybrid scheme are demonstrated by solving three different types of two‐dimensional time‐fractional convection–diffusion equations numerically. The convergent rates are calculated for different meshing within the boundary element technique. Numerical results are given by graphs and tables for solutions and different type of error norms.

Regional output feedback stabilization of semilinear time‐fractional diffusion systems in a parallelepipedon with control constraints

SummaryThis article is concerned with the regional output feedback stabilization problems for semilinear time‐fractional diffusion systems in a 1≤n−dimensional parallelepipedon with control inequality constraints. For this, the spectrum decomposition method is used to derive a finite‐dimensional fractional ordinary differential equation (ODE) system that captures the dominant dynamics of the considered system. With this ODE system, we propose a finite‐dimensional fractional compensator to guarantee that the constrained closed‐loop semilinear systems are Mittag‐Leffler stable in some subregions of their evolution domains. An example is finally included to illustrate our results.

Observer-based output feedback control design for a coupled system of fractional ordinary and reaction–diffusion equations

Abstract In this paper, we investigate the stabilization problem of a cascade of a fractional ordinary differential equation (FODE) and a fractional reaction–diffusion (FRD) equation where the interconnections are of Neumann type. We exploit the partial differential equation backstepping method for designing a controller, which guarantees the Mittag–Leffler stability of the FODE-FRD cascade. Moreover, we propose an observer that is Mittag–Leffler convergent. Also, we propose an output feedback boundary controller, and we prove that the closed-loop FODE-FRD system is Mittag–Leffler stable in the sense of the corresponding norm. Finally, numerical simulations are presented to verify the results.

Real life application of Caputo fractional derivative for measles epidemiological autonomous dynamical system

Abstract Various infectious diseases primarily contain the characteristics of memory and non-locality. Keeping this in view; we have, in the present paper, proposed a new epidemiological system for the measles epidemic via Caputo fractional order operator which is not only a non-local operator but also contains all characteristics concerned with memory of the epidemic. The proposed system is autonomous in nature which has been made physically meaningful by dimensional homogeneity among the fractional order Caputo derivative and the biological parameters used in the system. Caputo fractional order parameter “τ” and the disease transmission rate “β” are fitted with nonlinear least-squares curve fitting technique while using real confirmed measles incidence cases for May-December, 2018 in Pakistan as reported by World Health Organization (WHO). Coming to mathematical analysis, the system is found to have unique solution under fixed point theory with biologically feasible region which is shown as positively invariant. Steady-states are determined to be locally asymptotically stable under different conditions on the basic reproduction number whereas the least and the most influential parameters for the system are computed via forward sensitivity index analysis. Finally, the Caputo system is simulated using Adams technique devised for finding approximate solutions of fractional Caputo ordinary differential equations. Comparative analysis is carried out and effects of different biological parameters on dynamics and transmission of the measles epidemic have been thoroughly investigated.

Some high order formulae for approximating Caputo fractional derivatives

In order to approximate the Caputo fractional derivative of order α, 0<α<1, we construct here a new class of formulae on the basis of B-spline interpolation. These new formulae called S1, S2 and S3 have 2−α, 3−α and 4−α order of convergence, respectively. The proposed formulae are as simple as the well-known L1 formula and the main advantage of them lies in the fact that their accuracy is fixed in the whole interval of integration while the previous formulae such as L1-2 have lower accuracy at the beginning of the interval. Hence in comparison with the previous formulae, new ones have better accuracy and their computational costs are comparable. We then modify S2 and S3 formulae for approximating the Caputo fractional derivative of order α, 1<α<2. Some numerical examples as well as two applications in solving fractional ordinary and partial differential equations (PDEs) are provided to demonstrate the applicability and accuracy of the new formulae.

Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term

In this paper, numerical solutions of space-fractional advection-diffusion equations, involving the Riemann-Liouville derivative with a nonlinear source term, are presented. We propose a procedure that combines the fractional Lie symmetries analysis, to reduce the original fractional partial differential equations into fractional ordinary differential equations, with a numerical method. By adopting the Caputo definition of derivative, the reduced fractional ordinary equations are solved by applying the implicit trapezoidal method. The numerical results confirm the applicability and the efficiency of the proposed approach.

High-Order Methods for Systems of Fractional Ordinary Differential Equations and Their Application to Time-Fractional Diffusion Equations

Taking into account the regularity properties of the solutions of fractional differential equations, we develop a numerical method which is able to deal, with the same accuracy, with both smooth and nonsmooth solutions of systems of fractional ordinary differential equations of the Caputo-type. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples. Finally, we solve the time-fractional diffusion equation using a combination of the method of lines and the newly developed hybrid method.

Analytic numeric solution of coronavirus (COVID-19) pandemic model in fractional - order

In this paper, we consider the coronavirus (COVID-19) pandemic model. The fractional ordinary differential equations were defined in the sense of the Caputo derivative. Adams-type predictor-corrector method with α ∈ [0,1] is employed to compute an approximation to the solution of the model of fractional order. The obtained results are compared with the results by Atangana Baleanu derivative method. Basic reproduction number, R0, affects the model behaviour. We used R0 to establish the stability and existence conditions at the equilibrium points. The results obtained show that the method is highly applicable and also an efficient approach for solving fractional ordinary differential equations of such order.

Algorithm for solving the Cauchy problem for stationary systems of fractional order linear ordinary differential equations

A new simplified analytical formula is given for solving the Cauchy problem for a homogeneous system of fractional order linear differential equations with constant coefficients (SFOLDECC). The matrix exponential function in this formula is re- placed by a Taylor series. Next, an analytical expression of the integral is obtained, with the help of which, for the transition matrix, a relation is obtained that allows one to obtain a solution of the Cauchy problem with high accuracy. The results also apply to the case of inhomogeneous systems with constant perturbations and are illustrated by numerical examples.

Controlled Singular Volterra Integral Equations and Pontryagin Maximum Principle

This paper is concerned with a class of controlled singular Volterra integral equations, which could be used to describe problems involving memories. The well-known fractional order ordinary differential equations of the Riemann--Liouville or Caputo types are strictly special cases of the equations studied in this paper. Well-posedness and some regularity results in proper spaces are established for such kind of questions. For the associated optimal control problem, by using a Liapounoff's type theorem and the spike variation technique, we establish a Pontryagin's type maximum principle for optimal controls. Different from the existing literature, our method enables us to deal with the problem without assuming regularity conditions on the controls, the convexity condition on the control domain, and some additional unnecessary conditions on the nonlinear terms of the integral equation and the cost functional.