Ordinary Differential Equations Of Fractional Order Research Articles

Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.

A novel numerical scheme for fractional differential equations using extreme learning machine

In this paper, we propose a neural network-based approach with an Extreme Learning Machine (ELM) for solving fractional differential equations. The solution procedure for the linear and nonlinear fractional differential equations has been derived. Also the convergence and stability of the proposed method is provided. Then we examine the numerical solution of several fractional-order ordinary and partial differential equations. As a last example the Burgers equation without an explicit exact solution. The effect of changing the number of neurons on the accuracy of the solution is obtained graphically.

Comparison principles for systems of Caputo fractional order ordinary differential equations

The comparison principle for systems of Caputo fractional order ordinary differential equations still remains blank. By applying the Caputo solution continuation and Vainikko’s definition, and reconstructing the framework of system comparison, here we state and prove general comparison principles for systems of Caputo fractional order ordinary differential equations, and illustrate them by examples with numerical simulations.

Tumor treatment with chemo-virotherapy and MEK inhibitor: A mathematical model of Caputo fractional differential operator

Mitogen-activated protein kinase (MEK) inhibitors and oncolytic virotherapy are identified as promising cancer therapies that can enhance the efficacy of other cancer treatments. A few studies demonstrate that cancer cells proliferate when exposed to virotherapy with MEK inhibitors in an integer order model or without them in a fractional order model. None of them are intended to investigate tumor cell growth under the combined treatment strategy of chemo-virotherapy with a MEK inhibitor in a fractional order model. In this paper, a mathematical model based on fractional order ordinary differential equations (ODEs) is developed for the mutual interactions among tumor cells, as well as a therapeutic combination of chemotherapy, oncolytic viruses and the functional consequence of MEK inhibitor, to investigate how virotherapy could enhance chemotherapy under the action of MEK inhibitor. The numerical results show that virus burst size and MEK inhibitors have a noticeable impact on regulating the trend of tumor cell proliferation. While virotherapy responses to tumor cell proliferation are undoubtedly quicker than chemotherapeutic treatment responses, MEK intensity clearly affects the success of the treatment regimen. The results of this study can contribute to the development of a therapeutic strategy that combines MEK inhibitor functional monitoring with tumor cell growth control.

Compatibility of the Paraskevopoulos’s algorithm with operational matrices of Vieta–Lucas polynomials and applications

In this study, the numerically stable operational matrices are proposed to approximate the Caputo fractional-order derivatives by introducing an algorithm. The proposed operational matrices are named fractional Vieta–Lucas differentiation matrices that are constructed by using the basis of shifted Vieta–Lucas polynomials (VLPs) and Caputo-fractional derivatives. In addition, we numerically solve the multi-order linear and nonlinear fractional-order differential equations by introducing a new numerical algorithm that is based on Paraskevopoulos’s algorithm together with the newly proposed operational matrices of shifted VLPs. The applicability of Paraskevopoulos’s algorithm was previously studied with the Adomian decomposition technique to solve fractional-order ordinary differential equations. We extend its applicability to the operational matrices technique. However, to the best of our knowledge, no previous study has reported that discusses the applicability of Paraskevopoulos’s algorithm with the operational matrices of shifted VLPs. To demonstrate the advantages of the newly proposed numerical algorithm, the multi-order linear and nonlinear Caputo fractional-order differential equations are solved numerically. The solutions of the first, third, fourth, and fifth examples obtained by using the proposed algorithm are compared with the solutions obtained otherwise by using various numerical approaches including stochastic approach, Taylor matrix method, Bessel collocation method, shifted Jacobi collocation method, spectral Tau method, and Chelyshkov collocation method. It is shown that the proposed numerical algorithm and the fractional Vieta–Lucas differentiation matrices are highly efficient in solving all the aforementioned examples.

Stability for the Systems of Ordinary Differential Equations with Caputo Fractional Order Derivatives

Fractional calculus has paid much attention in recent years, because it plays an essential role in many fields of science and engineering, where the study of stability theory of fractional differential equations emerges to be very important. In this paper, the stability of fractional order ordinary differential equations will be studied and introduced the backstepping method. The Lyapunov function is easily found by this method. This method also gives a guarantee of stable solutions for the fractional order differential equations. Furthermore it gives asymptotically stable.

Adaptive Backstepping Method for Stabilizing Systems of Fractional Order Ordinary Differential Equations

In this paper the method of adaptive backstepping for stabilizing and solving system of ordinary and partial differential equations will be used and applied to investigate and study the stability linear systems of Caputo fractional order ordinary differential equations. The basic idea of this approach is to find a quadratic Lyapunov functions for stabilizing the subsystems.

New similarity variable to transform the fluid flow from PDEs into fractional-order ODEs: Numerical study

The recent research aims to study the fractional-order differential equations of the higher-order. The problems consist of momentum and thermal boundary layers. The similarity variables are introduced to convert the partial differential equations (PDEs) into the fractional-order ordinary differential equations (ODEs). The two nonlinear problems of fractional order ODEs are tackled through the (Fractional Differential equation-12) FDE-12 method. The obtained results show that the fractional-order problems are more nonlinear as compared to the integer-order differential equations. The fractional-order increasing the nonlinearity and consequently more efforts required to obtain the outputs. The impact of the physical parameters in the case of the integer-order and fractional-order are obtained and discussed. It is observed that the impact of the physical parameters more compact in the case of fractional order as compared to the integer-order case. The important features of the drag force and heat transfer rate in the case of the fractional-order boundary value problem also have been analyzed and discussed.

A Fractional Drift Diffusion Model for Organic Semiconductor Devices

Because charge carriers of many organic semiconductors (OSCs) exhibit fractional drift diffusion (Fr-DD) transport properties, the need to develop a Fr-DD model solver becomes more apparent. However, the current research on solving the governing equations of the Fr-DD model is practically nonexistent. In this paper, an iterative solver with high precision is developed to solve both the transient and steady-state Fr-DD model for organic semiconductor devices. The Fr-DD model is composed of two fractional-order carriers (i.e., electrons and holes) continuity equations coupled with Poisson’s equation. By treating the current density as constants within each pair of consecutive grid nodes, a linear Caputo’s fractional-order ordinary differential equation (FrODE) can be produced, and its analytic solution gives an approximation to the carrier concentration. The convergence of the solver is guaranteed by implementing a successive over-relaxation (SOR) mechanism on each loop of Gummel’s iteration. Based on our derivations, it can be shown that the Scharfetter–Gummel discretization method is essentially a special case of our scheme. In addition, the consistency and convergence of the two core algorithms are proved, with three numerical examples designed to demonstrate the accuracy and computational performance of this solver. Finally, we validate the Fr-DD model for a steady-state organic field effect transistor (OFET) by fitting the simulated transconductance and output curves to the experimental data.

Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics

PurposeFractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.Design/methodology/approachThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=∑i=0nai(DξαG/G)i/∑i=0nbi(DξαG/G)i.FindingsAchieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.Originality/valueThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

Closed-form wave structures of the space-time fractional Hirota–Satsuma coupled KdV equation with nonlinear physical phenomena

AbstractThe present paper applies the variation of(G^{\prime} /G)-expansion method on the space-time fractional Hirota–Satsuma coupled KdV equation with applications in physics. We employ the new approach to receive some closed form wave solutions for any nonlinear fractional ordinary differential equations. First, the fractional derivatives in this research are manifested in terms of Riemann–Liouville derivative. A complex fractional transformation is applied to transform the fractional-order ordinary and partial differential equation into the integer order ordinary differential equation. The reduced equations are then solved by the method. Some novel and more comprehensive solutions of these equations are successfully constructed. Besides, the intended approach is simplistic, conventional, and able to significantly reduce the size of computational work associated with other existing methods.

Numerical solution of Bagley–Torvik equations using Legendre artificial neural network method

In this article, we have used the Legendre artificial neural network to find the solution of the Bagley–Torvik equation, which is a fractional-order ordinary differential equation. Caputo fractional derivative has been considered throughout the presented work to handle the fractional order differential equation. The training of optimal weights of the network has been carried out using a simulated annealing optimization technique. Here we have presented three examples to exhibit the precision and relevance of the proposed technique with comparison to the other numerical methods with error analysis. The proposed technique is an easy, highly efficient, and robust technique for finding the approximate solution of fractional-order ordinary differential equations.

A semi-analytical approach to Caputo type time-fractional modified anomalous sub-diffusion equations

This article is devoted to a new semi-analytical algorithm for solving time-fractional modified anomalous sub-diffusion equations (FMASDEs). In this method first, the main problem is reduced to a system of fractional-order ordinary differential equations (FODEs) under known initial value conditions by using the Chebyshev collocation procedure. After that, to solve this system, some auxiliary initial value problems are defined. Next, we find an optimal linear combination of some particular solutions for these problems and finally we use this linear combination to construct a semi-analytical approximate solution for the main problem. To demonstrate the convergence property of the new method, a residual error analysis is performed in details. Some test problems are investigated to show reliability and accuracy of the proposed method. Besides, convergence order's indicators are evaluated for all test problems and are compared with ones of the other methods. Moreover, a comparison between our computed numerical results and the reported results of the other numerical schemes in the literature exhibits that the proposed technique is more precise and reliable.In summary advantages of the proposed method are: high accuracy, easy programming, high experimental convergence order, and solving another types of fractional differential equations.

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.

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.

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.

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.

Dynamical Analysis of a Fractional-Order Hantavirus Infection Model

Abstract This paper considers a Hantavirus infection model consisting of a system of fractional-order ordinary differential equations with logistic growth. The fractional-order model describes the spread of Hantavirus infection in a system consisting of a population of susceptible and infected mice. The existence, uniqueness, non-negativity and boundedness of the solutions are established. In addition, the local and global asymptotic stability of the equilibrium points of the fractional order system and the basic reproduction number are studied. The impact of basic reproduction number and carrying capacity on the stability of the fractional order system are also theoretically and numerically investigated.

Numerical Solution of Fractional-Order Ordinary Differential Equations Using the Reformulated Infinite State Representation

Abstract In this paper, we propose a novel approach for the numerical solution of fractional-order ordinary differential equations. The method is based on the infinite state representation of the Caputo fractional differential operator, in which the entire history of the state of the system is considered for correct initialization. The infinite state representation contains an improper integral with respect to frequency, expressing the history dependence of the fractional derivative. The integral generally has a weakly singular kernel, which may lead to problems in numerical computations. A reformulation of the integral generates a kernel that decays to zero at both ends of the integration interval leading to better convergence properties of the related numerical scheme. We compare our method to other schemes by considering several benchmark problems.

Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator

Abstract This research study shows the impacts of deforestation on wildlife species using a newly proposed four dimensional nonlinear mathematical model based upon fractional-order ordinary differential equations. Being a nonlinear model, some theorems using fixed point theory have been proved showing the existence and uniqueness properties for the solution of the fractional-order model. Using an explicit version of Adams–Bashforth–Moulton method devised for the fractional-order ordinary differential equations with convergence order p = min ( 1 + τ , 2 ) , where τ is the order of the differential equations used in the model; some numerical simulations in the form of graphical illustrations have been carried out depicting the better performance of the fractional-order model for being capable enough to capture all the history information of the system under consideration which is a phenomenon not found in the classical (integer-order) differential equations. Varying values for both the fractional-order parameter τ and parameters of the model itself are used during the required numerical simulations.