Variable-order Fractional Derivative Research Articles

A novel wavelets operational matrix method for the time variable-order fractional mobile–immobile advection–dispersion model

A novel computational technique for the solution of the variable-order fractional mobile–immobile advection–dispersion equation has been presented in this paper. Firstly, operational integration matrices and variable-order fractional derivatives were deduced using Boubaker wavelets to implement this proposed technique. Utilizing Boubaker wavelets basis for functions approximations and the operational matrices of integration and variable-order fractional derivative along with collocation points, the variable-order fractional mobile–immobile advection–dispersion equation is reduced into the system of algebraic equations. In addition, to determine the convergence analysis and error estimate of the proposed numerical technique, some useful theorems are discussed. Finally, to analyze the computational efficiency and applicability of the proposed numerical method, several numerical examples are presented.

Convergence analysis of the hp-version spectral collocation method for a class of nonlinear variable-order fractional differential equations

In this paper, a general class of nonlinear initial value problems involving a Riemann-Liouville fractional derivative and a variable-order fractional derivative is investigated. An existence result of the exact solution is established by using Weissinger's fixed point theorem and Gronwall-Bellman lemma. An hp-version spectral collocation method is presented to solve the problem in numerical frames. The collocation method employs the Legendre-Gauss interpolations to conquer the influence of the nonlinear term and variable-order fractional derivative. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The error estimates under the H1-norm for smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived. Numerical results are given to support the theoretical conclusions.

Approximation and application of the Riesz-Caputo fractional derivative of variable order with fixed memory

In this paper, the Riesz-Caputo fractional derivative of variable order with fixed memory is considered. The studied non-integer differential operator is approximated by means of modified basic rules of numerical integration. The three proposed methods are based on polynomial interpolation: piecewise constant, piecewise linear, and piecewise quadratic interpolation. The errors generated by the described methods and the experimental rate of convergence are reported. Finally, an application of the Riesz-Caputo fractional derivative of space-dependent order in continuum mechanics is depicted.

A variable-order fractional derivative creep constitutive model of salt rock based on the damage effect

A variable-order fractional derivative creep constitutive model of salt rock based on the damage effect

Approximate Solution of 2-Dimensional VO Linear Fractional Partial Differential Equation

The non-polynomial spline method has been used to solving 2-dimensional variable-order(VO) fractional partial differential equations (FPDE). For VO fractional derivative, described in the sense of the Caputo. The main objective of this study and advantage of the proposed method is to investigate a public approximation for the frequency of the trigonometric functions of the non-polynomial part of the spline function. The powerful algorithm of the proposed method gives high accuracy results.

Variable order fractional grey model and its application

The use of constant order differential equations to describe the evolution of complex systems is often unable to describe some of the changing characteristics of the systems accurately. Variable order fractional derivatives provide us with new tools to solve such problems. In this paper, the accumulation and derivative orders of the classic grey model are expanded from constants to functions, and a variable order fractional grey model is established to describe the evolution process of complex systems. Firstly, this paper defines the variable order fractional accumulation generation sequence. On the basis of this sequence, a variable order fractional derivative grey model is established, the parameters of the model are estimated using the least square method, and the quantum particle swarm optimization algorithm is used to solve the order of fractional derivative and accumulation. Sadik transform and Laplace transform are adopted to obtain the analytical solution of the new model. Lastly, the effectiveness of the new model is verified through four cases. Compared with other models, the variable order fractional model can describe the development process of complex systems more accurately.

Optimal bang-bang control for variable-order dengue virus; numerical studies

IntroductionDengue and Malaria are the most important mosquito-borne viral diseases affecting humans. Fever is transmitted between human hosts by infected female aedes mosquitoes. The modeling study of viral infections is very useful to show how the virus replicates in an infected individual and how the human antibody response acts to control that replication, which antibody playing a key role in controlling infection. ObjectivesOptimal control of a novel variable-order nonlinear model of dengue virus is studied in the present work. Bang-bang control is suggested to minimize the viral infection as well as quick clearance of the virus from the host. Necessary conditions for the control problem are given. The variable-order derivatives are given in the sense of Caputo. Moreover, the parameters of the proposed model are dependent on the same variable-order fractional power. Two numerical schemes are constructed for solving the optimality systems. Comparative studies and numerical simulations are implemented. The variable-order fractional derivative can be describe the effects of long variable memory of time dependent systems than the integer order and fractional order derivatives. MethodsBoth the nonstandard generalized fourth order Runge-Kutta and the nonstandard generalized Euler methods are presented. ResultsWe have successfully applied a kind of Pontryagin’s maximum principle with bang-bang control and were able to reduce the viraemia level by adding the dose of DI particles. The nonstandard generalized fourth order Runge-Kutta method has the best results than nonstandard generalized Euler method. ConclusionThe combination of the variable-order fractional derivative and bang-bang control in the Dengue mathematical model improves the dynamics of the model. The nonstandard generalized Euler method and the nonstandard generalized fourth order Runge-Kutta method can be used to study the variable order fractional optimal control problem simply.

Synchronization in finite time for variable-order fractional complex dynamic networks with multi-weights and discontinuous nodes based on sliding mode control strategy

This paper is concerned with the global synchronization in finite time for variable-order fractional complex dynamic networks with multi-weights, where the dynamic nodes are modeled to be discontinuous, and subject to the local Hölder nonlinear growth in a neighborhood of continuous points. Firstly, an inequality with respect to variable-order fractional derivative for convex functions is proposed. On the basis of the proposed inequality, a global convergence principle in finite time for absolutely continuous functions is developed. Secondly, based on proposed convergence principle in finite time, a new sliding mode surface is presented, and an appropriate sliding mode control law is designed to drive the trajectory of the error system to the prescribed sliding mode surface in finite time and remain on it forever. In addition, on the basis of differential inclusions theory and Lur’e Postnikov-type convex Lyapunov function approach, the sufficient conditions with respect to the global stability in finite time are established in terms of linear matrix inequalities for the error system on designed sliding mode surface. Moreover, the upper bound of the settling time is explicitly evaluated. Finally, the effectiveness and correction of synchronization strategies are illustrated through two simulation experiments.

Existence, uniqueness, Ulam–Hyers stability and numerical simulation of solutions for variable order fractional differential equations in fluid mechanics

In this paper, a numerical approximation is shown to solve solutions of time fractional linear differential equations of variable order in fluid mechanics where the considered fractional derivatives of variable order are in the Caputo sense. Existence, uniqueness of solutions and Ulam–Hyers stability results are displayed. To solve the considered equations a numerical approximation based on the shifted Legendre polynomials are proposed. To perform the method, an operational matrix of fractional derivative with variable-order is derived for the shifted Legendre polynomials to be applied for developing the unknown function. By substituting the aforesaid operational matrix into the considered equations and using the properties of the shifted Legendre polynomials together with the collocation points, the main equations are reduced to a system of algebraic equations. The approximate solution is calculated by solving the obtained system which is technically easier for checking. We also study the error analysis for the approximate solution yielded by the introduced method. Finally, the accuracy and performance of the proposed method are checked by some illustrative examples. The illustrative examples results establish the applicability and usefulness of the proposed method.

A hybrid method based on the orthogonal Bernoulli polynomials and radial basis functions for variable order fractional reaction-advection-diffusion equation

In this paper, the variable order fractional derivative in the Heydari-Hosseininia sense is employed to define a new fractional version of 2D reaction-advection-diffusion equation. An efficient and accurate hybrid method based on the shifted orthogonal Bernoulli polynomials and radial basis functions is developed for solving this equation. The presented method converts solving the problem under consideration into solving a system of algebraic equations. The applicability and accuracy of the proposed method are investigated by solving some numerical examples.

The meshless approach for solving 2D variable-order time-fractional advection–diffusion equation arising in anomalous transport

In this paper, with the aim of extending an elegant and straightforward numerical approximation to describe one of the most common physical phenomena has been undertaken. In this regard, the generalization of advection–diffusion equation, namely, the time-fractional advection–diffusion equation with understanding sense variable-order fractional derivative, is taken into consideration. An efficient and accurate approach is relying on the Kansa scheme and finite difference method to provide a mathematical framework to treat the spatial discretization and temporal term, respectively. The meshless collocation approach is utilized for interior scattered points and those on the boundary. Thus, the problem under consideration is reduced to a system of linear algebraic equations. The use of the radial basis function as shape function brings many advantages for proposal numerical method in terms of improved accuracy by setting an appropriate shape parameter and applied for solving high-dimensional models without extra cost. The validity and accuracy of the proposed approach is investigated by four various examples involving three benchmark examples and a practical application of pollution transfer phenomena.

Vieta-Lucas polynomials for the coupled nonlinear variable-order fractional Ginzburg-Landau equations

In this article, the non-singular variable-order fractional derivative in the Heydari-Hosseininia concept is used to formulate the variable-order fractional form of the coupled nonlinear Ginzburg-Landau equations. To solve this system, a numerical scheme is constructed based upon the shifted Vieta-Lucas polynomials. In this method, with the help of classical and fractional derivative matrices of the shifted Vieta-Lucas polynomials (which are extracted in this study), solving the studied problem is transformed into solving a system of nonlinear algebraic equations. The convergence analysis and the truncation error of the shifted Vieta-Lucas polynomials in two dimensions are investigated. Numerical problems are demonstrated to confirm the convergence rate of the presented algorithm.

A Numerical Scheme Based on the Chebyshev Functions to Find Approximate Solutions of the Coupled Nonlinear Sine-Gordon Equations with Fractional Variable Orders

In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.

New Numerical Algorithm to Solve Variable-Order Fractional Integrodifferential Equations in the Sense of Hilfer-Prabhakar Derivative

In this article, a numerical technique based on the Chebyshev cardinal functions (CCFs) and the Lagrange multiplier technique for the numerical approximation of the variable-order fractional integrodifferential equations are shown. The variable-order fractional derivative is considered in the sense of regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives. To solve the problem, first, we obtain the operational matrix of the regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives of CCFs. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional integrodifferential equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate its accuracy and efficiency.

Hereditary Riccati Equation with Fractional Derivative of Variable Order

The Riccati differential equation with a fractional derivative of variable order is considered. A derivative of variable fractional order in the original equation implies the hereditary property of the medium, i.e., the dependence of the current state of a dynamic system on its previous states. A software called Numerical Solution of a Fractional-Differential Riccati Equation (briefly NSFDRE) is created; it allows one to compute a numerical solution of the Cauchy problem for the Riccati differential equation with a derivative of variable fractional order. The numerical algorithm implemented in the software is based on the approximation of the variable-order derivative by finite differences and the subsequent solution of the corresponding nonlinear algebraic system. New distribution modes depending on the specific type of variable order of the fractional derivative were obtained. We also show that some distribution curves are specific for other hereditary dynamic systems.

On the development of variable-order fractional hyperchaotic economic system with a nonlinear model predictive controller

Abstract Mathematical modelling plays an indispensable role in our understanding of systems and phenomena. However, most mathematical models formulated for systems either have an integer order derivate or posses constant fractional-order derivative. Hence, their performance significantly deteriorates in some conditions. For the first time in the current paper, we develop a model of an economic system with variable-order fractional derivatives. Our underlying assumption is that the values of fractional derivatives are time-varying functions instead of constant parameters. The effects of variable-order time derivative into the economic system is studied. The dependency of the system's behaviour on the value of the fractional-order derivative is investigated. Afterwards, a nonlinear model predictive controller (NMPC) for hyperchaotic control of the system is suggested. The necessary optimality and sufficient conditions for solving the nonlinear optimal control problem (NOCP) of the NMPC in the form of fractional calculus with variable-order which is formulated as a two-point boundary value problem (TPBVP) are derived. Since the proposed methodology is a robust controller, the efficiency of the proposed controller in the presence of external bounded disturbances is examined. Simulation results show that not only does the presented control approach suppresses the related chaotic behaviour and stabilizes the close-loop system, but it also rejects the external bounded disturbances.

A numerical method for solving variable order fractional optimal control problems

This study is devoted to introducing a computational technique based on Bernstein polynomials to solve variable order fractional optimal control problems (VO-FOCPs). This class of problems generated by dynamical systems describe with variable order fractional derivatives in the Caputo sense. In the proposed method, the Bernstein operational matrix of the fractional variable-order derivatives will be derived. Then, this matrix is used to obtain an approximate solution to mentioned problems. With the use of Gauss-Legendre quadrature rule and the mentioned operational matrix, the considered VO-FOCPs are reduced to a system of equations that are solved to get approximate solutions. The obtained results show the accuracy of the numerical technique.

The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.

Numerical solution of variable order fractional differential equations by using shifted Legendre cardinal functions and Ritz method

In this paper, we present a new numerical method for solving variable order fractional differential equations, which is based on shifted Legendre cardinal functions. First, we obtain the pseudo-operational matrix of the variable order fractional derivative by applying the properties mentioned in the Caputo derivative of fractional variable order. Then, using Ritz method, the pseudo-operational matrix and collocation method, the problem is reduced to a system of algebraic equations that is solved by Newton’s iterative method. Illustrative examples are included to demonstrate the efficiency and accuracy of the proposed method.

Numerical solution of variable-order space-time fractional KdV–Burgers–Kuramoto equation by using discrete Legendre polynomials

In this paper, a new version of the nonlinear space-time fractional KdV–Burgers–Kuramoto equation has been generated via the variable-order (VO) fractional derivatives defined in the Caputo type. A numerical method has been developed based on the discrete Legendre polynomials (LPs) and the collocation scheme for solving this equation. First, the solution of the problem is expanded in terms of the shifted discrete LPs. Then, this expansion and its derivatives, including the classical partial derivatives and the VO fractional partial derivatives are replaced in the equation. Eventually, the operational matrices of the shifted discrete LPs, including the classical derivatives and the VO fractional derivatives (which are derived in this study), and the collocation method are employed to convert the approximated problem into an algebraic system of equations. Some numerical results are given to illustrate the accuracy of the method.