Variable-order Fractional Differential Equations Research Articles

Generalized shifted Chebyshev polynomials: Solving a general class of nonlinear variable order fractional PDE

We introduce a new general class of nonlinear variable order fractional partial differential equations (NVOFPDE). The NVOFPDE contains, as special cases, several partial differential equations, such as the nonlinear variable order (VO) fractional equations usually denoted as Klein-Gordon, diffusion-wave and convection-diffusion-wave. To find the numerical solution of the NVOFPDE, we formulate a novel class of basis functions called generalized shifted Chebyshev polynomials (GSCP) that includes the shifted Chebyshev polynomials as a particular case. The solution of the NVOFPDE is expanded following the GSCP and the corresponding operational matrices of VO fractional derivatives (VO-FD), in the Caputo type, are obtained. An optimization method based on the GSCP and the Lagrange multipliers converts the problem into a system of nonlinear algebraic equations. The convergence analysis is guaranteed through a theorem concerning the GSCP and several numerical examples confirm the precision of the method.

A new approach for solving linear fractional integro-differential equations and multi variable order fractional differential equations

In the sequel, the numerical solution of linear fractional integrodifferential equations (LFIDEs) and multi variable order fractional differential equations (MVOFDEs) are found by Bezier curve method (BCM) and operational matrix. Some numerical examples are stated and utilized to evaluate the good and accurate results.

On three-dimensional variable order time fractional chaotic system with nonsingular kernel

Abstract We use the Adams-Bashforth-Moulton scheme (ABMS) to determine the approximate solution of a variable order fractional three-dimensional chaotic process. The derivative is defined in the fractional sense of variable order Atangana-Baleanu-Caputo (ABC). Numerical examples show that to solve these variable-order fractional differential equations easily and efficiently, the Adams-Bashforth-Moulton method can be implemented. Lastly, simulation results demonstrate the proposed robust control’s effectiveness.

Legendre-collocation spectral solver for variable-order fractional functional differential equations

A numerical method for the variable-order fractional functional differential equations (VO-FFDEs) has been developed. This method is based on approximation with shifted Legendre polynomials. The properties of the latter were stated, first. These properties, together with the shifted Gauss-Legendre nodes were then utilized to reduce the VO-FFDEs into a solution of matrix equation. Sequentially, the error estimation of the proposed method was investigated. The validity and efficiency of our method were examined and verified via numerical examples.

A novel Jacobi operational matrix for numerical solution of multi-term variable-order fractional differential equations

In this article, we introduce a numerical technique for solving a class of multi-term variable-order fractional differential equation. The method depends on establishing a shifted Jacobi operational matrix (SJOM) of fractional variable-order derivatives. By using the constructed (SJOM) in combination with the collocation technique, the main problem is reduced to an algebraic system of equations that can be solved numerically. The bound of the error estimate for the suggested method is investigated. Numerical examples are introduced to illustrate the applicability, generality, and accuracy of the proposed technique. Moreover, many physical applications problems that have the multi-term variable-order fractional differential equation formulae such as the damped mechanical oscillator problem and Bagley-Torvik equation can be solved via the presented method. Furthermore, the proposed method will be considered as a generalization of many numerical techniques.

Study on Numerical Solution of a Variable Order Fractional Differential Equation based on Symmetric Algorithm

As the class of fractional differential equations with changing order has attracted more attention and attention in the fields of research and engineering, it is important to study its numerical solutions. Numerical solution algorithm for a class of fractional differential equations with transformed arrays based on the proposed symmetry algorithm. The symmetry classification is used for the class of values of the boundary problem of the fractional differential equation with the order of change. A fully symmetric classification of the boundary value problem for a class of fractional differential equations with variable sequences is determined by using a fully symmetric differential sequence sorting algorithm. The problem of the boundary value of the fractional differential equation with the transformed order is reduced to the initial value of the ordinary differential equation. The Legendre polynomial method is used to solve the numerical solution of the starting value of the differential equation. The common differential equation is transformed into a matrix series product by a different operator matrix. The matrix products are converted to algebraic equations by discrete variables. By solving the equations, the numerical solution of the starting value of the common differential equation is obtained.

King algorithm: A novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems

Abstract In this study, a new optimization algorithm, called King, is introduced for solving variable order fractional optimal control problems (VO-FOCPs). The variable order fractional derivative is portrayed in the Caputo sense through the dynamics of the system as variable order fractional differential equation (VO-FDE). To this end, firstly, the VO-FOCP is converted into a system of VO-FDEs. Then, according to the fact that the VO-FDE is equivalent to a Volterra integral equation, the system of VO-FDEs is transformed into an equivalent system of variable order fractional integro-differential equations. In the next step, both the context of minimization of total error and a joint application of Banach’s fixed-point theorem are used to solve a nonlinear optimization problem. Actually, using the existing mechanism, the synchronization problem is recast to an optimization problem. Due to the discretization and its board range of arbitrary nodes, the proposed method provides a very adjustable framework for direct trajectory optimization. Finally, the proposed algorithm is verified using several common optimization functions and a chaotic financial system. Also, through simulation results, the proposed method is compared with some popular methods. Simulation results demonstrate the appropriate performance of the introduced method.

A Novel Lagrange Operational Matrix and Tau-Collocation Method for Solving Variable-Order Fractional Differential Equations

The main result achieved in this paper is an operational Tau-Collocation method based on a class of Lagrange polynomials. The proposed method is applied to approximate the solution of variable-order fractional differential equations (VOFDEs). We achieve operational matrix of the Caputo’s variable-order derivative for the Lagrange polynomials. This matrix and Tau-Collocation method are utilized to transform the initial equation into a system of algebraic equations. Also, we discuss the numerical solvability of the Lagrange-Tau algebraic system in the case of a variable-order linear equation. Error estimates are presented. Some examples are provided to illustrate the accuracy and computational efficiency of the present method to solve VOFDEs. Moreover, one of the numerical examples is concerned with the shape-memory polymer model.

Neural minimization methods (NMM) for solving variable order fractional delay differential equations (FDDEs) with simulated annealing (SA).

To enrich any model and its dynamics introduction of delay is useful, that models a precise description of real-life phenomena. Differential equations in which current time derivatives count on the solution and its derivatives at a prior time are known as delay differential equations (DDEs). In this study, we are introducing new techniques for finding the numerical solution of fractional delay differential equations (FDDEs) based on the application of neural minimization (NM) by utilizing Chebyshev simulated annealing neural network (ChSANN) and Legendre simulated annealing neural network (LSANN). The main purpose of using Chebyshev and Legendre polynomials, along with simulated annealing (SA), is to reduce mean square error (MSE) that leads to more accurate numerical approximations. This study provides the application of ChSANN and LSANN for solving DDEs and FDDEs. Proposed schemes can be effortlessly executed by using Mathematica or MATLAB software to get explicit solutions. Computational outcomes are depicted, for various numerical experiments, numerically and graphically with error analysis to demonstrate the accuracy and efficiency of the methods.

Numerical solution of variable-order fractional integro-partial differential equations via Sinc collocation method based on single and double exponential transformations

This paper addresses the numerical solution of the multi-dimensional variable-order fractional integro-partial differential equations. The upwind scheme and a piecewise linear interpolation, are proposed to approximate the Coimbra variable-order fractional derivatives and integral term with kernel, respectively. Two new approaches via the Sinc collocation method based on single and double exponential transformations are adopted for the temporal and spatial discretizations, respectively. The convergence behaviour of the methods is analysed and the error bounds are provided. In addition, four test problems illustrate the validity and effectiveness of the proposed algorithms.

A novel method to solve variable-order fractional delay differential equations based in lagrange interpolations

Abstract In this work, we present a novel numerical method based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation to solve numerically fractional delay differential equations. We focus on the fractional derivative with power-law, exponential decay and Mittag-Leffler kernel of Liouville-Caputo type with constant and variable-order. The numerical methods were applied to simulate the Duffing attractor, El-Nino/Southern-Oscillation, and Ikeda systems.

Analysis and numerical solution of a nonlinear variable-order fractional differential equation

We prove the wellposedness of a nonlinear variable-order fractional differential equation and the regularity of its solutions. The regularity of the solutions is determined solely by the values of the variable order and its high-order derivatives at time t = 0 (in addition to the usual regularity assumptions on the variable order and the coefficients). If the variable-order reduces to an integer order at t = 0, then the solution has full regularity as the solution to a first-order ordinary differential equation. In this case, we prove that the corresponding finite difference scheme discretized on a uniform mesh has an optimal-order convergence rate. However, if the variable order does not reduce to an integer order at t = 0, then the solution has a singularity at time t = 0, as Stynes et al. proved in [15] for the constant-order time-fractional diffusion equations. The corresponding finite difference scheme discretized on a uniform mesh has only a suboptimal-order convergence rate. Instead, we prove that the finite difference scheme discretized on a graded mesh determined by the value of the variable order at time t = 0 has an optimal-order convergence rate in terms of the number of the time steps. Numerical experiments substantiate these theoretical results.

Unique Existence Result of Approximate Solution to Initial Value Problem for Fractional Differential Equation of Variable Order Involving the Derivative Arguments on the Half-Axis

The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results.

Wellposedness and regularity of a nonlinear variable-order fractional wave equation

We prove the wellposedness of a nonlinear variable-order fractional wave differential equation that describes the vibration of a single particle attached to a viscoelastic material. We then prove that the high-order regularity of its solution depends on the behavior of the variable orders α(t) and β(t) of the fractional damping terms at t=0. In particular, we prove that the solution to the model recovers full regularity when the fractional damping terms exhibit (the physically relevant) elastic restoration force and integer-order damping at the initial time t=0, which, thus, eliminates the incompatibility between the nonlocal differential operators and the locality of classical initial condition.

The minimax approach for a class of variable order fractional differential equation

This paper introduces an approximate solution for Liouville‐Caputo variable order fractional differential equations with order 0 < α(t) ≤ 1. The solution is adapted using a family of fractional‐order Chebyshev functions with unknown coefficients. These coefficients have been obtained by using an optimization approach based on minimax technique and the least pth optimization function. Several linear and nonlinear fractional‐order differential equations are discussed using the proposed technique for fixed and variable order fractional‐order derivatives. Moreover, the response of RC charging circuit with variable order fractional capacitor is studied for different cases. Several comparisons with related published techniques have been added to illustrate the accuracy of the proposed approach.

A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel

In this work we present a numerical method based on the Adams-Bashforth-Moulton scheme to solve numerically fractional delay differential equations. We focus in the fractional derivative with Mittag-Leffler kernel of type Liouville-Caputo with variable-order and the Liouville-Caputo fractional derivative with variable-order. Numerical examples are presented to show the applicability and efficiency of this novel method.

A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation

This paper is concerned with the moving least squares (MLS) meshless approach for the numerical solution of two-dimensional (2D) variable-order time fractional nonlinear diffusion-wave equation (V-OTFND-WE) on arbitrary domains. The variable-order fractional derivative of this equation is expressed in the Caputo type. The proposed method is based on the MLS approximation in conjunction with the finite difference technique. For spatial derivatives, the MLS approximation and for temporal derivative, the finite difference technique are utilized to discretize the equation. The aim of the present paper is to show that the meshfree method based on the MLS approximation and collocation scheme is suitable for solving the variable-order fractional partial differential equations (PDEs). The proposed method is validated, using some different nonlinear numerical examples with complex geometries. The results obtained demonstrate the accuracy and easy implementation of the method for nonlinear variable-order time fractional PDEs.

A numerical technique for variable-order fractional functional nonlinear dynamic systems

This paper provides an efficient technique to discretize the variable-order fractional functional nonlinear differential equations. The proposed technique is based on a piecewise integro quadratic spline interpolation and finite difference approximation. To reveal the performance and accuracy of the proposed method, the behavioral responses of the suitcase with two wheels and pantograph models with variable-order fractional order are analyzed.

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Abstract Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.