Variable-order Fractional Differential Equations Research Articles

Computational technique for simulating variable-order fractional Heston model with application in US stock market

In this paper, a numerical technique is developed to discretize variable-order fractional Heston differential equation. The proposed strategy is followed by an optimization technology, genetic algorithm, for tuning the unknown parameters in the proposed model. The performance of the model is analyzed to profit and loss 500 close index from the US stock markets. Simulations illustrate the application of the proposed technique.

Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives

We extended the nutrient–phytoplankton–zooplankton model involving variable-order fractional differential operators of Liouville–Caputo, Caputo–Fabrizio and Atangana–Baleanu. Variable-order fractional operators permits model and describe accurately real world problems, for example, diffusion or spread of nutrients or species in different states. Particularly, we model the interaction of nutrient phytoplankton and its predator zooplankton. The variable-order fractional numerical scheme based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation was consider. Numerical simulation results are provided for illustrating the effectiveness and applicability of the algorithm to solve variable-order fractional differential equations.

FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law

This paper presents the simulation and control implementation on a Field Programmable Gate Array (FPGA) for a class of variable-order fractional chaotic systems by using sliding mode control strategy. Four different fractional variable-order chaotic systems via Atangana–Baleanu–Caputo fractional-order derivative were considered; Dadras, Aizawa, Thomas and 4 Wings attractors. A methodology has been developed to construct variable-order fractional chaotic systems using LabVIEW® software for its implementation in the National Instruments myRio-1900 (Xilinx FPGA Z-7010)® device. The variable-order fractional differential equations and the control law were solved using the variable-order Adams algorithm. Finally, simulation results show that FPGA provides high-speed realizations with the desired accuracy and demonstrate the effectiveness of the proposed sliding mode control.

A numerical solution of variable order fractional functional differential equation based on the shifted Legendre polynomials

This paper deals with the numerical approach of a class of variable order fractional functional boundary value problems. The fractional derivative is described in the variable order Caputo sense. The method is based upon shifted Legendre polynomials. The properties of shifted Legendre polynomials are utilized to reduce the given fractional differential equation problem to the system of algebraic equations. To show the performance of the method some numerical examples are provided and the results are compared with two other methods.

A space–time spectral approximation for solving nonlinear variable-order fractional sine and Klein–Gordon differential equations

In this paper, we propose an efficient spectral numerical method for solving sine and Klein–Gordon nonlinear variable-order fractional differential equations with the initial and Dirichlet boundary conditions. The approach is based on the shifted Legendre–Gauss and Chebyshev–Gauss collocation methods. The Caputo fractional derivative of variable order is adopted, and the original problems are reduced to systems of algebraic equations. The validity and effectiveness of the method is demonstrated by means of several numerical examples.

An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations

The article is devoted to a new computational algorithm based on the Gegenbauer wavelets (GWs) to solve the linear and nonlinear variable-order fractional differential equations. The novel operational matrices for derivatives of positive integer and variable order are derived. New piecewise functions are introduced to obtain the said operational matrices. In the proposed method, the given problem via Gegenbauer wavelets is transformed to a system of algebraic equations. The obtained solutions are endorsing the accuracy and efficiency of the suggested method and are in excellent agreement with the existing literature. The convergence and error bound analysis are presented in our study to show the credibility of the computational method and support the mathematical formulation of the algorithm. The discussed problems reconfirm the appropriateness of said algorithm and perceived that the proposed algorithm is an efficient tool to tackle the nonlinear fractional order problems of complex nature.

Novel numerical method for solving variable-order fractional differential equations with power, exponential and Mittag-Leffler laws

Variable-order differential operators can be employed as a powerful tool to modeling nonlinear fractional differential equations and chaotical systems. In this paper, we propose a new generalize numerical schemes for simulating variable-order fractional differential operators with power-law, exponential-law and Mittag-Leffler kernel. The numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes were applied to simulate the chaotic financial system and memcapacitor-based circuit chaotic oscillator. Numerical examples are presented to show the applicability and efficiency of this novel method.

Fractional collocation methods for multi-term linear and nonlinear fractional differential equations with variable coefficients on the half line

ABSTRACTIn this paper, two new and efficient fractional spectral collocation schemes are proposed to solve the multi-term linear and nonlinear fractional differential equations (FDEs) with variable coefficients on the half line, both for the Riemann–Liouville and Caputo fractional operators. Our main task is to find corresponding fractional pseudospectral differentiation matrices (F-PSDMs) which are computed by two different methods. The first one is based on the three-term recurrence relations and the derivative recurrence relations of the generalized Laguerre polynomials. The second one is based on the direct fractional integral and derivative relations of the generalized Laguerre polynomials. Based on the F-PSDMs, the corresponding spectral collocation schemes are developed. By taking suitable parameter involved in the methods, we can significantly enhance the performance of these schemes. Moreover, both schemes are easy to extend to solve the variable-order FDEs. Numerical examples are presented, which show the efficiency and spectral accuracy of our methods.

Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations

In this paper, we analyze an alcoholism model which involves the impact of Twitter via Liouville–Caputo and Atangana–Baleanu–Caputo fractional derivatives with constant- and variable-order. Two fractional mathematical models are considered, with and without delay. Special solutions using an iterative scheme via Laplace and Sumudu transform were obtained. We studied the uniqueness and existence of the solutions employing the fixed point postulate. The generalized model with variable-order was solved numerically via the Adams method and the Adams–Bashforth–Moulton scheme. Stability and convergence of the numerical solutions were presented in details. Numerical examples of the approximate solutions are provided to show that the numerical methods are computationally efficient. Therefore, by including both the fractional derivatives and finite time delays in the alcoholism model studied, we believe that we have established a more complete and more realistic indicator of alcoholism model and affect the spread of the drinking.

A non-linear thermo-viscoelastic rheological model based on fractional derivatives for high temperature creep in concrete

• Fractional viscoelastic creep model is proposed for concrete at high temperatures. • Model parameters are calibrated using existing experimental data. • Accurate representation of experimental data is achieved with only few parameters. • Time-varying stresses and temperatures can be efficiently modelled. • Model can be applied to concrete under fire or high temperature working conditions. In this paper, a novel non-linear thermo-viscoelastic rheological model based on fractional derivatives for high temperature creep in concrete is proposed. The rheological model consists of a linear springpot unit placed in series with a second springpot used for non-linear creep which activates under high stress and temperature. The model parameters which include the dynamic viscosities of the springpots and the fractional exponent are calibrated using existing experimental data of basic creep strain in concrete under constant stress and temperatures for various aggregate types. The power law form of the naturally resulting creep compliance allows an accurate representation of experimental data with the use of only a few model parameters. Furthermore, the variable-order fractional differential stress-strain equation provides a compact method for analytical and numerical modelling of basic creep under conditions of time-varying stress and temperature. In addition, applications of the proposed model to determine axial deformations in columns and transverse deflections in beams under constant and varying temperatures are demonstrated.

New Operational Matrix for Solving Multiterm Variable Order Fractional Differential Equations

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

A numerical solution for a variable-order reaction–diffusion model by using fractional derivatives with non-local and non-singular kernel

A reaction–diffusion system can be represented by the Gray–Scott model. The reaction–diffusion dynamic is described by a pair of time and space dependent Partial Differential Equations (PDEs). In this paper, a generalization of the Gray–Scott model by using variable-order fractional differential equations is proposed. The variable-orders were set as smooth functions bounded in (0,1] and, specifically, the Liouville–Caputo and the Atangana–Baleanu–Caputo fractional derivatives were used to express the time differentiation. In order to find a numerical solution of the proposed model, the finite difference method together with the Adams method were applied. The simulations results showed the chaotic behavior of the proposed model when different variable-orders are applied.

An efficient numerical method for variable order fractional functional differential equation

In this paper, we consider a new technique for variable order fractional functional differential equations (FDE for short). The proposed method relies on the reproducing kernel splines method (RKSM). The method can lessen computation cost and provide highly precise approximate solutions. Numerical results demonstrate that the algorithm is more effective and efficient.

Analytic Approximate Solutions for a Class of Variable Order Fractional Differential Equations Using The Polynomial Least Squares Method

Abstract In this paper a new way to compute analytic approximate polynomial solutions for a class of nonlinear variable order fractional differential equations is proposed, based on the Polynomial Least Squares Method (PLSM). In order to emphasize the accuracy and the efficiency of the method several examples are included.

Non-Standard Finite Difference Schemes for Solving Variable-Order Fractional Differential Equations

A non-standard finite difference (NSFD) methodology of Mickens is a popular method for the solution of differential equations. In this paper, we discusses how we can generalize NSFD schemes for solving variable-order fractional problems. The variable-order fractional derivatives are described in the Riemann–Liouville and Grunwald–Letinkov sense. Special attention is given to the Grunwald–Letinkov definition which is used to approximate the variable-order fractional derivatives. Some applications of the variable-order fractional in viscous-viscoelasticity oscillator model and chaotic financial system are included to demonstrate the validity and applicability of the proposed technique.

A variable-order fractional differential equation model of shape memory polymers

A shape-memory polymer (SMP) is capable of memorizing its original shape, and can acquire a temporary shape upon deformation and returns to its permanent shape in response to an external stimulus such as a temperature change. SMPs have been widely used industrial and medical applications. Previously, differential equation models were developed to describe SMPs and their applications. However, these models are often of very complicated form, which require accurate numerical simulations.In this paper we argue that a variable-order fractional differential equation model of the shape-memory behavior is more suitable than constant-order fractional differential equation models in terms of modeling the memory behavior of SMPs. We develop a numerical method to simulate the variable-order model and, in particular, to identify the unknown variable order of the model. Numerical experiments are presented to show the utility of the method.

An integro quadratic spline approach for a class of variable-order fractional initial value problems

This paper develops a technique for the approximate solution of a class of variable-order fractional differential equations useful in the area of fluid dynamics. The method adopts a piecewise integro quadratic spline interpolation and is used in the study of the variable-order fractional Bagley–Torvik and Basset equations. The accuracy of the proposed algorithm is verified by means of illustrative examples.

An Optimization Wavelet Method for Multi Variable-order Fractional Differential Equations

An Optimization Wavelet Method for Multi Variable-order Fractional Differential Equations

On solutions of variable-order fractional differential equations

Numerical calculation of the fractional integrals and derivatives is the code tosearch fractional calculus and solve fractional differential equations. The exactsolutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhanced numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give two examples to demonstrate how efficiently our theory can be implemented in practice.

A new numerical method for variable order fractional functional differential equations

In this letter, a high order numerical scheme is proposed for solving variable order fractional functional differential equations. Firstly, the problem is approximated by an integer order functional differential equation. The integer order differential equation is then solved by the reproducing kernel method. Numerical examples are given to demonstrate the theoretical analysis and verify the efficiency of the proposed method.