A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations

Moghaddam, Behrouz Parsa; Machado, J. A. Tenreiro

doi:10.1016/j.camwa.2016.07.010

Cited by 77 publications

(21 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hafez and Youssri [43] developed the shifted Jacobi collocation method to solve the VO fractional linear subdiffusion and nonlinear reaction-subdiffusion equations. Based on the linear B-spline approximation with the Caputo sense and the Du Fort-Frankel algorithm, Moghaddam and Machado [72] proposed a stable three-level explicit FDM for conducting the nonlinear time VO-FDEs. Tayebi et al [115] proposed an accurate and robust meshless method to solve the VO time fractional advection-diffusion equation model on 2D arbitrary domains.…”

Section: Numerical Methods For Time Fdesmentioning

confidence: 99%

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

Sun

Chang

Zhang

et al. 2019

FCAA

265

138

View full text Add to dashboard Cite

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.

show abstract

Section: Numerical Methods For Time Fdesmentioning

confidence: 99%

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

Sun

Chang

Zhang

et al. 2019

FCAA

265

138

View full text Add to dashboard Cite

show abstract

“…The fractional calculus (FC), as a branch of mathematical analysis, investigates the extension of deriva-tives and integrals to non-integer orders [1][2][3]. Nowadays, the new aspects of the FC are growing fast and its applications are found in different areas like chaos synchronization [4], diffusion equation [5], biology [6], control [7] and economic [8]. Additionally, the utility of the FC in the optimal control problems (OCPs) has attracted the attention of many researchers.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2018

View full text Add to dashboard Cite

The aim of this paper is to design an efficient numerical method to solve a class of time fractional optimal control problems. In this problem formulation, the fractional derivative operator is consid-ered in three cases with both singular and non-singular kernels. The necessary conditions are derived for the optimality of these problems and the proposed method is evaluated for different choices of derivative operators. Simulation results indicate that the suggested technique works well and pro-vides satisfactory results with considerably less computational time than the other existing methods. Comparative results also verify that the fractional operator with Mittag-Leffler kernel in the Caputo sense improves the performance of the controlled system in terms of the transient response compared to the other fractional and integer derivative operators.

show abstract

“…Schemes such as the adomian decomposition [14], homotopy perturbation [15], homotopy analysis [16], and reduced differential transform [17] were proposed to solve the initial value problem of the Burgers-Huxley equation. Moreover, some authors considered the initial boundary value problem of this equation and used spectral collocation [18], finite-difference [19,20], Haar wavelet [21] and modified cubic B-spline differential quadrature [22] methods for its solution.…”

Section: Introductionmentioning

confidence: 99%

Highly Accurate Scheme for the Cauchy Problem of the Generalized Burgers-Huxley Equation

2016

APH

View full text Add to dashboard Cite

In this paper, a weighted algorithm, based on the reduced differential transform method, is introduced. The new approach is adopted in the approximate analytical solution of the Cauchy problem for the Burgers-Huxley equation. The proposed scheme considers the initial and boundary conditions simultaneously for obtaining a solution of the equation. Several examples are discussed demonstrating the performance of the algorithm.

show abstract

A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations

Cited by 77 publications

References 20 publications

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

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

Untitled

Highly Accurate Scheme for the Cauchy Problem of the Generalized Burgers-Huxley Equation

Contact Info

Product

Resources

About