An efficient cubic spline approximation for variable-order fractional differential equations with time delay

2016 ◽  
Vol 87 (2) ◽  
pp. 815-826 ◽  
Author(s):  
Shole Yaghoobi ◽  
Behrouz Parsa Moghaddam ◽  
Karim Ivaz
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Spline Approximation ◽  
Cubic Spline ◽  
Variable Order ◽  
Fractional Differential

scholarly journals Hermite Cubic Spline Collocation Method for Nonlinear Fractional Differential Equations with Variable-Order

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 872
Author(s):  
Tinggang Zhao ◽  
Yujiang Wu
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
Cubic Spline ◽  
Variable Order ◽  
Spline Collocation ◽  
Nonlinear Fractional Differential Equations ◽  
Spline Collocation Method ◽  
Fractional Differential ◽  
Cubic Spline Collocation

In this paper, we develop a Hermite cubic spline collocation method (HCSCM) for solving variable-order nonlinear fractional differential equations, which apply C1-continuous nodal basis functions to an approximate problem. We also verify that the order of convergence of the HCSCM is about O(hmin{4−α,p}) while the interpolating function belongs to Cp(p≥1), where h is the mesh size and α the order of the fractional derivative. Many numerical tests are performed to confirm the effectiveness of the HCSCM for fractional differential equations, which include Helmholtz equations and the fractional Burgers equation of constant-order and variable-order with Riemann-Liouville, Caputo and Patie-Simon sense as well as two-sided cases.

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Differential Equations ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Variable Order ◽  
Polynomial Solutions ◽  
Fractional Differential ◽  
Approximate Polynomial

AbstractIn 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.

New Operational Matrix for Solving Multiterm Variable Order Fractional Differential Equations

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
A. M. Nagy ◽  
N. H. Sweilam ◽  
Adel A. El-Sayed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Algebraic System ◽  
Order Differential Equation ◽  
Fundamental Problem ◽  
Operational Matrix ◽  
Variable Order ◽  
Engineering Problems ◽  
Fractional Differential ◽  
Fourth Kind

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.

scholarly journals A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications

2019 ◽  
Vol 22 (1) ◽  
pp. 27-59 ◽  
Author(s):  
HongGuang Sun ◽  
Ailian Chang ◽  
Yong Zhang ◽  
Wen Chen
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Relevant Literature ◽  
Physical Models ◽  
Variable Order ◽  
Mathematical Tool ◽  
Fractional Differential ◽  
Mathematical Foundations ◽  
Selection Of

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.

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