scholarly journals Benchmark problems for Caputo fractional-order ordinary differential equations

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Dingyü Xue ◽  
Lu Bai
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Block Diagram ◽  
Numerical Algorithms ◽  
Benchmark Problems

AbstractThere are many numerical algorithms for solving the fractional-order ordinary differential equations (FODEs). They are usually very different in nature, and it is difficult to compare their performances. To solve this problem, a set of five benchmark problems of different categories of FODEs with known analytical solution are designed and proposed, they can be used as benchmark problems for testing the numerical algorithms. A Simulink block diagram scheme is used for solving these benchmark problems, with computing errors and the running times reported.

scholarly journals Solution of Fractional Autonomous Ordinary Differential Equations

2021 ◽  
Author(s):  
Rami AlAhmad ◽  
Qusai AlAhmad ◽  
Ahmad Abdelhadi
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Singular Kernel ◽  
Autonomous Differential Equations ◽  
Implicit Analytical Solution

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.

Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation

2019 ◽  
Vol 22 (5) ◽  
pp. 1321-1350 ◽  
Author(s):  
Matthias Hinze ◽  
André Schmidt ◽  
Remco I. Leine
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Benchmark Problems ◽  
State Representation ◽  
Weakly Singular Kernel ◽  
Integration Interval ◽  
Novel Approach ◽  
Infinite State

Abstract In this paper, we propose a novel approach for the numerical solution of fractional-order ordinary differential equations. The method is based on the infinite state representation of the Caputo fractional differential operator, in which the entire history of the state of the system is considered for correct initialization. The infinite state representation contains an improper integral with respect to frequency, expressing the history dependence of the fractional derivative. The integral generally has a weakly singular kernel, which may lead to problems in numerical computations. A reformulation of the integral generates a kernel that decays to zero at both ends of the integration interval leading to better convergence properties of the related numerical scheme. We compare our method to other schemes by considering several benchmark problems.

A semi-analytic method for fractional-order ordinary differential equations: Testing results

2018 ◽  
Vol 21 (6) ◽  
pp. 1598-1618 ◽  
Author(s):  
Sergiy Reutskiy ◽  
Zhuo-Jia Fu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Benchmark Problems ◽  
Analytic Method

Abstract The paper presents the testing results of a semi-analytic collocation method, using five benchmark problems published in a paper by Xue and Bai in Fract. Calc. Appl. Anal., Vol. 20, No 5 (2017), pp. 1305–1312, DOI: 10.1515/fca-2017-0068.

Application of He’s Homotopy Perturbation Method to Stiff Systems of Ordinary Differential Equations

2008 ◽  
Vol 63 (1-2) ◽  
pp. 19-23 ◽  
Author(s):  
Mohammad Taghi Darvishi ◽  
Farzad Khani
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
Homotopy Perturbation Method ◽  
Nonlinear Ordinary Differential Equations ◽  
Stiff Systems ◽  
Homotopy Perturbation ◽  
Linear And Nonlinear

We propose He’s homotopy perturbation method (HPM) to solve stiff systems of ordinary differential equations. This method is very simple to be implemented. HPM is employed to compute an approximation or analytical solution of the stiff systems of linear and nonlinear ordinary differential equations.

AUTOMATIC PARALLELIZATION OF ELECTRO-MECHANICAL SYSTEMS GOVERNED BY ODEs

2002 ◽  
Vol 26 (3) ◽  
pp. 347-365
Author(s):  
C.A. Rabbath ◽  
A. Ait El Cadi ◽  
M. Abdonne ◽  
N. Lechevin ◽  
S. Lapierre ◽  
...  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Block Diagram ◽  
Time Integration ◽  
Mechanical Systems ◽  
Automatic Parallelization ◽  
Iteration Step ◽  
Control Laws ◽  
Loop Control ◽  
Novel Method

The paper proposes an effective approach for the automatic parallelization of models of electro-mechanical systems governed by ordinary differential equations. The novel method takes a nominal mathematical model, expressed in block diagram language, and portions in parallel the code to be executed on a set of standard microprocessors. The integrity of the simulations is preserved, the computing resources available are efficiently used, and the simulations are compliant with real-time constraints; that is, the time integration of the ordinary differential equations is performed within restricted time limits at each iteration step. The proposed method is applied to a two-degree-of-freedom revolute joint robotic system that includes an induction motor and two inner-outer loop control laws. Numerical simulations validate the proposed approach.

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