Numerical Solution of Ordinary Differential Equations of Fractional Order Using Variable Step Size Method

2018 ◽  
Vol 00 (1) ◽  
pp. 143-149
Author(s):  
Fadhel S. Fadhel ◽  
◽  
Hayder H. Khayoon ◽  
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Variable Step Size ◽  
Step Size ◽  
Variable Step

Variable-step-size second-order-derivative multistep method for solving first-order ordinary differential equations in system simulation

2020 ◽  
Vol 11 (01) ◽  
pp. 2050001
Author(s):  
Lei Zhang ◽  
Chaofeng Zhang ◽  
Mengya Liu
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Multistep Method ◽  
Second Order ◽  
Order Derivative ◽  
Variable Step Size ◽  
Variable Order ◽  
Step Size ◽  
Variable Step ◽  
The Stability

According to the relationship between truncation error and step size of two implicit second-order-derivative multistep formulas based on Hermite interpolation polynomial, a variable-order and variable-step-size numerical method for solving differential equations is designed. The stability properties of the formulas are discussed and the stability regions are analyzed. The deduced methods are applied to a simulation problem. The results show that the numerical method can satisfy calculation accuracy, reduce the number of calculation steps and accelerate calculation speed.

scholarly journals A Quantitative Comparison of Numerical Method for Solving Stiff Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computation Time ◽  
Quantitative Comparison ◽  
Initial Value ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Variable Step

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

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.

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