On two improved numerical algorithms for vibration analysis of systems involving fractional derivatives

Zhang and Shimizu (1998) proposed a numerical algorithm based on Newmark method to calculate the dynamic response of mechanical systems involving fractional derivatives. On the basis of Runge-Kutta-Nyström method and Newmark method, the present study proposes two new numerical algorithms, namely, the improved Newmark algorithm using the second order derivative and the improved Runge-Kutta-Nyström algorithm using the second order derivative to solve the fractional differential equations of vibration systems. The accuracy of new algorithms is investigated in detail by numerical simulation. The simulation result demonstrated that the Runge-Kutta-Nyström algorithm using the second order derivative for the vibration analysis of systems involving fractional derivatives is more effective than the Newmark algorithm of Zhang and Shimizu.

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A Universal Predictor-corrector Algorithm for Numerical Simulation of Generalized Fractional Differential Equations

10.21203/rs.3.rs-375580/v1 ◽

2021 ◽

Author(s):

Zaid Odibat

Keyword(s):

Numerical Simulation ◽

Fractional Derivatives ◽

Fractional Integral ◽

Differential Operators ◽

Numerical Solutions ◽

Fractional Operators ◽

Initial Value ◽

Generalized Fractional Integral ◽

Fractional Differential ◽

Predictor Corrector

Abstract This study introduces some remarks on generalized fractional integral and differential operators, that generalize some familiar fractional integral and derivative operators, with respect to a given function. We briefly explain how to formulate representations of generalized fractional operators. Then, mainly, we propose a predictor-corrector algorithm for the numerical simulation of initial value problems involving generalized Caputo-type fractional derivatives with respect to another function. Numerical solutions of some generalized Caputo-type fractional derivative models have been introduced to demonstrate the applicability and efficiency of the presented algorithm. The proposed algorithm is expected to be widely used and utilized in the field of simulating fractional-order models.

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A new second-order midpoint approximation formula for Riemann–Liouville derivative: algorithm and its application

IMA Journal of Applied Mathematics ◽

10.1093/imamat/hxx019 ◽

2017 ◽

Vol 82 (5) ◽

pp. 909-944 ◽

Cited By ~ 3

Author(s):

Hengfei Ding ◽

Changpin Li ◽

Qian Yi

Keyword(s):

Fractional Derivatives ◽

Second Order ◽

Stability And Convergence ◽

Approximation Formula ◽

Finite Difference Schemes ◽

First Order ◽

Liouville Derivative ◽

Two Space Dimensions ◽

Fractional Differential ◽

Nicolson Scheme

Abstract Compared to the classical first-order Grünwald–Letnikov formula at time $t_{k+1}\; (\text{or}\; t_{k})$, we firstly propose a second-order numerical approximate formula for discretizing the Riemann–Liouvile derivative at time $t_{k+\frac{1}{2}}$, which is very suitable for constructing the Crank–Nicolson scheme for the fractional differential equations with time fractional derivatives. The established formula has the following form RLD0,tαu(t)| t=tk+12=τ−α∑ℓ=0kϖℓ(α)u(tk−ℓτ)+O(τ2),k=0,1,…,α∈(0,1), where the coefficients $\varpi_{\ell}^{(\alpha)}$$(\ell=0,1,\ldots,k)$ can be determined via the following generating function G(z)=(3α+12α−2α+1αz+α+12αz2)α,|z|<1. Next, applying the formula to the time fractional Cable equations with Riemann–Liouville derivative in one and two space dimensions. Then the high-order compact finite difference schemes are obtained. The solvability, stability and convergence with orders $\mathcal{O}(\tau^2+h^4)$ and $\mathcal{O}(\tau^2+h_x^4+h_y^4)$ are shown, where $\tau$ is the temporal stepsize and $h$, $h_x$, $h_y$ are the spatial stepsizes, respectively. Finally, numerical experiments are provided to support the theoretical analysis.

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Symmetric and symplectic methods for gyrocenter dynamics in time-independent magnetic fields

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962316500082 ◽

2016 ◽

Vol 07 (02) ◽

pp. 1650008 ◽

Cited By ~ 4

Author(s):

Beibei Zhu ◽

Zhenxuan Hu ◽

Yifa Tang ◽

Ruili Zhang

Keyword(s):

Numerical Simulation ◽

Hamiltonian System ◽

Second Order ◽

Symplectic Methods ◽

Kutta Method ◽

Numerical Accuracy ◽

Runge Kutta Method ◽

Simulation Results ◽

Runge Kutta

We apply a second-order symmetric Runge–Kutta method and a second-order symplectic Runge–Kutta method directly to the gyrocenter dynamics which can be expressed as a noncanonical Hamiltonian system. The numerical simulation results show the overwhelming superiorities of the two methods over a higher order nonsymmetric nonsymplectic Runge–Kutta method in long-term numerical accuracy and near energy conservation. Furthermore, they are much faster than the midpoint rule applied to the canonicalized system to reach given precision.

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