Differential operator multiplication method for fractional differential equations

Tang, Shaoqiang; Ying, Yuping; Lian, Yuan; Lin, Stephen; Yang, Yibo; Wagner, Gregory J.; Liu, Wing Kam

doi:10.1007/s00466-016-1320-0

Cited by 17 publications

(11 citation statements)

References 11 publications

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“…Actually, these initial conditions are to be determined except x(0). Even worse, sometimes there would be singularities for initial conditions from Dx(0) to Dr-1x(0) as shown in Tang et al. (2016) as well as in the following numerical examples.…”

Section: Solution Methodsmentioning

confidence: 99%

“…Following Tang et al. (2016), a complementary differential operator is constructed as where A is a square matrix of dimension n×n, as mentioned above.…”

Section: Solution Methodsmentioning

confidence: 99%

“…It should be pointed out that the original DOM (Tang et al., 2016) was applied to solve scalar rather than vector equations such as equation (1) with n≥2, and the FD was based on the Riemann–Liouville definition. Basically, the DOM can be straightforwardly applied to vector equations with the Caputo FDs due to the equality between AiD(m-i-1)αAx(t) and Ai+1D(m-i-2)αDαx(t).…”

Section: Solution Methodsmentioning

confidence: 99%

“…(2003) for solving linear viscoelastic dynamic systems with memory. More recently, a novel approach, namely differential operator multiplication (DOM) method, was initiated by Tang et al. (2016), wherein a certain class of FDEs can be transformed exactly into ODEs.…”

Section: Introductionmentioning

confidence: 99%

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Transforming linear FDEs with rational orders into ODEs by modified differential operator multiplication method

Chen

Liu

2018

Journal of Vibration and Control

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This paper presents a modified differential operator multiplication (DOM) method for solving a certain class of fractional differential equations (FDEs), with emphasis on linear oscillators subjected to periodic excitation. The main idea of DOM is to transform the considered FDEs with rational order r/ m into rth-order ordinary differential equations (ODEs), herein r and m are positive integers. The transformation is realized by differentiating the FDEs stepwise with fractional-order r/m, until the rth-order derivative is reached due to the accumulative property for FDs. In the modified method, differently, first-order ODEs are deduced by transforming the FDEs with order r/m into 1 /m. In addition, we introduce auxiliary state space variables to represent the FDs of inhomogeneous terms such as triangle functions. Exact and explicit first-order ODEs are deduced and solved by the Runge–Kutta algorithm. To validate the modified DOM, we solve the fractional Kelvin-Voigt equation and harmonically excited oscillators with visco-elastic dampings. The presented method can provide highly accurate solution, at the expense of linearly increasing computational resources as the solution domain expands. Such high computation accuracy and efficiency would potentially enable the presented method to become a benchmark comparison for solving certain FDEs, especially harmonically excited fractional oscillators.

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Section: Solution Methodsmentioning

confidence: 99%

“…Following Tang et al. (2016), a complementary differential operator is constructed as where A is a square matrix of dimension n×n, as mentioned above.…”

Section: Solution Methodsmentioning

confidence: 99%

Section: Solution Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transforming linear FDEs with rational orders into ODEs by modified differential operator multiplication method

Chen

Liu

2018

Journal of Vibration and Control

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“…In fact, the computational efficiency has been enhanced to just the same as solving ODEs. As we know, such high computational efficiency for solving FDEs has been shared by several other approaches such as spectrum based and memory-free algorithms [26,27], and a newly initiated method named as differential operator multiplication method [29,30]. To the best of our knowledge, an outstanding merit of the presented method is its applicability to various different equations by maintaining such high computational efficiency.…”

Section: Conclusion and Remarksmentioning

confidence: 97%

A Novel Method for Solving the Bagley-Torvik Equation as Ordinary Differential Equation

Liu

et al. 2019

Journal of Computational and Nonlinear Dynamics

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We present a novel method to solve the Bagley-Torvik equation by transforming it into ordinary differential equations (ODEs). This method is based on the equivalence between the Caputo-type fractional derivative (FD) of order 3/2 and the solution of a diffusion equation subjected to certain initial and boundary conditions. The key procedure is to approximate the infinite boundary condition by a finite one, so that the diffusion equation can be solved by separation of variables. By this procedure, the Bagley-Torvik and the diffusion equations together are transformed to be a set of ODEs, which can be integrated numerically by the Runge-Kutta scheme. The presented method is tested by various numerical cases including linear, nonlinear, nonsmooth, or multidimensional equations, respectively. Importantly, high computational efficiency is achieved as this method is at the expense of linearly increasing computational cost with the solution domain being enlarged.

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A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

Lin

Wang

2017

Comput Mech

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Differential operator multiplication method for fractional differential equations

Cited by 17 publications

References 11 publications

Transforming linear FDEs with rational orders into ODEs by modified differential operator multiplication method

Transforming linear FDEs with rational orders into ODEs by modified differential operator multiplication method

A Novel Method for Solving the Bagley-Torvik Equation as Ordinary Differential Equation

A finite element formulation preserving symmetric and banded diffusion stiffness matrix characteristics for fractional differential equations

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