A windowing waveform relaxation method for time-fractional differential equations

Ding, Xiaoli; Jiang, Yu-Gang

doi:10.1016/j.cnsns.2015.06.017

Cited by 7 publications

(3 citation statements)

References 25 publications

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“…Many real dynamical systems are better characterized using a fractional order dynamic model based on fractional calculus or, differentiation or integration of fractional order [1,2]. Finding analytical and numerical solutions for fractional differential equations has been considered as a strong topic of study; these methods include the Legendre spectral tau method [9], the waveform relaxation method [10], the collocation method [11], the explicit iterations method [12] and others [13,14]. Finding analytical and numerical solutions for fractional differential equations has been considered as a strong topic of study; these methods include the Legendre spectral tau method [9], the waveform relaxation method [10], the collocation method [11], the explicit iterations method [12] and others [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…Fractional calculus provides a powerful mathematical tool for the description of physical systems in many fields, such as, among others, physics, engineering and bioengineering , mechanical systems , stochastic systems , mobile sensor , and control systems . Finding analytical and numerical solutions for fractional differential equations has been considered as a strong topic of study; these methods include the Legendre spectral tau method , the waveform relaxation method , the collocation method , the explicit iterations method and others . An interesting active research area of fractional calculus is that of fractional optimal control (FOC), in which the dynamical system as well as cost function involve not only integer order derivatives but also fractional order derivatives or integrals .…”

Section: Introductionmentioning

confidence: 99%

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An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

Nemati

Yousefi

Soltanian

et al. 2016

Asian Journal of Control

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This paper deals with the Ritz spectral method to solve a class of fractional optimal control problems (FOCPs). The developed numerical procedure is based on the function approximation by the Bernstein polynomials along with fractional operational matrix usage. The approximation method is computationally consistent and moreover, has a good flexibility in the sense of satisfying the initial and boundary conditions of the optimal control problems. We construct a new fractional operational matrix applicable in the Ritz method to estimate the fractional and integer order derivatives of the basis. As a result, we achieve an unconstrained optimization problem. Next, by applying the necessary conditions of optimality, a system of algebraic equations is obtained. The resultant problem is solved via Newton's iterative method. Finally, the convergence of the proposed method is investigated and several illustrative examples are added to demonstrate the effectiveness of the new methodology.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

Nemati

Yousefi

Soltanian

et al. 2016

Asian Journal of Control

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show abstract

“…The main properties of the method are flexibility and good parallelism, which is used to solve differential equations [20,21], fractional differential equations [22,23], delay differential equations [24,25], and differential-algebraic equations [26][27][28]. The waveform relaxation method based on multisplitting is proposed to improve parallelism [29,30].…”

Section: Introductionmentioning

confidence: 99%

Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations

Liu¹,

Fu²,

Li³

2017

Discrete Dynamics in Nature and Society

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Fractional order delay differential-algebraic equations have the characteristics of time lag and memory and constraint limit. These yield some difficulties in the theoretical analysis and numerical computation. In this paper, we are devoted to solving them by the waveform relaxation method. The corresponding convergence results are obtained, and some numerical examples show the efficiency of the method.

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Symmetry analysis of some nonlinear generalised systems of space–time fractional partial differential equations with time-dependent variable coefficients

Kumar

Kour

2019

Pramana - J Phys

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A windowing waveform relaxation method for time-fractional differential equations

Cited by 7 publications

References 25 publications

An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

An Efficient Numerical Solution of Fractional Optimal Control Problems by using the Ritz Method and Bernstein Operational Matrix

Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations

Symmetry analysis of some nonlinear generalised systems of space–time fractional partial differential equations with time-dependent variable coefficients

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