Definitions of Real Order Integrals and Derivatives Using Operator Approach

Andriambololona, Raoelina

doi:10.11648/j.pamj.20130201.11

Cited by 10 publications

(13 citation statements)

References 5 publications

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“…The fractional order integral and derivatives of Riemann-Liouville operator of order µ ∈ C, Re(µ) > 0, are given (see [14,37,38]), respectively, as…”

Section: Preliminariesmentioning

confidence: 99%

On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

et al. 2020

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The generalization of Rodrigues’ formula for orthogonal matrix polynomials has attracted the attention of many researchers. This generalization provides new integral and differential representations in addition to new mathematical results that are useful in theoretical and numerical computations. Using a recently studied operational matrix for shifted Legendre polynomials with the variable coefficients fractional differential equations, the present work introduces the shifted Legendre-type matrix polynomials of arbitrary (fractional) orders utilizing some Rodrigues matrix formulas. Many interesting mathematical properties of these matrix polynomials are investigated and reported in this paper, including recurrence relations, differential properties, hypergeometric function representation, and integral representation. Furthermore, the orthogonality property of these polynomials is examined in some particular cases. The developed results provide a matrix framework that generalizes and enhances the corresponding scalar version and introduces some new properties with proposed applications. Some of these applications are explored in the present work.

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“…The fractional order integral and derivatives of Riemann-Liouville operator of order µ ∈ C, Re(µ) > 0, are given (see [14,37,38]), respectively, as…”

Section: Preliminariesmentioning

confidence: 99%

On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

et al. 2020

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“…where H is written as H N /H D . While we obtained H by converting from state space to a transfer function, it should be noted that Equation 19 still holds true if the transfer function H was obtained via another method. This latter characteristic will be the basis for the model synthesis approach discussed in §4.…”

Section: Model Order Reduction From I-mdof To F-sdofmentioning

confidence: 99%

“…This latter characteristic will be the basis for the model synthesis approach discussed in §4. Equation 19 can be made a function of frequency by substituting s = iω. Note that Equation 19 provides a fractional order α which guarantees an exact match of the transfer functions of the two systems.…”

Section: Model Order Reduction From I-mdof To F-sdofmentioning

confidence: 99%

“…Equation 19 can be made a function of frequency by substituting s = iω. Note that Equation 19 provides a fractional order α which guarantees an exact match of the transfer functions of the two systems. Therefore, as far as the individual functions H D and H N are exact, the response of the equivalent F-SDOF is an exact match of the initial I-MDOF.…”

Section: Model Order Reduction From I-mdof To F-sdofmentioning

confidence: 99%

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Model-order reduction of lumped parameter systems via fractional calculus

Hollkamp¹,

Sen²,

Semperlotti³

2018

Journal of Sound and Vibration

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This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional integer order approach to the simulation of non-homogeneous systems dictates the use of numerical solutions and often imposes stringent compromises between accuracy and computational performance. Fractional calculus provides an alternative approach where complex dynamical systems can be modeled with compact fractional equations that not only can still guarantee analytical solutions, but can also enable high levels of order reduction without compromising on accuracy. Different approaches are explored in order to transform the integer order model into a reduced order fractional model able to match the dynamic response of the initial system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting fractional differential models have both a complex and frequency-dependent order of the differential operator. The implications of this type of approach for both model order reduction and model synthesis are discussed.

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“…In references [12], [13], we study another method to define real and complex order derivatives and integrals by using operator approach in the case of more general functions than power functions.…”

Section: Conclusion and Comparison Of The Two Definitionsmentioning

confidence: 99%

Two Definitions of Fractional Derivative of Powers Functions

Andriambololona¹,

Hanitriarivo²,

Ranaivoson³

et al. 2013

PAMJ

Self Cite

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We consider the set of powers functions defined on and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.

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Definitions of Real Order Integrals and Derivatives Using Operator Approach

Cited by 10 publications

References 5 publications

On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

On the Fractional Order Rodrigues Formula for the Shifted Legendre-Type Matrix Polynomials

Model-order reduction of lumped parameter systems via fractional calculus

Two Definitions of Fractional Derivative of Powers Functions

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