Fractional calculus of periodic distributions

Khan, Khaula Naeem; Lamb, Wilson; McBride, Adam C.

doi:10.2478/s13540-011-0016-6

Cited by 7 publications

(4 citation statements)

References 7 publications

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“…371] and which, in turn, coincides with the Marchaud derivative [20,Theorem 20.2]. See also [19] for an approach in the context of periodic distributions. ) Given a closed linear operator A in X, we say that A is sectorial if A satisfies the following conditions (i) D…”

Section: Preliminariesmentioning

confidence: 76%

Normal periodic solutions for the fractional abstract Cauchy problem

Bravo

Lizama

2021

Bound Value Probl

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We show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ t 0 ≥ 0 and $M>0$ M > 0 such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ { ( i m ) α } | m | > t 0 ⊂ ρ ( A ) , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ ∥ ( i m ) α ( A + ( i m ) α I ) − 1 ∥ ≤ M for all | m | > t 0 , m ∈ Z , then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ 1 p < α ≤ 2 p and $1< p < 2$ 1 < p < 2 , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ { D t α G L u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , 2 π ) ; u ( 0 ) = u ( 2 π ) , where $_{GL}D^{\alpha }$ D α G L denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ f ∈ L 2 π p ( R , X ) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ ϕ A ∈ ( 0 , α π / 2 ) and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ ∫ 0 2 π f ( t ) d t ∈ Ran ( A ) .

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Section: Preliminariesmentioning

confidence: 76%

Normal periodic solutions for the fractional abstract Cauchy problem

Bravo

Lizama

2021

Bound Value Probl

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“…The fractional derivative or integral does not reflect merely the nature or quantity of a local area or a single point, but a way of comprehensive consideration, which is more appropriate to describe these problems than an integer order model. Therefore, fractional calculus has been widely used in dynamic systems such as vibration control, [13][14][15][16] viscoelastic material [17][18][19] modeling and so on. So far, the solution of fractional order dynamic system and the analysis of dynamic characteristics become very important.…”

Section: Introductionmentioning

confidence: 99%

Analysis of resonance and bifurcation in a fractional order nonlinear Duffing system

Bai

Yang

Xie

et al. 2022

Math Methods in App Sciences

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In this paper, resonance and bifurcation of a nonlinear damped fractional‐order Duffing system are studied. The amplitude and phase of the steady‐state response of system are obtained by means of the average method, the stability is analyzed, and then the amplitude‐frequency characteristic curves of the system with different parameters are drawn based on the implicit function equation of amplitude. Grunwald–Letnikov fractional derivative is used to discretize the system numerically, the response curve and phase trajectory of the system under different parameters are obtained; meanwhile, the dynamic behavior is analyzed. The bifurcation and saddle bifurcation behavior of the system is studied through numerical simulation with different parameters.

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“…Nevertheless, when we consider derivatives or integrals of non integer order, this fact is not true [15]. Periodicity is also important in the context of fractional calculus [15,16,17,18,19].…”

Section: Introductionmentioning

confidence: 99%

On quasi-periodicity properties of fractional integrals and fractional derivatives of periodic functions

Area

Losada

Nieto

2015

Integral Transforms and Special Functions

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Abstract. This paper is devoted to the study of quasi-periodic properties of fractional order integrals and derivatives of periodic functions. Considering Riemann-Liouville and Caputo definitions, we discuss when the fractional derivative and when the fractional integral of a certain class of periodic functions satisfies particular properties. We study concepts close to the well known idea of periodic function, such as S-asymptotically periodic, asymptotically periodic or almost periodic function. Boundedness of fractional derivative and fractional integral of a periodic function is also studied.

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Fractional calculus of periodic distributions

Cited by 7 publications

References 7 publications

Normal periodic solutions for the fractional abstract Cauchy problem

Normal periodic solutions for the fractional abstract Cauchy problem

Analysis of resonance and bifurcation in a fractional order nonlinear Duffing system

On quasi-periodicity properties of fractional integrals and fractional derivatives of periodic functions

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