Principles of Nabla Fractional Calculus on Time Scales with Inequalities

2011 ◽  
pp. 711-729
Author(s):  
George A. Anastassiou
Keyword(s):  
Fractional Calculus ◽  
Time Scales

scholarly journals Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Usama Hanif ◽  
Ammara Nosheen ◽  
Rabia Bibi ◽  
Khuram Ali Khan ◽  
Hamid Reza Moradi
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Fractional Integrals ◽  
Discrete Fractional Calculus ◽  
Hardy Inequalities ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Superquadratic Functions

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.

scholarly journals Bayesian Estimation in Delta and Nabla Discrete Fractional Weibull Distributions

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
M. Ganji ◽  
F. Gharari
Keyword(s):  
Fractional Calculus ◽  
Bayesian Estimation ◽  
Time Scales ◽  
Discrete Distributions ◽  
Weibull Distributions ◽  
Discrete Fractional Calculus

We use discrete fractional calculus for showing the existence of delta and nabla discrete distributions and then apply time scales for definitions of delta and nabla discrete fractional Weibull distributions. Also, we study the Bayesian estimation of the functions of parameters of these distributions.

scholarly journals Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Yanning Wang ◽  
Jianwen Zhou ◽  
Yongkun Li
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Differential Equation ◽  
Time Scales ◽  
Fractional Derivatives ◽  
Point Theory ◽  
Existence Results ◽  
Recent Approach ◽  
Equation Boundary ◽  
Fractional Differential

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap-Laplacian conformable fractional differential equation boundary value problem on time scaleT:  Tα(Tαup-2Tα(u))(t)=∇F(σ(t),u(σ(t))),Δ-a.e.  t∈a,bTκ2,u(a)-u(b)=0,Tα(u)(a)-Tα(u)(b)=0,whereTα(u)(t)denotes the conformable fractional derivative ofuof orderαatt,σis the forward jump operator,a,b∈T,  0<a<b,  p>1, andF:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.

scholarly journals Principles of delta fractional calculus on time scales and inequalities

2010 ◽  
Vol 52 (3-4) ◽  
pp. 556-566 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Fractional Calculus ◽  
Time Scales

scholarly journals A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration

2015 ◽  
Vol 107 ◽  
pp. 230-237 ◽  
Author(s):  
Nadia Benkhettou ◽  
Artur M.C. Brito da Cruz ◽  
Delfim F.M. Torres
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Fractional Integration ◽  
Fractional Differentiation ◽  
Arbitrary Time

scholarly journals A conformable fractional calculus on arbitrary time scales

2016 ◽  
Vol 28 (1) ◽  
pp. 93-98 ◽  
Author(s):  
Nadia Benkhettou ◽  
Salima Hassani ◽  
Delfim F.M. Torres
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Arbitrary Time

Fractional calculus on time scales with Taylor’s theorem

2012 ◽  
Vol 15 (4) ◽  
Author(s):  
Paul Williams
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Power Functions ◽  
Arbitrary Time ◽  
Basic Properties ◽  
Use Of Time ◽  
Integrability Conditions ◽  
Definition Of

AbstractWe present a definition of the Riemann-Liouville fractional calculus for arbitrary time scales through the use of time scales power functions, unifying a number of theories including continuum, discrete and fractional calculus. Basic properties of the theory are introduced including integrability conditions and index laws. Special emphasis is given to extending Taylor’s theorem to incorporate our theory.

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