A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

Fan, Qiang; Wu, Guo–Cheng; Fu, Hui

doi:10.1007/s44198-021-00021-w

Cited by 48 publications

(28 citation statements)

References 13 publications

(13 reference statements)

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“…Before continuing, it is necessary to mention that due to the large number of fractional operators that may exist [1][2][3][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], some sets must be defined to fully characterize elements of fractional calculus. It is worth mentioning that characterizing elements of fractional calculus through sets is the main idea behind of the methodology known as fractional calculus of sets [24,25].…”

Section: Sets Of Fractional Operatorsmentioning

confidence: 99%

Sets of Fractional Operators and Some of Their Applications

Torres-Hernandez¹,

Brambila-Paz²,

Ramirez-Melendez³

2023

Operator Theory - Recent Advances, New Perspectives and Applications

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This chapter presents one way to define Abelian groups of fractional operators isomorphic to the group of integers under addition through a family of sets of fractional operators and a modified Hadamard product, as well as one way to define finite Abelian groups of fractional operators through sets of positive residual classes less than a prime number. Furthermore, it is presented one way to define sets of fractional operators which allow generalizing the Taylor series expansion of a vector-valued function in multi-index notation, as well as one way to define a family of fractional fixed-point methods and determine their order of convergence analytically through sets.

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Section: Sets Of Fractional Operatorsmentioning

confidence: 99%

Sets of Fractional Operators and Some of Their Applications

Torres-Hernandez¹,

Brambila-Paz²,

Ramirez-Melendez³

2023

Operator Theory - Recent Advances, New Perspectives and Applications

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show abstract

“…The primary problem with fractional operators and their generalized counterparts is accurately defining them in the appropriate function space. In the article [ 58 – 60 ], the authors have discussed the generalized fractional derivative. Using some specific function in the introduced general fractional derivative, we can get the standard Caputo fractional derivative, Hadamard, Katugampola, and exponential-type fractional derivatives.…”

Section: Future Challengesmentioning

confidence: 99%

Fractional dynamical probes in COVID-19 model with control interventions: a comparative assessment of eight most affected countries

Pitchaimani

Devi

2022

Eur. Phys. J. Plus

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The ultimate aim of the article is to predict COVID-19 virus inter-cellular behavioral dynamics using an infection model with a quarantine compartment. Internal viral dynamics and stability attributes are thoroughly investigated around stable equilibrium states to probe possible ways in reducing rapid spread by incorporating fractional-order components into epidemic systems. Furthermore, a fractional optimal problem was built and studied with three control measures to restrict the widespread of COVID-19 infections and exhibit perfect protection. It is found that by following of control strategies can eradicate the infectives. Furthermore, the time frame of sixteen months has been divided into four short periods to grasp the pandemic, as the pandemic’s parameters change over time. Finally, using real data, we estimated the parameters of the model system and the expression of the basic reproduction number for the most affected countries, China, USA, UK, Italy, France, Germany, Spain, and Iran.

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“…(17) used in studying reactions under anomalous diffusion of subdiffusive type, can clearly be identified as a tempered fractional derivative, while similar operators are also seen in Cartea et al 37, Section III.A and Henry et al 38, Section III.B in the analysis of continuous time random walks. On the other hand, the operators of fractional calculus with respect to an arbitrary monotonic function have also been used very recently 39,40 in studying continuous time random walks. Thus, it is suggested that combining both ideas, tempered fractional calculus and

\Psi

‐fractional calculus, may be a useful endeavour.…”

Section: Introductionmentioning

confidence: 99%

On tempered fractional calculus with respect to functions and the associated fractional differential equations

Mali

Kucche

Fernandez

et al. 2022

Math Methods in App Sciences

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The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the Ψ-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the Ψ-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann-Liouville-type and Caputo-type cases of these operators. Furthermore, we study some non-linear fractional differential equations involving Ψ-tempered derivatives, proving existence-uniqueness theorems by using the Banach contraction principle and proving stability results by using Grönwall type inequalities.

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A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

Cited by 48 publications

References 13 publications

Sets of Fractional Operators and Some of Their Applications

Sets of Fractional Operators and Some of Their Applications

Fractional dynamical probes in COVID-19 model with control interventions: a comparative assessment of eight most affected countries

On tempered fractional calculus with respect to functions and the associated fractional differential equations

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