Weakly absolutely continuous functions without weak, but fractional weak derivatives

Salem, Hussein A. H.

doi:10.1007/s11868-019-00274-6

Cited by 5 publications

(1 citation statement)

References 23 publications

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“…Remark For this definition, requiring

f \in {scriptC}^{n - 1} false(false[a, b false] false)

would actually be too strong; for example, for

α \in false[1,2 false)

, the function would only need to be absolutely continuous, that is,

f \in scriptA scriptC false(false[a, b false] false)

; for this, compare also Salem [23]. However, for the proof of the mass conservation, we require differentiability, so that we will keep this condition.…”

Section: Methodsmentioning

confidence: 99%

A numerical method for space‐fractional diffusion models with mass‐conserving boundary conditions

Schäfer

Götz

2023

Math Methods in App Sciences

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The study of the spread of epidemics has gained significant attention in recent years, due to ongoing and recurring outbreaks of diseases such as COVID‐19, dengue, Ebola, and West Nile virus. In particular, modeling the spatial spread of these epidemics is crucial. This article explores the use of fractional diffusion as a means of describing non‐local infection spread. The Grünwald–Letnikov formulation of fractional diffusion is presented, along with several mass‐conserving boundary conditions, that is, we aim to design the boundary conditions in a mass‐conserving way, by not allowing gain or loss of the total population. The stationary points of the model for both sticky and reflecting boundary conditions are discussed, with numerical examples provided to illustrate the results. It is shown that reflecting boundary conditions are more reasonable, as the stationary point for sticky boundary conditions is infinite at the boundaries, while reflecting boundary conditions only have the trivial stationary point, given sufficiently fine discretization. The numerical results were applied to an SI model with fractional diffusion, highlighting the dependence of the system on the value of the fractional derivative. Results indicate that as the order of the derivative increases, the diffusivity also increases, accompanied by a slight increase in the average number of infected individuals. These models have the potential to provide valuable insights into the dynamics of disease spread and aid in the development of effective control strategies.

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“…Remark For this definition, requiring

f \in {scriptC}^{n - 1} false(false[a, b false] false)

would actually be too strong; for example, for

α \in false[1,2 false)

, the function would only need to be absolutely continuous, that is,

f \in scriptA scriptC false(false[a, b false] false)

; for this, compare also Salem [23]. However, for the proof of the mass conservation, we require differentiability, so that we will keep this condition.…”

Section: Methodsmentioning

confidence: 99%

A numerical method for space‐fractional diffusion models with mass‐conserving boundary conditions

Schäfer

Götz

2023

Math Methods in App Sciences

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On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems

Cichoń

Salem

2020

J. Pseudo-Differ. Oper. Appl.

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Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems

2023

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In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals.We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.

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Weakly absolutely continuous functions without weak, but fractional weak derivatives

Cited by 5 publications

References 23 publications

A numerical method for space‐fractional diffusion models with mass‐conserving boundary conditions

A numerical method for space‐fractional diffusion models with mass‐conserving boundary conditions

On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems

Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems

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