Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions

Li, Yulong

doi:10.1216/jie.2021.33.327

J. Integral Equations Applications

2021

DOI: 10.1216/jie.2021.33.327

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Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions

Yulong Li¹

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Cited by 4 publications

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Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

2021

Rend. Circ. Mat. Palermo, II. Ser

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Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

2021

Rend. Circ. Mat. Palermo, II. Ser

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Analysis of a class of completely non-local elliptic diffusion operators

Li,

Çelik,

Telyakovskiy

2024

Fract Calc Appl Anal

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This work explores the possibility of developing the analog of some classic results from elliptic PDEs for a class of fractional ODEs involving the composition of both left- and right-sided Riemann-Liouville (R-L) fractional derivatives, $${D^\alpha _{a+}}{D^\beta _{b-}}$$ D a + α D b - β , $$1<\alpha +\beta <2$$ 1 < α + β < 2 . Compared to one-sided non-local R-L derivatives, these composite operators are completely non-local, which means that the evaluation of $${D^\alpha _{a+}}{D^\beta _{b-}}u(x)$$ D a + α D b - β u ( x ) at a point x will have to retrieve the information not only to the left of x all the way to the left boundary but also to the right up to the right boundary, simultaneously. Therefore, only limited tools can be applied to such a situation, which is the most challenging part of the work. To overcome this, we do the analysis from a non-traditional perspective and eventually establish elliptic-type results, including Hopf’s Lemma and maximum principles. As $$\alpha \rightarrow 1^-$$ α → 1 - or $$\alpha ,\beta \rightarrow 1^-$$ α , β → 1 - , those operators reduce to the one-sided fractional diffusion operator and the classic diffusion operator, respectively. For these reasons, we still refer to them as “elliptic diffusion operators", however, without any physical interpretation.

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On the Dirichlet BVP of fractional diffusion advection reaction equation in bounded interval: Structure of solution, integral equation and approximation

Ginting

2023

Journal of Computational and Applied Mathematics

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Raising the regularity of generalized Abel equations in fractional Sobolev spaces with homogeneous boundary conditions

Cited by 4 publications

References 12 publications

Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

Integral representation bound of the true solution to the BVP of double-sided fractional diffusion advection reaction equation

Analysis of a class of completely non-local elliptic diffusion operators

On the Dirichlet BVP of fractional diffusion advection reaction equation in bounded interval: Structure of solution, integral equation and approximation

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