Fractional calculus for power functions and eigenvalues of the fractional Laplacian

By Bartłlomiej Dyda. 2012

132 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

fPINNs: Fractional Physics-Informed Neural Networks (2018, background): “The analytical forms of the space-fractional and the time-fractional derivatives of the fabricated solutions are given in [8] and [12], respectively.”
Ten equivalent definitions of the fractional laplace operator (2015, background): “Explicit expressions for the fractional Laplace operator can be found in [5, 9, 11, 19, 20, 26, 38].”
Fractional superharmonic functions and the Perron method for nonlinear integro-differential equations (2016, background): “[16], and also [34, 39] for related applications, and [11] for explicit computations.”
What is the fractional Laplacian? A comparative review with new results (2018, unknown): “The Riesz fractional Laplacians of these functions are [107] f1(x) = (− Riesz)u1(x) = 2 (α 2 + 1 )2 , (75)”
Fractional Laplace Operator and Meijer G-function (2015, background): “(36) Theorems 1 and 2 readily imply the following explicit expressions, which extend those previously derived in [2, 8] (see also [15]).”
A logistic equation with nonlocal interactions (2016, background): “This follows, for instance, from 1 formulas (9) and (10) in [7].”
Getting Acquainted with the Fractional Laplacian (2017, background): “For this formula, and in fact even more general ones, see [46].”
Some observations on the Green function for the ball in the fractional Laplace framework (2015, background): “The more general case of this result (more precisely, for the function upxq “ p1 ́ |x|(2)qp` for any p ą ́1) was proved in [4] and [5].”
A new look at the fractional Poisson problem via the logarithmic Laplacian (2019, unknown): “[17,20].”
Nonlocal Diffusion and Applications (2015, background): “See [46] for a list of functions whose fractional Laplacian can be explicitly computed (unfortunately, differently from the classical cases, explicit computations in the fractional setting are available only for very few functions).”
Overdetermined problems with fractional Laplacian (2013, background): “We point out that the precise nonnegative and nontrivial function solving (∗) when Ω is a ball were calculated in [5] (see also [15]).”