A couple of fractional powers of Hankel-type integral transformations and pseudo-differential operators

SeMA Journal ◽  
2016 ◽  
Vol 74 (2) ◽  
pp. 181-211 ◽  
Author(s):  
Akhilesh Prasad ◽  
P. K. Maurya
Keyword(s):  
Differential Operators ◽  
Fractional Powers ◽  
Integral Transformations ◽  
Pseudo Differential Operators ◽  
Type Integral

DERIVATIVES WITH TWISTS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 887-895
Author(s):  
ARTHUR JAFFE
Keyword(s):  
Differential Operators ◽  
Fractional Powers ◽  
Pseudo Differential Operators ◽  
Twisting Angle

We study derivatives on an interval of length ℓ (or the associated circle of the same length), and certain pseudo-differential operators that arise as their fractional powers. We compare different translations across the interval (around the circle) that are characterized by a twisting angle. These results have application in

scholarly journals A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 65
Author(s):  
Benjamin Akers ◽  
Tony Liu ◽  
Jonah Reeger
Keyword(s):  
Radial Basis Function ◽  
Hilbert Transform ◽  
Basis Function ◽  
Differential Operators ◽  
Convergence Rates ◽  
Traveling Wave Solutions ◽  
Algebraic Decay ◽  
Pseudo Differential Operators ◽  
Radial Basis ◽  
The Hilbert Transform

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on R, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

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