scholarly journals On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge Morales Paredes ◽  
Félix Humberto Soriano Méndez
Keyword(s):  
Fractional Derivative ◽  
Sobolev Spaces ◽  
Hilbert Transform ◽  
Well Posedness ◽  
Cauchy Problems ◽  
Anisotropic Sobolev Spaces ◽  
The Hilbert Transform

<p style='text-indent:20px;'>In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated with a family of equations of ZK-KP-type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} u_{t} = u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x}, \cr u(0) = \psi \in Z \end{cases} $\end{document} </tex-math> </disp-formula></p><p style='text-indent:20px;'>in anisotropic Sobolev spaces, where <inline-formula><tex-math id="M1">\begin{document}$ 1\le \alpha \le 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \mathscr{H} $\end{document}</tex-math></inline-formula> is the Hilbert transform and <inline-formula><tex-math id="M3">\begin{document}$ D_{x}^{\alpha} $\end{document}</tex-math></inline-formula> is the fractional derivative, both with respect to <inline-formula><tex-math id="M4">\begin{document}$ x $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Generalized multidimensional Hilbert transforms in Clifford analysis

2006 ◽  
Vol 2006 ◽  
pp. 1-19 ◽  
Author(s):  
Fred Brackx ◽  
Bram De Knock ◽  
Hennie De Schepper
Keyword(s):  
Sobolev Spaces ◽  
Hilbert Transform ◽  
Clifford Analysis ◽  
Hilbert Transforms ◽  
Singular Operators ◽  
The Hilbert Transform

Two specific generalizations of the multidimensional Hilbert transform in Clifford analysis are constructed. It is shown that though in each of these generalizations some traditional properties of the Hilbert transform are inevitably lost, new bounded singular operators emerge on Hilbert or Sobolev spaces ofL2-functions.

scholarly journals The Cauchy problem for a two-dimensional generalized Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces

2019 ◽  
Vol 18 (03) ◽  
pp. 469-522
Author(s):  
Wei Yan ◽  
Yongsheng Li ◽  
Jianhua Huang ◽  
Jinqiao Duan
Keyword(s):  
Cauchy Problem ◽  
Sobolev Spaces ◽  
Well Posedness ◽  
Two Dimensional ◽  
Anisotropic Sobolev Spaces ◽  
The Cauchy Problem ◽  
Kp Equations ◽  
Fifth Order ◽  
Well Posed

The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation [Formula: see text] is locally well-posed in the anisotropic Sobolev spaces [Formula: see text] with [Formula: see text] and [Formula: see text]. Second, we prove that the Cauchy problem is globally well-posed in [Formula: see text] with [Formula: see text] if [Formula: see text]. Finally, we show that the Cauchy problem is globally well-posed in [Formula: see text] with [Formula: see text] if [Formula: see text] Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. 79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl. 90 (2008) 338–352].

Model of multicomponent narrow-band periodical non-stationary random signal

2020 ◽  
Vol 2020 (48) ◽  
pp. 17-24
Author(s):  
I.M. Javorskyj ◽  
◽  
R.M. Yuzefovych ◽  
P.R. Kurapov ◽  
◽  
...  
Keyword(s):  
Time Series ◽  
Real Time ◽  
Hilbert Transform ◽  
Spectral Properties ◽  
Narrow Band ◽  
Analytic Signal ◽  
Random Signal ◽  
Hilbert Transformation ◽  
The Hilbert Transform

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

A Bayesian View on the Hilbert Transform and the Kramers-Kronig Transform of Electrochemical Impedance Data: Probabilistic Estimates and Quality Scores

2020 ◽  
Author(s):  
Jiapeng Liu ◽  
Ting Hei Wan ◽  
Francesco Ciucci
Keyword(s):  
Power Generation ◽  
Electrochemical Impedance Spectroscopy ◽  
Hilbert Transform ◽  
Electrochemical Impedance ◽  
The Self ◽  
Impedance Data ◽  
The Hilbert Transform ◽  
Validation Tests ◽  
Self Consistency ◽  
Mathematical Relations

<p>Electrochemical impedance spectroscopy (EIS) is one of the most widely used experimental tools in electrochemistry and has applications ranging from energy storage and power generation to medicine. Considering the broad applicability of the EIS technique, it is critical to validate the EIS data against the Hilbert transform (HT) or, equivalently, the Kramers–Kronig relations. These mathematical relations allow one to assess the self-consistency of obtained spectra. However, the use of validation tests is still uncommon. In the present article, we aim at bridging this gap by reformulating the HT under a Bayesian framework. In particular, we developed the Bayesian Hilbert transform (BHT) method that interprets the HT probabilistic. Leveraging the BHT, we proposed several scores that provide quick metrics for the evaluation of the EIS data quality.<br></p>

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.

scholarly journals The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

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