scholarly journals Asymptotically linear problems driven by fractional Laplacian operators

2015 ◽  
Vol 38 (16) ◽  
pp. 3551-3563 ◽  
Author(s):  
Alessio Fiscella ◽  
Raffaella Servadei ◽  
Enrico Valdinoci
Keyword(s):  
Fractional Laplacian ◽  
Asymptotically Linear ◽  
Linear Problems ◽  
Asymptotically Linear Problems ◽  
Laplacian Operators

Asymptotically Linear Problems and Antimaximum Principle for the Square Root of the Laplacian

2012 ◽  
Vol 12 (4) ◽  
Author(s):  
David Arcoya ◽  
Eduardo Colorado ◽  
Tommaso Leonori
Keyword(s):  
Positive Solutions ◽  
Square Root ◽  
Asymptotically Linear ◽  
Linear Problems ◽  
Asymptotically Linear Problems

AbstractThis work deals with bifurcation of positive solutions for some asymptotically linear problems, involving the square root of the Laplacian (−Δ)with Ω ⊂ ℝ

scholarly journals Perturbed asymptotically linear problems

2012 ◽  
Vol 193 (1) ◽  
pp. 89-101 ◽  
Author(s):  
Rossella Bartolo ◽  
Anna Maria Candela ◽  
Addolorata Salvatore
Keyword(s):  
Asymptotically Linear ◽  
Linear Problems ◽  
Asymptotically Linear Problems

Decoupled Fréchet kernels based on a fractional viscoacoustic wave equation

Geophysics ◽  
2021 ◽  
pp. 1-42
Author(s):  
Guangchi Xing ◽  
Tieyuan Zhu
Keyword(s):  
Wave Equation ◽  
Fractional Laplacian ◽  
Waveform Inversion ◽  
Full Waveform Inversion ◽  
Physical Processes ◽  
Full Waveform ◽  
Adjoint State ◽  
The Finite Difference Method ◽  
Adjoint State Method ◽  
Laplacian Operators

We formulate the Fréchet kernel computation using the adjoint-state method based on a fractional viscoacoustic wave equation. We first numerically prove that both the 1/2- and the 3/2-order fractional Laplacian operators are self-adjoint. Using this property, we show that the adjoint wave propagator preserves the dispersion and compensates the amplitude, while the time-reversed adjoint wave propagator behaves identically as the forward propagator with the same dispersion and dissipation characters. Without introducing rheological mechanisms, this formulation adopts an explicit Q parameterization, which avoids the implicit Q in the conventional viscoacoustic/viscoelastic full waveform inversion ( Q-FWI). In addition, because of the decoupling of operators in the wave equation, the viscoacoustic Fréchet kernel is separated into three distinct contributions with clear physical meanings: lossless propagation, dispersion, and dissipation. We find that the lossless propagation kernel dominates the velocity kernel, while the dissipation kernel dominates the attenuation kernel over the dispersion kernel. After validating the Fréchet kernels using the finite-difference method, we conduct a numerical example to demonstrate the capability of the kernels to characterize both velocity and attenuation anomalies. The kernels of different misfit measurements are presented to investigate their different sensitivities. Our results suggest that rather than the traveltime, the amplitude and the waveform kernels are more suitable to capture attenuation anomalies. These kernels lay the foundation for the multiparameter inversion with the fractional formulation, and the decoupled nature of them promotes our understanding of the significance of different physical processes in the Q-FWI.

scholarly journals Existence results for fractional p-Laplacian problems via Morse theory

2016 ◽  
Vol 9 (2) ◽  
Author(s):  
Antonio Iannizzotto ◽  
Shibo Liu ◽  
Kanishka Perera ◽  
Marco Squassina
Keyword(s):  
Phase Transition ◽  
Game Theory ◽  
Morse Theory ◽  
Fractional Laplacian ◽  
Existence Results ◽  
Nonlocal Problems ◽  
Finite Multiplicity ◽  
Topological Methods ◽  
Multiplicity Results ◽  
Linear Problems

AbstractWe investigate a class of quasi-linear nonlocal problems, including as a particular case semi-linear problems involving the fractional Laplacian and arising in the framework of continuum mechanics, phase transition phenomena, population dynamics and game theory. Under different growth assumptions on the reaction term, we obtain various existence as well as finite multiplicity results by means of variational and topological methods and, in particular, arguments from Morse theory.

Modeling viscoacoustic wave propagation using a new spatial variable-order fractional Laplacian wave equation

Geophysics ◽  
2021 ◽  
pp. 1-74
Author(s):  
Xinru Mu ◽  
Jianping Huang ◽  
Lei Wen ◽  
Subin Zhuang
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Order ◽  
Fractional Laplacian ◽  
Spatial Variable ◽  
Variable Order ◽  
New Wave ◽  
Dissipation Term ◽  
Laplacian Operators ◽  
Dispersion Term

We propose a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for the proposed wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then the proposed viscoacoustic wave equation can be directly solved using the pseudospectral method (PSM). We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of the proposed approximate dispersion term and approximate dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results show that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q < 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we demonstrate the accuracy of HPSFDM in heterogeneous media by several numerical examples.

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