Helmholtz decomposition with a scalar Poisson equation in elastic anisotropic media

P-and S-wave separation plays an important role in elastic reverse-time migration.It can reduce the artifacts caused by crosstalk between different modes and improve image quality.In addition,P-and Swave separation can also be used to better understand and distinguish wave types in complex media.At present,the methods for separating wave modes in anisotropic media mainly include spatial nonstationary filtering,low-rank approximation,and vector Poisson equation.Most of these methods require multiple Fourier transforms or the calculation of large matrices,which require high computational costs for problems with large scale.In this paper,an efficient method is proposed to separate the wave mode for anisotropic media by using a scalar anisotropic Poisson operator in the spatial domain.For 2D problems,the computational complexity required by this method is 1/2 of the methods based on solving a vector Poisson equation.Therefore,compared with existing methods based on pseudoHelmholtz decomposition operators,this method can significantly reduce the computational cost.Numerical examples also show that the P and S waves decomposed by this method not only have the correct amplitude and phase relative to the input wavefield but also can reduce the computational complexity significantly.

STUDY OF EXPLICIT ANALYTIC SOLUTIONS FOR THE NONLINEAR COUPLED SCALAR FIELD EQUATIONS

By using two different transformations, several types of exact analytic solutions for a class of nonlinear coupled scalar field equation are obtained, which contain soliton solutions, singular solitary wave solutions and triangle function solutions. These results can be applied to other nonlinear equations. In addition, parts of conclusions in some references are corrected.

Higgs-like(pseudo)scalars in AdS_(4),marginal and irrelevant deformations in CFT_(3),and instantons on S^(3)

Employing a 4-form ansatz of 11-dimensional supergravity over a non-dynamical AdS_(4)×S^(7)/Z_(k)background and setting the internal space as an S1 Hopf fibration on CP3,we obtain a consistent truncation.The(pseudo)scalars,in the resulting scalar equations in Euclidean AdS_(4)space,may be considered to arise from(anti)M-branes wrapping around the internal directions in the(Wick-rotated)skew-whiffed M2-brane background(as the resulting theory is for anti-M2-branes),thus realizing the modes after swapping the three fundamental representations 8_(s),8_(c),and 8_(v) of SO(8).Taking the backreaction on the external and internal spaces,we obtain the massless and massive modes,corresponding to exactly marginal and marginally irrelevant deformations on the boundary CFT3,respectively.Subsequently,we obtain a closed solution for the bulk equation and compute its correction with respect to the background action.Next,considering the Higgs-like(breathing)mode m^(2)=18,having all supersymmetries as well as parity and scale-invariance broken,solving the associated bulk equation with mathematical methods,specifically the Adomian decomposition method,and analyzing the behavior near the boundary of the solutions,we realize the boundary duals in the SU(4)×U(1)-singlet sectors of the ABJM model.Then,introducing the new dual deformationΔ_(+)=3,6 operators made of bi-fundamental scalars,fermions,and U(1)gauge fields,we obtain the SO(4)-invariant solutions as small instantons on a three-sphere with the radius at infinity,which correspond to collapsing bulk bubbles leading to big-crunch singularities.

An artificial neural network based deep collocation method for the solution of transient linear and nonlinear partial differential equations

A combined deep machine learning(DML)and collocation based approach to solve the partial differential equations using artificial neural networks is proposed.The developed method is applied to solve problems governed by the Sine–Gordon equation(SGE),the scalar wave equation and elasto-dynamics.Two methods are studied:one is a space-time formulation and the other is a semi-discrete method based on an implicit Runge–Kutta(RK)time integration.The methodology is implemented using the Tensorflow framework and it is tested on several numerical examples.Based on the results,the relative normalized error was observed to be less than 5%in all cases.

Analysis of thickness-shear and thickness-twist modes of AT-cut quartz acoustic wave resonator and filter

The thickness-shear （TS） and thickness-twist （TT） vibrations of partially electroded AT-cut quartz plates for acoustic wave resonator and filter applications are theoretically studied. The plates have structural variations in one of the two in-plane directions of the plates only. The scalar differential equations derived by Tiersten and Smythe for electroded and unelectroded AT-cut quartz plates are used, resulting in free vibration resonant frequencies and mode shapes for both fundamental and overtone fam- ilies of modes. The trapped modes with vibrations, mainly confined in the electroded areas, are found to exist in both the resonator and the filter structures. The numerical results for the trapped modes are presented for different aspect ratios of electrodes and material properties, providing a reference to the design and optimization of quartz acous- tic wave resonators and filters.

THE STABILITY OF SCALAR AUTONOMOUS DIFFERENCE EQUATIONS

The general criteria of stability for equihbrium points of scalar autonomous difference equations are given, wich cover all the cases as long as the right-hand function has continuous derivatives up to the desired order. Thus, the stability problems of scalar autonomous difference equations are thoroughly solved. The proofs of the obtained criteria are mathematically rigorous and complete. Also, several exam pies are given to illustrate the obtained results.

Asymptotic Integration for Certain Scalar Functional Differential Equations

In this paper we present a type of asymptotic integration for a certain scalar time\|varying functional differential equations with L\+p\|integrable coefficients. The main theorem we obtain generalizes the result of Haddock & Sacker′s Conjecture with n=1.

CONVERGENCE ANALYSIS FOR SPECTRAL APPROXIMATION TO A SCALAR TRANSPORT EQUATION WITH A RANDOM WAVE SPEED

This paper is concerned with the initial-boundary value problems of scalar transport equations with uncertain transport velocities. It was demonstrated in our earlier works that regularity of the exact solutions in the random spaces （or the parametric spaces） can be determined by the given data. In turn, these regularity results are crucial to convergence analysis for high order numerical methods. In this work, we will prove the spectral conver- gence of the stochastic Galerkin and collocation methods under some regularity results or assumptions. As our primary goal is to investigate the errors introduced by discretizations in the random space, the errors for solving the corresponding deterministic problems will be neglected.

One-dimensional monotone nonautonomous dynamical systems

This work is devoted to the study of the dynamics of one-dimensional monotone nonautonomous(cocycle) dynamical systems. A description of the structures of their invariant sets, omega limit sets,Bohr/Levitan almost periodic and almost automorphic motions, global attractors, and pinched and minimalsets is given. An application of our general results is given to scalar differential and difference equations.

An Approximation for Rapid Simulation of Thin-Film Bulk Acoustic Resonators(FBARs)with Sandwich-Layered Structure

Thin-film bulk acoustic resonators(FBARs)operating with essentially thickness-extensional mode have been widely used in communication fields.In this paper,we provide a convenient means for analyzing FBARs with sandwich-layered structure by appropriately neglecting the high-order terms from 3D elasticity equations.First,for straight-crested waves,an approximate method is proposed,which can accurately describe the dispersion relation near the operating frequency range of an FBAR.Using the approximation,the optimum lateral size of a 2D model of frame-like FBAR is obtained,and the results are in good agreement with that obtained by commercial FEM software COMSOL.The approximation is further extended to variablecrested waves in order to analyze the 3D plate models for real devices.The mode shapes of 3D FBARs with and without frame-like structures are obtained.The results show that the approximation presented in this paper is of sufficient accuracy and can be used as an efficient tool for the analysis and design of FBARs.