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Existence of solutions for a higher order non-local equation appearing in crack dynamics
Hydraulic fractures Higher order equation Non-local equation Thin film equation Non-negative solutions periodic Hilbert transform
2015/10/15
In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modeling of hydraulic fractures. The equation is similar to th...
Existence and regularity of extremal solutions for a mean-curvature equation
Mean curvature minimal solution semi-stable solution extremal solution regularity
2015/10/15
We study a class of mean curvature equations −Mu = H +λup where M denotes the mean curvature operator and for p ≥ 1. We show that there exists an extremal parameter λ∗ such that this equat...
Growth of Sobolev norms in the cubic defocusing nonlinear Schrodinger equation
Sobolev norms cubic defocusing nonlinear Schrodinger equation
2015/9/25
We consider the cubic defocusing nonlinear Schr¨odinger equation in the two dimensional torus. Fix s > 1. Colliander, Keel, Staffilani, Tao and Takaoka proved in [CKS+10] the existence of solutions wi...
A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains
oundary integral equations Nystrom discretization Singular integrals Nearly singular integrals Fast solvers Laplace equation Stokes equation Navier equation Fast multipole method Fast Fourier transform
2015/7/14
We present a high-order boundary integral equation solver for 3D elliptic boundary value problems on domains with smooth boundaries. We use Nystro ¨m’s method for discretization, and combine it with s...
FAST MULTISCALE GAUSSIAN WAVEPACKET TRANSFORMS AND MULTISCALE GAUSSIAN BEAMS FOR THE WAVE EQUATION
ast multiscale Gaussian wavepacket transforms multiscale Gaussian beams wave equations
2015/7/14
We introduce a new multiscale Gaussian beam method for the numerical solution of the wave equation with smooth variable coefficients. The first computational question addressed in this paper is how to...
Sweeping Preconditioner for the Helmholtz Equation:Hierarchical Matrix Representation
Sweeping Preconditioner Helmholtz Equation Hierarchical Matrix Representation
2015/7/14
The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable-coefficient Helmholtz equation including veryhigh-frequency problems. The first cent...
Fast Gaussian wavepacket transforms and Gaussian beams for the Schrödinger equation
Fast Gaussian wavepacket transforms Gaussian beams Schrodinger equation
2015/7/14
This paper introduces a wavepacket-transform-based Gaussian beam method for solving the Schrödinger equation. We focus on addressing two computational issues of the Gaussian beam method: how to g...
Second kind integral equations for the first kind Dirichlet problem of the biharmonic equation in three dimensions
Second kind integral equation Dirichlet problem Biharmonic equation
2015/7/14
A Fredholm second kind integral equation (SKIE) formulation is constructed for the Dirichlet problem of the biharmonic equation in three dimensions. A fast numerical algorithm is developed based on th...
A Convergent Multiscale Gaussian-Beam Parametrix for the Wave Equation
Multiscale Gaussian beams Multiscale Gaussian wave packets Phase space transform Wave equations
2015/7/14
The Gaussian beam method is an asymptotic method for wave equations with highly oscillatory data. In a recent published paper by two of the authors, a multiscale Gaussian beam method was first propose...
Additive Sweeping Preconditioner for the Helmholtz Equation
Helmholtz equation perfectly matched layers preconditioners high frequency waves
2015/7/14
We introduce a new additive sweeping preconditioner for the Helmholtz equation based on the perfect matched layer (PML). This method divides the domain of interest into thin layers and proposes a new ...
Recursive Sweeping Preconditioner for the 3D Helmholtz Equation
Helmholtz equation perfectly matched layers preconditioners high frequency waves
2015/7/14
This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving...
Exponential decay for the fragmentation or cell-division equation
Exponential decay fragmentation cell-division equation
2015/7/14
We consider a classical integro-differential equation that arises in various applications as a model for cell-division or fragmentation. In biology, it describes the evolution of the density of cells ...
The quantum scattering limit for a regularized Wigner equation
quantum scattering limit regularized Wigner equation
2015/7/14
We consider a regularized Wigner equation with an oscillatory kernel, the regularization acts in the space variable to damp high frequencies. The oscillatory kernel is directly derived from the Schrod...
Time splitting for the Liouville equation in a random medium
Time splitting Liouville equation random medium
2015/7/14
We consider the Liouville equations with highly heterogeneous Hamiltonians and their numerical solution by a time splitting algorithm. Such equations model the density of particles evolving according ...
Limit of fluctuations of solutions of Wigner Equation
fluctuations of solutions Wigner Equation
2015/7/14
We consider fluctuations of the solution W ε(t, x, k) of the Wigner equation, which describes energy evolution of a solution of the Schr¨odinger equation with a random white noise in time potential. T...