Related papers: The Fourier method for the linearized Davey-Stewar…
A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete…
A new method to solve the Dirac equation on a 3D lattice is proposed, in which the variational collapse problem is avoided by the inverse Hamiltonian method and the fermion doubling problem is avoided by performing spatial derivatives in…
The classical solvability of the initial-boundary problem for the Davey-Stewartson-II type system of equations is proved.
We describe the Ozawa solution to the Davey--Stewartson II equation from the point of view of surface theory by presenting a soliton deformation of surfaces which is ruled by the Ozawa solution. The Ozawa solution blows up at certain moment…
We consider a noncommutative version of the Davey-Stewartson equations and derive two families of quasideterminant solution via Darboux and binary Darboux transformations. These solutions can be verified by direct substitution. We then…
We implement the Numerical Unified Transform Method to solve the Nonlinear Schr\"odinger equation on the half-line. For so-called linearizable boundary conditions, the method solves the half-line problems with comparable complexity as the…
We present a novel method to estimate the stability of the Marchenko equation for finite data-sets. We show that we can derive a recursion relationship for the Fourier expansion coefficients of the kernel which is solved by the Marchenko…
In this work, we describe, analyze, and implement a pseudospectral quadrature method for a global computer modeling of the incompressible surface Navier-Stokes equations on the rotating unit sphere. Our spectrally accurate numerical error…
The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed…
In this paper, we propose a linearized Fourier pseudo-spectral method, which preserves the total mass and energy conservation laws, for the damped nonlinear Schr\"{o}dinger equation in three dimensions. With the aid of the semi-norm…
This paper presents a solution to an initial value problem for the 1-dimensional wave equation on time scales through the application of a Fourier transform and its inverse via contour integrals. The time scale of the spatial dimension is…
An efficient high precision hybrid numerical approach for integrable Davey-Stewartson (DS) I equations for trivial boundary conditions at infinity is presented for Schwartz class initial data. The code is used for a detailed numerical study…
In general, adding a stochastic perturbation to a differential equation possessing an invariant manifold destroys the invariance as far as the It\^o formalism is used. In this article, we propose an invariantization method for perturbations…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the fourth paper, the application of the Fourier series multiscale method to the…
In this paper we study the existence of a unique solution for linear stochastic differential equations driven by a L\'evy process, where the initial condition and the coefficients are random and not necessarily adapted to the underlying…
Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one…
We study the initial-value problem for the nonlinear Schr\"odinger equation. Application of the inverse scattering transform method involves solving direct and inverse scattering problems for the Zakharov-Shabat system with complex…
In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of…
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the third paper, the analytical analysis of multiscale phenomena inherent in the…
We present the first numerical approach to D-bar problems having spectral convergence for real analytic rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation which is…