Related papers: The Initial Value Problem for Weakly Nonlinear PDE

Initial boundary value problem on a half-line for the Modified KdV equation is considered with the boundary conditions equal to zero at the origin and initial condition chosen arbitrary decreasing rapidly enough and this problem is plunged…

We present an approach for analyzing initial-boundary value problems which is formulated on the finite interval ($0\le x\le L$, where $L$ is a positive constant) for integrable equations whose Lax pairs involve $3\times 3$ matrices.…

This paper is concerned with the variational inequality problem (VIP) over the fixed point set of a quasi-nonexpansive operator. We propose, in particular, an algorithm which entails, at each step, projecting onto a suitably chosen…

We study initial boundary value problems for linear scalar partial differential equations with constant coefficients, with spatial derivatives of {\em arbitrary order}, posed on the domain $\{t>0, 0<x<L\}$. We first show that by analysing…

This paper focuses on proposing a deep learning initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an…

Explicit step-truncation tensor methods have recently proven successful in integrating initial value problems for high-dimensional partial differential equations (PDEs). However, the combination of non-linearity and stiffness may introduce…

The long time behavior of solutions to the defocusing modified Korteweg-de vries (MKdV) equation is established for initial conditions in some weighted Sobolev spaces. Our approach is based on the nonlinear steepest descent method of Deift…

We consider the stochastic variational inequality problem in which the map is expectation-valued in a component-wise sense. Much of the available convergence theory and rate statements for stochastic approximation schemes are limited to…

Conventional solvers are often computationally expensive for constrained optimization, particularly in large-scale and time-critical problems. While this leads to a growing interest in using neural networks (NNs) as fast optimal solution…

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. We consider the case in which the elastic interface is clamped at the lateral walls of a closed trough and thus encloses…

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general…

In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional…

We consider a characteristic initial value problem for a class of symmetric hyperbolic systems with initial data given on two smooth null intersecting characteristic surfaces. We prove existence of solutions on a future neighborhood of the…

In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original…

In this paper, we introduce a stochastic projected subgradient method for weakly convex (i.e., uniformly prox-regular) nonsmooth, nonconvex functions---a wide class of functions which includes the additive and convex composite classes. At a…

A M\"untz spectral collocation method is implemented for solving weakly singular Volterra integro-differential equations (VDIEs) with proportional delays. After constructing the numerical scheme to seek an approximate solution, we derive…

The ability to express a learning task in terms of a primal and a dual optimization problem lies at the core of a plethora of machine learning methods. For example, Support Vector Machine (SVM), Least-Squares Support Vector Machine…

A new method for the solution of initial-boundary value problems for \textit{linear} and \textit{integrable nonlinear} evolution PDEs in one spatial dimension was introduced by one of the authors in 1997 \cite{F1997}. This approach was…

We study the problem of estimating the coefficients in linear ordinary differential equations (ODE's) with a diverging number of variables when the solutions are observed with noise. The solution trajectories are first smoothed with local…

A new numerical method for solving a scalar ordinary differential equation with a given initial condition is introduced. The method is using a numerical integration procedure for an equivalent integral equation and is called in this paper…