Related papers: The initial-boundary value problem for Schr\"oding…
We consider the initial boundary value problem (IBVP) for a non-local scalar conservation laws in one space dimension. The non-local operator in the flux function is not a mere convolution product, but it is assumed to be aware of…
This article deals with a special case of the Sturm-Liouville boundary value problem (BVP), an eigenvalue problem characterized by the Sturm-Liouville differential operator with unknown spectra and the associated eigenfunctions. By…
This work studies the initial-boundary value problem of the two-dimensional nonlinear Schr\"odinger equation on the half-plane with initial data in Sobolev spaces and Neumann or Robin boundary data in appropriate Bourgain spaces. It…
We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schr\"odinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for…
Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix…
The initial-dynamic boundary value problem (idbvp) for the complex Ginzburg-Landau equation (CGLE) on bounded domains of $\mathbb{R}^N$ is studied by converting the given mathematical model into a Wentzell initial-boundary value problem…
In this paper, we study a class of initial-boundary value problems for the Korteweg-de Vries equation posed on a bounded domain $(0,L)$. We show that the initial-boundary value problem is locally well-posed in the classical Sobolev space…
We develop methods for the solution of inhomogeneous Robin type boundary value problems (BVPs) that arise for certain linear parabolic Partial Differential Equations (PDEs) on a half line, as well as a second order generalisation. We are…
We derive explicit solution representations for linear, dissipative, second-order Initial-Boundary Value Problems (IBVPs) with coefficients that are spatially varying, with linear, constant-coefficient, two-point boundary conditions. We…
In this paper, we study the initial-boundary value problem of the Navier-Stokes system in the half space. We prove the unique solvability of the weak solution on some short time interval (0, T) with the velocity in $C^{\alpha, \frac12…
In this paper, the initial boundary value problem of the Korteweg-de Vries Burger equation on the negative half-plane is analyzed. Initially, the well-posedness on $H^s(\R^-)$ for $s\geq 0$ of the IBVP is established to concentrate on the…
A first-order ordinary differential equation, solved with respect to derivative, is considered. It's right-hand side is defined and continuous on the set, consisting of a connected open subset of a two-dimensional Euclidean space and a part…
This work studies the initial-boundary value problem for both the linear Schr\"odinger equation and the cubic nonlinear Schr\"odinger equation on the half-space in higher dimensions ($n\ge 2$). First, the forced linear problem is solved on…
We study the Schwartz class of initial-boundary value (IBV) problems for the integrable Fokas-Lenells equation on the half-line via the Deift-Zhou's nonlinear descent method analysis of the corresponding Riemann-Hilbert problem such that…
The purpose of this paper is to study well-posedness of the initial value problem (IVP) for the inhomogeneous nonlinear Schr\"odinger equation (INLS) $$ i u_t +\Delta u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$,…
In this work we study the initial boundary value problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities, that appears in nonlinear optics}, on the half-line. We obtain local well-posedness for data {in…
This paper investigates a boundary-value problem for the Korteweg-de Vries (KdV) equation on a star-graph structure. We develop a unified framework introducing the notion of $s$-compatibility, which generalizes classical compatibility…
We consider the inhomogeneous Mixed-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions…
In this work, we present a numerical method for the initial-boundary value problem (IBVP) of first-order hyperbolic systems with source terms. The scheme directly solves the relaxation system using a relatively coarse mesh and captures the…
We consider the initial boundary value problem for the focusing nonlinear Schr\"odinger equation in the quarter plane $x>0,t>0$ in the case of decaying initial data (for $t=0$, as $x\to +\infty$) and the Robin boundary condition at $x=0$.…