On an initial-boundary value problem for the nonlinear Schrödinger equation

The Initial-boundary Value Problem for the 1d Nonlinear Schrödinger Equation on the Half-line

We prove, by adapting the method of Colliander-Kenig [9], local wellposedness of the initial-boundary value problem for the one-dimensional nonlinear Schrödinger equation i∂tu+∂ 2 x u+λu|u|α−1 = 0 on the half-line under low boundary regularity assumptions.

An analytic solution for a non-local initial-boundary value problem including a partial differential equation with variable coefficients

‎This paper considers a non-local initial-boundary value problem containing a first order partial differential equation with variable coefficients‎. ‎At first‎, ‎the non-self-adjoint spectral problem is derived‎. ‎Then its adjoint problem is calculated‎. ‎After that‎, ‎for the adjoint problem the associated eigenvalues and the subsequent eigenfunctions are determined‎. ‎Finally the convergence ...

Existence of positive solutions for a boundary value problem of a nonlinear fractional differential equation

This paper presents conditions for the existence and multiplicity of positive solutions for a boundary value problem of a nonlinear fractional differential equation. We show that it has at least one or two positive solutions. The main tool is Krasnosel'skii fixed point theorem on cone and fixed point index theory.

The Initial Boundary Value Problem for Einstein’s Vacuum Field Equation

We study the initial boundary value problem for Einstein’s vacuum field equation. We prescribe initial data on an orientable, compact, 3-dimensional manifold S with boundary 6 6= ∅ and boundary conditions on the manifold T = R0 × 6. We assume the boundaries 6 and {0} × 6 of S and T to be identified in the natural way. Furthermore, we prescribe certain gauge source functions which determine the ...

Triple Positive Solutions for Boundary Value Problem of a Nonlinear Fractional Differential Equation