Related papers: A Python Class for Higher-Dimensional Schr\"odinge…
We announce some Python classes for numerical solution of partial differential equations, or boundary value problems of ordinary differential equations. These classes are built on routines in \texttt{numpy} and \texttt{scipy.sparse.linalg}…
We present analytical results and numerical simulations for a class of nonlinear dispersive equations in two spatial dimensions. These equations are of (derivative) nonlinear Schr\"odinger type and have recently been obtained in \cite{DLS}…
We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schroedinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our…
This paper is dedicated to present an open-source program so-called \emph{AIMpy} built on Python language. \emph{AIMpy} is a solver for Schr\"{o}dinger-like differential equations using Asymptotic Iteration Method (AIM). To confirm the code…
We develop a high accuracy power series method for solving partial differential equations with emphasis on the nonlinear Schr\"odinger equations. The accuracy and computing speed can be systematically and arbitrarily increased to orders of…
We prove the existence of global analytic solutions to the nonlinear Schr\"odinger equation in one dimension for a certain type of analytic initial data in $L^2$.
In this paper, we introduce some analytical techniques to solve some classes of second order differential equations. Such classes of differential equations arise in describing some mathematical problems in Physics and Engineering.
A class of generalized Schr\"{o}dinger problems in bounded domain is studied. A complete overview of the set of solutions is provided, depending on the values assumed by parameters involved in the problem. In order to obtain the results, we…
We obtain exact solutions to the class of parabolic partial differential equations of arbitrary dimensionality and with arbitrary potentials. The solutions are presented in a compact-form: as explicit mathematical expressions consisting of…
We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the $d$-dimensional fractional equation is reformulated as a local partial differential equation in…
Recent releases of open-source research codes and solvers for numerically solving partial differential equations in Python present a great opportunity for educators to integrate these codes into the classroom in a variety of ways. The ease…
This article analyzes and develops a method to solve fractional ordinary differential equations using the Monte Carlo Method. A numerical simulation is performed for some differential equations, comparing the results with what exists in the…
We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\"odinger equations driven by additive It\^o noise. The class of nonlinearities of interest includes nonlocal…
We investigate a method to solve a class of Schr{\"o}dinger equation eigenvalue problems numerically to very high precision $P$ (from thousands to a million of decimals). The memory requirement, and the number of high precision algebraic…
In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution of the one-dimensional Schr\"odinger equation. The new methodology is based on the requirement of vanishing the…
A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati…
We present new approaches for solving constrained multicomponent nonlinear Schr\"odinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose…
This paper provides a small data global existence result for a class of quadratic derivative nonlinear Schr\"odinger systems in two space dimensions. This is an extension of the previous results by Li [Discrete Contin. Dyn. Syst., 32…
This paper presents \textsc{hankel}, a pure-python code for solving Hankel-type integrals and transforms. Such transforms are common in the physical sciences, especially appearing as the radial solution to angularly symmetric Fourier…
In this note, we generalize the nonlinearity-recovery result in [7] for classical cubic nonlinear Schr\"odinger equations to higher-order Schr\"odinger equations with a more general nonlinearity. More precisely, we consider a…