相关论文: Multiplicity of Solutions for Linear Partial Diffe…

Partial differential equations (PDEs) are central to scientific modeling. Modern workflows increasingly rely on learning-based components to support model reuse, inference, and integration across large computational processes. Despite the…

The method of separation of variables can be used to solve many separable linear partial differential equations (LPDEs). Moreover, variable separation solutions usually are some trigonometric series. In the paper, base on some ideas of this…

We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to…

This paper deals with the partial solution of the energy eigenvalue problem for generalized symmetric quartic oscillators. Algebraization of the problem is achieved by expressing the Schroedinger operator in terms of the generators of a…

We define a deformed kinetic energy operator for a discrete position space with a finite number of points. The structure may be either periodic or nonperiodic with well-defined end points. It is shown that for the nonperiodic case the…

In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary…

Foundation models, such as large language models, have demonstrated success in addressing various language and image processing tasks. In this work, we introduce a multi-modal foundation model for scientific problems, named PROSE-PDE. Our…

Partial Differential Equations (PDEs) are central to science and engineering. Since solving them is computationally expensive, a lot of effort has been put into approximating their solution operator via both traditional and recently…

In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there…

Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…

In this work, we study the existence of various classes of standing waves for a nonlinear Schr\"odinger system with quadratic interaction, along with a harmonic or partially harmonic potential. We establish the existence of ground-state…

Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…

Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…

We present a set of smooth infinite energy global solutions (without spatial symmetry) to the non-integrable, nonlinear Schr\"odinger equations on $\Bbb R$. These solutions are space-time quasi-periodic with two frequencies each. Previous…

Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis…

Physical processes evolving in both time and space are often modeled using Partial Differential Equations (PDEs). Recently, it has been shown how stability analysis and control of coupled PDEs in a single spatial variable can be more…

Partial Differential Equations (PDEs) describe several problems relevant to many fields of applied sciences, and their discrete counterparts typically involve the solution of sparse linear systems. In this context, we focus on the analysis…

An differential equation for wave functions is proposed, which is equivalent to Schr\"{o}dinger's wave equation and can be used to determine energy-level gaps of quantum systems. Contrary to Schr\"{o}dinger's wave equation, this equation is…

PDEs with periodic boundary conditions are frequently used to model processes in large spatial environments, assuming solutions to extend periodically beyond some bounded interval. However, solutions to these PDEs often do not converge to a…

Elliptic partial differential equations (PDEs) arise in many areas of computational sciences such as computational fluid dynamics, biophysics, engineering, geophysics and more. They are difficult to solve due to their global nature and…