Related papers: Extra Dimensions and Nonlinear Equations
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
We establish existence and uniqueness of generalized solutions to the initial-boundary value problem corresponding to an Euler-Bernoulli beam model from mechanics. The governing partial differential equation is of order four and involves…
In this work some nonlinear integral equation is studied. This equation has arisen in the (super)string field theory and cosmology. In this work it is proved that some boundary problem for this equation has a solution.
We discuss a method of solving $n^{th}$ order scalar ordinary differential equations by extending the ideas based on the Prelle-Singer (PS) procedure for second order ordinary differential equations. We also introduce a novel way of…
The aim of this paper is a quantitative analysis of the solution set of a system of polynomial nonlinear differential equations, both in the ordinary and partial case. Therefore, we introduce the differential counting polynomial, a common…
Master character of the multidimensional homogeneous Euler equation is discussed. It is shown that under restrictions to the lower dimensions certain subclasses of its solutions provide us with the solutions of various hydrodynamic type…
In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -\Delta u=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial \nu}} = V(x)u+b \vert…
Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equation. We apply this principle by finding dilatations and…
In this work we obtain sufficient conditions for the existence of bounded solutions of a resonant multi-point second-order boundary value problem, with a fully differential equation. The noninvertibility of the linear part is overcome by a…
A self-consistent treatment of two and three point functions in models with trilinear interactions forces them to have opposite anomalous dimensions. We indicate how the anomalous dimension can be extracted nonperturbatively by solving and…
We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.
Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…
This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling…
It is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain, with Dirichlet boundary conditions, are unique up to multiplication by a positive constant.
We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…
We generalize the method of obtaining the fundamental linear partial differential equations such as the diffusion and Schrodinger equation, Dirac and telegrapher's equation from a simple stochastic consideration to arrive at certain…
We consider the Euler system set on a bounded convex planar domain, endowed with impermeability boundary conditions. This system is a model for the barotropic mode of the Primitive Equations on a rectangular domain. We show the existence of…
Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not…
In this paper we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and…
This paper on the whole concerns with the duality of Mayer problem for k-th order differential inclusions, where k is an arbitrary natural number. Thus, this work for constructing the dual problems to differential inclusions of any order…