scholarly journals Physics-informed neural networks method in high-dimensional integrable systems

2021 ◽  
Author(s):  
Zheng Wu Miao ◽  
Yong Chen
Keyword(s):  
Neural Networks ◽  
Integrable Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Data Driven ◽  
High Dimensional ◽  
Classical Solutions ◽  
Dimensional System ◽  
Kp Equation ◽  
Initial Boundary Value

In this paper, the physics-informed neural networks (PINNs) are applied to high-dimensional system to solve the [Formula: see text]-dimensional initial-boundary value problem with [Formula: see text] hyperplane boundaries. This method is used to solve the most classic (2+1)-dimensional integrable Kadomtsev–Petviashvili (KP) equation and (3+1)-dimensional reduced KP equation. The dynamics of (2+1)-dimensional local waves such as solitons, breathers, lump and resonance rogue are reproduced. Numerical results display that the magnitude of the error is much smaller than the wave height itself, so it is considered that the classical solutions in these integrable systems are well obtained based on the data-driven mechanism.

The evolution of travelling waves in reaction-diffusion equations with monotone decreasing diffusivity. I. Continuous diffusivity

1995 ◽  
Vol 350 (1694) ◽  
pp. 335-360 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Propagation Speed ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Classical Solutions ◽  
Initial Boundary Value

We examine the effects of a concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics. The diffusivity is taken as a continuous monotone, a decreasing function of concentration that has compact support, of the form that arises in polymerization processes. We consider piecewise-classical solutions to an initial-boundary value problem. The existence of a family of permanent form travelling wave solutions is established, and the development of the solution of the initial-boundary value problem to the travelling wave of minimum propagation speed is considered. It is shown that an interface will always form in finite time, with its initial propagation speed being unbounded. The interface represents the surface of the expanding polymer matrix.

The effects of variable diffusivity on the development of travelling waves in a class of reaction-diffusion equations

1994 ◽  
Vol 348 (1687) ◽  
pp. 229-260 ◽  
Keyword(s):  
Boundary Value Problem ◽  
Travelling Waves ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Travelling Wave ◽  
Reaction Diffusion Equations ◽  
Classical Solutions ◽  
Initial Boundary Value

In this paper we examine the effects of concentration dependent diffusivity on a reaction-diffusion process which has applications in chemical kinetics and ecology. We consider piecewise classical solutions to an initial boundary-value problem. The existence of a family of permanent form travelling wave solutions is established and the development of the solution of the initial boundary-value problem to the travelling wave of minimum propagation speed is considered. For certain types of initial data, ‘waiting time’ phenomena are encountered.

scholarly journals On global classical solutions to one-dimensional compressible Navier–Stokes/Allen–Cahn system with density-dependent viscosity and vacuum

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Menglong Su
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Large Time Behavior ◽  
Energy Estimates ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Time Behavior ◽  
One Dimensional ◽  
Global Classical Solutions ◽  
Initial Boundary Value

AbstractIn this paper, by using the energy estimates, the structure of the equations, and the properties of one dimension, we establish the global existence and uniqueness of strong and classical solutions to the initial boundary value problem of compressible Navier–Stokes/Allen–Cahn system in one-dimensional bounded domain with the viscosity depending on density. Here, we emphasize that the time does not need to be bounded and the initial vacuum is still permitted. Furthermore, we also show the large time behavior of the velocity.

scholarly journals Some Properties of Solutions for the Sixth-Order Cahn-Hilliard-Type Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Zhao Wang ◽  
Changchun Liu
Keyword(s):  
Boundary Value Problem ◽  
Finite Time ◽  
Blow Up ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sixth Order ◽  
Classical Solutions ◽  
Oil Water ◽  
Initial Boundary Value

We study the initial boundary value problem for a sixth-order Cahn-Hilliard-type equation which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. We show that the solutions might not be classical globally. In other words, in some cases, the classical solutions exist globally, while in some other cases, such solutions blow up at a finite time. We also discuss the existence of global attractor.

scholarly journals Global existence of classical solutions to the two-dimensional compressible Boussinesq equations in a square domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xucheng Huang ◽  
Zhaoyang Shang ◽  
Na Zhang
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Boussinesq Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Priori Estimates ◽  
Constant Viscosity ◽  
Initial Boundary Value

Abstract In this paper, we consider the initial boundary value problem of two-dimensional isentropic compressible Boussinesq equations with constant viscosity and thermal diffusivity in a square domain. Based on the time-independent lower-order and time-dependent higher-order a priori estimates, we prove that the classical solution exists globally in time provided the initial mass $\|\rho _{0}\|_{L^{1}}$ ∥ ρ 0 ∥ L 1 of the fluid is small. Here, we have no small requirements for the initial velocity and temperature.

Sign in / Sign up

Export Citation Format

Share Document