Application of the Chimera Method to Poisson’s Equation with the Homogeneous Dirichlet Boundary Condition

Small perturbations in the type of boundary conditions for an elliptic operator

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

Generic bifurcation of steady-state solutions

International audience

We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one-dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville–Bratu–Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element (FEM) methods. We show that the proposed numerical machine learning method outperforms in terms of numerical accuracy both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids; both methods (ELM and FEM) outperform the FD scheme. Furthermore, the computational times required with the proposed machine learning scheme were comparable and in particular slightly smaller than the ones required with FEM.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

The collocation solution of Poisson problems based on approximate Fekete points

We study the large time behavior of the solution of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Dirichlet or Neumann boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term.

The Construction of Solutions for Boundary Value Problems by Function Theoretic Methods

By the maximum principle the solution of the homogeneous heat equation with homogeneous Dirichlet boundary conditions is nonnegative for positive time if the initial values are nonnegative. In recent work it has been shown that this does not hold for the standard spatially discrete and fully discrete piecewise linear finite element methods. However, for the corresponding semidiscrete and Backward Euler Lumped Mass methods, nonnegativity of initial data is preserved, provided the underlying triangulation is of Delaunay type. In this paper, we study the corresponding problems where the homogeneous Dirichlet boundary conditions are replaced by Neumann and Robin boundary conditions, and show similar results, sometimes requiring more refined technical arguments.

Application of volume integrals to the solution of partial differential equations

Regularity refers to the properties of the solution, including the smoothness, symmetry, and asymptotic of the solution. It is an important part of the theoretical study of partial differential equations. It plays a key role in the existence, uniqueness, stability, and smoothness of the theoretical solutions of partial differential equations. It is an important basis for understanding the nature of partial differential equations and their corresponding physical reality. This paper studies the boundary regularity of elliptic partial differential equations, including the problem of the oblique boundary of completely nonlinear equations. It is well known that the regularity of the solution at the region boundary depends not only on the equation, but also on the geometric properties of the region boundary. This is why boundary regularity is complicated. This paper is to obtain the regularity of the solution at the region boundary under different boundary conditions.

The nonlinear Schrödinger equation in the finite line

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

Introduction. Among the theorems which deal with the functional properties of the solutions of elliptic linear partial differential equations, the most important ones are perhaps the following: (a) The solutions of equations with analytic coefficients are analytic. (b) The solutions of equations with indefinitely differentiable (i.d.) coefficients are i.d. In Part I of this paper we prove a result which connects the above mentioned theorems. Qualitatively we prove that the i.d. solutions of elliptic differential systems with i.d. coefficients have the same distance from analyticity as have the coefficients. More precisely, we define classes of i.d. functions and show, under certain assumptions, that if the coefficients belong to a certain class, then so do the solutions. In particular, we give a new proof to Theorem (a). In Part II we define another kind of classes of i.d. functions and prove, for some kinds of elliptic equations, results similar to that of Part I. All functions in this paper are real functions. I should like to express my gratitude to Dr. S. Agmon for his help and encouragement during the preparation of Part I of this paper.