Related papers: Solving PDEs with Incomplete Information
Measurements on a subset of the boundary are common in electrical impedance tomography, especially any electrode model can be interpreted as a partial boundary problem. The information obtained is different to full-boundary measurements as…
In the present paper we investigate the inverse problem of identifying simultaneously the diffusion matrix, source term and boundary condition as well as the state in the Neumann boundary value problem for an elliptic partial differential…
We prove an existence and uniqueness result for Neumann boundary problem of a parabolic partial differential equation (PDE for short) with a singular nonlinear divergence term which can only be understood in a weak sense. A probabilistic…
We provide an a priori analysis of collocation methods for solving elliptic boundary value problems. They begin with information in the form of point values of the data and utilize only this information to numerically approximate the…
A new method is introduced for studying boundary value problems for a class of linear PDEs with {\it variable} coefficients. This method is based on ideas recently introduced by the author for the study of boundary value problems for PDEs…
We consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson-Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of…
In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
We consider a specific type of nonlinear partial differential equations (PDE) that appear in mathematical finance as the result of solving some optimization problems. We review some existing in the literature examples of such problems, and…
An important problem that arises in many engineering applications is the boundary value problem for ordinary differential equations. There have been many computational methods proposed for dealing with this problem. The convergence of the…
The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main…
The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to…
The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson-Neumann partial differential equations (PDEs) with Neumann boundary conditions.…
The paper is concerned with the mathematical theory and numerical approximation of systems of partial differential equations (pde) of hyperbolic, pseudo-parabolic type. Some mathematical properties of the initial-boundary-value problem…
We prove the existence and uniqueness of weak solution of a Neumann boundary problem for an elliptic partial differential equation (PDE for short) with a singular divergence term which can only be understood in a weak sense. A probabilistic…
This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for…
For a class of partial differential algebraic equations (PDAEs) of quasi-linear type which include nonlinear terms of convection type a possibility to determine a time and spatial index is considered. As a typical example we investigate an…
In this article we introduce a simple straightforward and powerful method involving symbolic manipulation, Picard iteration, and auxiliary variables for approximating solutions of partial differential boundary value problems. The method is…
We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we…
Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately…