Related papers: Solving PDEs with Incomplete Information
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…
We study the problem of identifying unknown processes embedded in time-dependent partial differential equation (PDE) using observational data, with an application to advection-diffusion type PDE. We first conduct theoretical analysis and…
Solving nonlinear optimal control problems is a challenging task, particularly for high-dimensional problems. We propose algorithms for model-based policy iterations to solve nonlinear optimal control problems with convergence guarantees.…
Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient…
Partial differential equations (PDEs) with Dirichlet boundary conditions defined on boundaries with simple geometry have been succesfuly treated using sigmoidal multilayer perceptrons in previous works. This article deals with the case of…
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 consider the Dirichlet boundary value problem for nonlinear triharmonic equation. Due to the reduction of the nonlinear boundary value problem to operator equation for the nonlinear term and the unknown second normal…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
We consider an abstract framework for the numerical solution of optimal control problems (OCPs) subject to partial differential equations (PDEs). Examples include not only the distributed control of elliptic PDEs such as the Poisson…
Results about existence and uniqueness of solutions of initial value problem for certain types of partial differential equations are recalled as well as iterative scheme and an error estimate for approximate solutions obtained using this…
We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing…
In a multidimensional infinite layer bounded by two hyperplanes, the Poisson equation with the polynomial right-hand side is considered. It is shown that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value…
We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are H\"{o}lder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results…
Neural networks can be trained to solve partial differential equations (PDEs) by using the PDE residual as the loss function. This strategy is called "physics-informed neural networks" (PINNs), but it currently cannot produce high-accuracy…
Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation.…
Partial differential equations (PDEs) are a fundamental tool in the modeling of many real world phenomena. In a number of such real world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are…
The very weak solution of the Poisson equation with $L^2$ boundary data is defined by the method of transposition. The finite element solution with regularized boundary data converges in the $L^2(\Omega)$-norm with order $1/2$ in convex…
This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones…