Related papers: Nonlinear PDEs risen when solving some optimizatio…
The concept of the path-dependent partial differential equation (PPDE) was first introduced in the context of path-dependent derivatives in financial markets. Its semilinear form was later identified as a non-Markovian backward stochastic…
Two types of second-order in time partial differential equations (PDEs), namely semilinear wave equations and semilinear beam equations are considered. To solve these equations with exponential integrators, we present an approach to compute…
The paper describes a number of simple but quite effective methods for constructing exact solutions of PDEs, that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
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.
Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general…
We investigate partial symmetry of solutions to semi-linear and quasi-linear elliptic problems with convex nonlinearities, in domains that are either axially symmetric or radially symmetric.
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.…
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling…
We present a novel method for using Neural Networks (NNs) for finding solutions to a class of Partial Differential Equations (PDEs). Our method builds on recent advances in Neural Radiance Field research (NeRFs) and allows for a NN to…
The problem of finding roots or solutions of a nonlinear partial differential equation may be formulated as the problem of minimizing a sum of squared residuals. One then defines an evolution equation so that in the asymptotic limit a…
We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any…
In this paper, we study nonlinear differential equations arising from Eulerian polynomials and their applications. From our study of nonlinear differential equations, we derive some new and explicit identities involving Eulerian and…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
This paper is concerned with the problem of counting solutions of stationary nonlinear Partial Differential Equations (PDEs) when the PDE is known to admit more than one solution. We suggest tackling the problem via a sampling-based…
We develop methods for the solution of inhomogeneous Robin type boundary value problems (BVPs) that arise for certain linear parabolic Partial Differential Equations (PDEs) on a half line, as well as a second order generalisation. We are…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
We present a short review of the evolution of the methodology of the Method of simplest equation for obtaining exact particular solutions of nonlinear partial differential equations (NPDEs) and the recent extension of a version of this…
Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we…
Partial differential equations (PDEs) serve as the cornerstone of mathematical physics. In recent years, Physics-Informed Neural Networks (PINNs) have significantly reduced the dependence on large datasets by embedding physical laws…