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Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE.…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…
In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the…
We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives,…
In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are…
In this paper we study the existence of solutions to an isotropic differential inclusion.
High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE…
Solutions of nonlinear multi-component Euler-Monge partial differential equations are constructed in n spatial dimensions by dimension-doubling, a method that completely linearizes the problem. Nonlocal structures are an essential feature…
In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) scheme to adapt them to systems of ODEs. This leads to exact schemes in the linear case, and also improve the accuracy in the nonlinear…
We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an…
In this paper, we study non-linear differential equations associated with Legendre polynomials and their applications. From our study of non- linear differential equations, we derive some new and explicit identities for Legendre…
Modelling real world systems frequently requires the solution of systems of nonlinear equations. A number of approaches have been suggested and developed for this computational problem. However, it is also possible to attempt solutions…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the…
Our purpose is to obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First we derive uniform gradient estimates for a certain semi-linear PDEs based on the coupling method introduced in Wang…
Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links…
We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental…
We develop a new integration technique allowing one to construct a rich manifold of particular solutions to multidimensional generalizations of classical $C$- and $S$-integrable Partial Differential Equations (PDEs). Generalizations of…
This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting…