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While linear systems are well-understood, no explicit solution for general nonlinear systems exists. A classical approach to make the understanding of linear system available in the nonlinear setting is to represent a nonlinear system by a…
In this note we extend to the random, stationary ergodic setting previous results of periodic homogenization for a particular family of nonlinear nonlocal "elliptic" equations with oscillatory coefficients. Such equations include, but are…
Since the advent of the ``Neural Ordinary Differential Equation (Neural ODE)'' paper, learning ODEs with deep learning has been applied to system identification, time-series forecasting, and related areas. Exploiting the diffeomorphic…
The solution of the Ornstein-Zernike equation with various closure approximations is studied. This problem is rewritten as an integral equation that can be solved iteratively on a grid. The convergence of the fixed point iterations is…
The purpose of this note is to present a formulation of a given nonlinear ordinary differential equation into an equivalent system of linear ordinary differential equations. It is evident that the easiness of a such procedure would be able…
In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the…
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and…
This paper is devoted to the investigation of long-time behaviour of solutions to wave equations with quadratic nonlinearity and cubic Dirac equations with Hartree-type nonlinearity. We consider the nonlinearity here with enough simplicity…
In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear…
A non-linear Boltzmann equation describing the time evolution of a partonic system in the central rapidity region after a heavy ion collision is solved numerically. A particular model of the collinear logarithmic divergences due to small…
In this article, we introduce a novel parallel-in-time solver for nonlinear ordinary differential equations (ODEs). We state the numerical solution of an ODE as a root-finding problem that we solve using Newton's method. The affine…
In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with…
Multivariate process satisfying affine stochastic recurrence equation with generic diagonal matrices is considered. We prove that the stationary solution is regularly varying. The results are applicable to diagonal autoregressive models.
In this second paper on the method of deriving linearizing transformations for nonlinear ODEs, we extend the method to a set of two coupled second order nonlinear ODEs. We show that besides the conventional point, Sundman and generalized…
Solving the indefinite Helmholtz equation is not only crucial for the understanding of many physical phenomena but also represents an outstandingly-difficult benchmark problem for the successful application of numerical methods. Here we…
Backward stochastic differential equations extend the martingale representation theorem to the nonlinear setting. This can be seen as path-dependent counterpart of the extension from the heat equation to fully nonlinear parabolic equations…
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial…
We study a homogenisation problem for problems of mixed type in the framework of evolutionary equations. The change of type is highly oscillatory. The numerical treatment is done by a discontinuous Galerkin method in time and a continuous…
The growth-fragmentation equation arises in many different contexts, ranging from cell division, protein polymerization, biopolymers, neurosciences etc. Direct observation of temporal dynamics being often difficult, it is of main interest…
The non-integrability of the Hill problem makes that its global dynamics must be necessarily approached numerically. However, the analytical approach is feasible in the computation of relevant solutions. In particular, the nonlinear…