Related papers: On approximate implicit Taylor methods for ordinar…
This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known…
We introduce a provably stable variant of neural ordinary differential equations (neural ODEs) whose trajectories evolve on an energy functional parametrised by a neural network. Stable neural flows provide an implicit guarantee on…
A trust-region algorithm using inexact function and derivatives values is introduced for solving unconstrained smooth optimization problems. This algorithm uses high-order Taylor models and allows the search of strong approximate minimizers…
We provide of a method to integrate first order non-linear systems of differential equations with variable coefficients. It determines approximate solutions given initial or boundary conditions or even for Sturm-Liouville problems. This…
This paper deals with the index reduction problem for the class of quasi-regular DAE systems. It is shown that any of these systems can be transformed to a generically equivalent first order DAE system consisting of a single purely…
Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods…
The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential $V(x) =\beta x^{2}+x^{2m}$ ($m>0$). The…
The main aim of this study is to introduce a 2-layered Artificial Neural Network (ANN) for solving the Black-Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed…
This short communication develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
Ordinary differential equations (ODEs) are widely used to describe the time evolution of natural phenomena across various scientific fields. Estimating the parameters of these systems from data is a challenging task, particularly when…
Partial differential equations can be used to model many problems in several fields of application including, e.g., fluid mechanics, heat and mass transfer, and electromagnetism. Accurate discretization methods (e.g., finite element or…
This article provides a general iterative approximation to partial differential equations, and thus establish existence of smooth solution. The heart of the method is to contract (or expand) the boundary conditions uniformly in the domain,…
We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods…
In this study, we introduce a refined method for ascertaining error estimations in numerical simulations of dynamical systems via an innovative application of composition techniques. Our approach involves a dual application of a basic…
The question about asymptotical behaviour of solutions for the system $\dot x=A_\nu x+f$ for big values of the parameter $\nu\in\frak A$ is considered. An approach to the reduction of a large class of problems to easily solvable problem…
The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given…
Neural ordinary differential equations (ODEs) are an emerging class of deep learning models for dynamical systems. They are particularly useful for learning an ODE vector field from observed trajectories (i.e., inverse problems). We here…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
A new numerical method for solving a scalar ordinary differential equation with a given initial condition is introduced. The method is using a numerical integration procedure for an equivalent integral equation and is called in this paper…