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While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation…
High index differential algebraic equations (DAEs) are ordinary differential equations (ODEs) with constraints and arise frequently from many mathematical models of physical phenomenons and engineering fields. In this paper, we generalize…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing…
Neural differential equations are a promising new member in the neural network family. They show the potential of differential equations for time series data analysis. In this paper, the strength of the ordinary differential equation (ODE)…
Neural Ordinary Differential Equations (Neural ODEs) represent continuous-time dynamics with neural networks, offering advancements for modeling and control tasks. However, training Neural ODEs requires solving differential equations at…
In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of…
The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coefficients with special…
We present a parameter estimation method in Ordinary Differential Equation (ODE) models. Due to complex relationships between parameters and states the use of standard techniques such as nonlinear least squares can lead to the presence of…
We aim to identify the generating, ordinary differential equation (ODE) from a set of trajectories of a partially observed system. Our approach does not need prescribed basis functions to learn the ODE model, but only a rich set of Neural…
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a…
Let y''' = f(x, y, y', y'') be a 3rd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates.
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…
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…
The main objective of this paper is to introduce an algorithm for solving fractional and classical differential equations based on a new generalized fractional power series. The algorithm relies on expanding the solution of an FDE or an ODE…
Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…
Ordinary differential equations (ODE's) are widespread models in physics, chemistry and biology. In particular, this mathematical formalism is used for describing the evolution of complex systems and it might consist of high-dimensional…
Let $dx_i/dt=f_i(x_1,\cdots,x_n)$, ($i=1,\cdots,n$) be a system of $n$ first order autonomous ordinary differential equations. We use E. Cartan's equivalence method to study the invariants of this system under diffeomorphisms of the form…
Recently, a remarkable correspondence has been unveiled between a certain class of ordinary linear differential equations (ODE) and integrable models. In the first part of the report, we survey the results concerning the 2nd order…
Viewing optimization methods as numerical integrators for ordinary differential equations (ODEs) provides a thought-provoking modern framework for studying accelerated first-order optimizers. In this literature, acceleration is often…