Related papers: A New Method of Solving Third Order Non-Linear Ord…
In quantum mechanics, the Rosen-Zener model represents a two-level quantum system. Its generalization to multiple degenerate sets of states leads to larger non-autonomous linear system of ordinary differential equations (ODEs). We propose a…
We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard…
In this paper, two numerical approaches based on the Newton iteration method with spectral algorithms are introduced to solve the Thomas-Fermi equation. That Thomas-Fermi equation is a nonlinear singular ordinary differential equation (ODE)…
Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and…
In this paper we consider an alternative approach to "un-reduction". This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions…
This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling…
Differential equations arise in mathematics, physics,medicine, pharmacology, communications, image processing and animation, etc. An Ordinary Differential Equation (ODE) is a differential equation if it involves derivatives with respect to…
We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [Comput. Methods…
In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation with this noise with a highly irregular coefficient. We employ a new method to…
We introduce a new overlapping Domain Decomposition Method (DDM) to solve the fully nonlinear Monge-Amp\`ere equation. While DDMs have been extensively studied for linear problems, their application to fully nonlinear partial differential…
In this paper, we propose and analyze a third-order dynamical system for solving a generalized inverse mixed variational inequality problem in a Hilbert space H. We establish the existence and uniqueness of the trajectories generated by the…
A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning…
A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to…
This paper proposes an explicit computational method for solving a three-dimensional system of nonlinear elastodynamic sine-Gordon equations subject to appropriate initial and boundary conditions. The time derivative is approximated by…
Partial differential equations (PDE) often involve parameters, such as viscosity or density. An analysis of the PDE may involve considering a large range of parameter values, as occurs in uncertainty quantification, control and…
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…
By interpreting the forward dynamics of the latent representation of neural networks as an ordinary differential equation, Neural Ordinary Differential Equation (Neural ODE) emerged as an effective framework for modeling a system dynamics…
In the present paper, an algorithm for the numerical solution of the external Dirichlet generalized harmonic problem for a sphere by the method of probabilistic solution (MPS) is given, where generalized indicates that a boundary function…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…