Related papers: Splitting Methods for differential equations
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
We present a set of new, efficient high-order symplectic methods designed for Hamiltonian systems with cubic or quartic potentials. By demonstrating that polynomial potentials require fewer order conditions, we develop schemes that…
Nonlinear elliptic problems arise in many fields, including plasma physics, astrophysics, and optimal transport. In this article, we propose a novel operator-splitting/finite element method for solving such problems. We begin by introducing…
The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a…
In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The…
We present a methodology for numerically integrating ordinary differential equations containing rapidly oscillatory terms. This challenge is distinct from that for differential equations which have rapidly oscillatory solutions: here the…
We consider a splitting approach for the Kadomtsev--Petviashvili equation with periodic boundary conditions and show that the necessary interpolation procedure can be efficiently implemented. The error made by this numerical scheme is…
We propose a family of reliable symplectic integrators adapted to the Discrete Non-Linear Schr\"odinger equation; based on an idea of Yoshida (H. Yoshida, Construction of higher order symplectic integrators, Physics Letters A, 150, 5,6,7,…
Neural integral equations are deep learning models based on the theory of integral equations, where the model consists of an integral operator and the corresponding equation (of the second kind) which is learned through an optimization…
We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…
We propose a high order numerical homogenization method for dissipative ordinary differential equations (ODEs) containing two time scales. Essentially, only first order homogenized model globally in time can be derived. To achieve a high…
In this paper we present splitting methods which are based on iterative schemes and applied to stochastic nonlinear Schroedinger equation. We will design stochastic integrators which almost conserve the symplectic structure. The idea is…
We investigate the equivalence of different operator-splitting schemes for the integration of the Langevin equation. We consider a specific problem, so called the directed percolation process, which can be extended to a wider class of…
In the context of the analysis of measured data, one is often faced with the task to differentiate data numerically. Typically, this occurs when measured data are concerned or data are evaluated numerically during the evolution of partial…
We introduce a high-order numerical scheme for fractional ordinary differential equations with the Caputo derivative. The method is developed by dividing the domain into a number of subintervals, and applying the quadratic interpolation on…
The phase-integral method (PIM) is an asymptotic method of the geometrical optics or semi-classical type for solving approximately, but in many cases very accurately, a wide class of differential equations in physics. Unlike the related…
The efficient numerical integration of large-scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications. Standard numerical methods applied to such problems require an unduly…
In this paper, we contribute operator-splitting methods improved by the Zassenhaus product for the numerical solution of linear partial differential equations. We address iterative splitting methods, that can be improved by means of the…
In the present work, an attempted was made to develop a numerical algorithm by the use of new orthogonal hybrid functions formed from hybrid of piecewise constant orthogonal sample-and-hold functions and piecewise linear orthogonal…