Related papers: Folding Difference and Differential Systems into H…
This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be factorized using a pair of mutually…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. Closed form of its solution is derived by means of newly defined delayed matrix sine/cosine using the…
The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate.…
A higher order difference equation may be generally defined in an arbitrary nonempty set S as: \[ f_{n}(x_{n},x_{n-1},...,x_{n-k})=g_{n}(x_{n},x_{n-1},...,x_{n-k}) \] where $f_{n},g_{n} :S^{k+1}\rightarrow S$ are given functions for…
The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…
Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
The paper deals with systems of ordinary differential equations containing in the right-hand side controls which are discontinuous in phase variables. These controls cause the occurrence of sliding modes. If one uses one of the well-known…
A consequent approach is proposed to construct symplectic force-gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations. Within this approach the basic…
The identification of a mathematical dynamics model is a crucial step in the designing process of a controller. However, it is often very difficult to identify the system's governing equations, especially in complex environments that…
The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive…
Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this…
The Lie linearizability criteria are extended to complex functions for complex ordinary differential equations. The linearizability of complex ordinary differential equations is used to study the linearizability of corresponding systems of…
It is often unnoticed that the predominant way to use collocation methods is fundamentally flawed when applied to optimal control in robotics. Such methods assume that the system dynamics is given by a first order ODE, whereas robots are…
Dynamical systems with quadratic or polynomial drift exhibit complex dynamics, yet compared to nonlinear systems in general form, are often easier to analyze, simulate, control, and learn. Results going back over a century have shown that…
Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…
This paper is devoted to the construction of order reduced method of fourth order problems. A framework is presented such that a problem on a high-regularity space can be deduced in a constructive way to an equivalent problem on three…
A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely…
A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations,…