Related papers: Integrating factors for second order ODEs
A systematic algorithm for building integrating factors of the form mu(x,y') or mu(y,y') for non-linear second order ODEs is presented. When such an integrating factor exists, the algorithm determines it without solving any differential…
An update of the ODEtools Maple package, for the analytical solving of 1st and 2nd order ODEs using Lie group symmetry methods, is presented. The set of routines includes an ODE-solver and user-level commands realizing most of the relevant…
An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ xi = F(x), eta = P(x) y + Q(x) ], where xi d/dx + eta d/dy is the symmetry generator - is presented. To these {\it linear} symmetries one can…
We describe a solving semi-decision method based on examination of the rational structures of the generalized integrating factors of first-order ODEs. We propose a conjecture that for some family of equations of the type…
Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the…
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a…
Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and non-local symmetries)…
An alternative proof of Lie's approach for linearization of scalar second order ODEs is derived using the relationship between $\lambda$-symmetries and first integrals. This relation further leads to a new $\lambda$-symmetry linearization…
The principle of finding an integrating factor for a none exact differential equations is extended to a class of third order differential equations. If the third order equation is not exact, under certain conditions, an integrating factor…
In the first part of planned series of papers the formal general solutions to selection of 80 examples of different types of second order nonlinear PDEs in two independent variables with constant parameters are given. The main goal here is…
We present two algorithms for computing what we call the absolute factorization of a difference operator. We also give an algorithm to solve third order difference equations in terms of second order equations, together with applications to…
This paper is a sequel of our previous work in which we introduced the MapDE algorithm to determine the existence of analytic invertible mappings of an input (source) differential polynomial system (DPS) to a specific target DPS, and…
The integrability problem of rational first-order ODEs $y^{\prime}=\frac{M(x,y)}{N(x,y)}$, where $M,N \in \mathbb{R}[x,y]$ is a long-term research focus in the area of dynamical systems, physics, etc. Although the computer algebra system…
We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE…
This paper discusses an efficient implementation of the generation of order conditions for the construction of exponential integrators like exponential splitting and Magnus-type methods in the computer algebra system Maple. At the core of…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
A method of determining two factors of an odd integer without need of multiplication or division operation in iterative portion of computation is presented. It is feasible for an implementing algorithm to use only integer addition and…
A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the…
Here we present an efficient method for finding and using a nonlocal symmetry admitted by a rational second order ordinary differential equation (rational 2ODE) in order to find a Liouvillian first integral (belonging to a vast class of…
This paper is the first of a series in which we develop exact and approximate algorithms for mappings of systems of differential equations. Here we introduce the MapDE algorithm and its implementation in Maple, for mappings relating…