Quasi-Lie schemes for PDEs

The theory of quasi-Lie systems, i.e. systems of first-order ordinary differential equations that can be related via a generalized flow to Lie systems, is extended to systems of partial differential equations (PDEs) and its applications to obtain [Formula: see text]-dependent superposition rules, and integrability conditions are analyzed. We develop a procedure of constructing quasi-Lie systems through a generalization to PDEs of the so-called theory of quasi-Lie schemes. Our techniques are illustrated with the analysis of Wess–Zumino–Novikov–Witten models, generalized Abel differential equations, Bäcklund transformations, as well as other differential equations of physical and mathematical relevance.

Polynomial Solutions of Equivariant Polynomial Abel Differential Equations

Let {a(x)} be non-constant and let {b_{j}(x)} , for {j=0,1,2,3} , be real or complex polynomials in the variable x. Then the real or complex equivariant polynomial Abel differential equation {a(x)\dot{y}=b_{1}(x)y+b_{3}(x)y^{3}} , with {b_{3}(x)\neq 0} , and the real or complex polynomial equivariant polynomial Abel differential equation of the second kind {a(x)y\dot{y}=b_{0}(x)+b_{2}(x)y^{2}} , with {b_{2}(x)\neq 0} , have at most 7 polynomial solutions. Moreover, there exist equations of this type having this maximum number of polynomial solutions.

A counterexample to the composition condition conjecture for polynomial Abel differential equations

Polynomial Abel differential equations are considered a model problem for the classical Poincaré center–focus problem for planar polynomial systems of ordinary differential equations. In the last few decades, several works pointed out that all centers of the polynomial Abel differential equations satisfied the composition conditions (also called universal centers). In this work we provide a simple counterexample to this conjecture.

The Integrating Factors for Riccati and Abel Differential Equations

We can recast the Riccati and Abel differential equationsinto new forms in terms of introduced integrating factors.Therefore, the Lie-type systems endowing with transformation Lie-groups$SL(2,{\mathbb R})$ can be obtained.The solution of second-order linearhomogeneous differential equation is an integrating factorof the corresponding Riccati differential equation.The numerical schemes which are developed to fulfil the Lie-group property have better accuracy and stability than other schemes.We demonstrate that upon applying the group-preserving scheme (GPS) to the logistic differential equation, it is not only qualitatively correct for all values of time stepsize $h$, and is also the most accurate one among all numerical schemes compared in this paper.The group-preserving schemes derived for the Riccati differential equation, second-order linear homogeneous and non-homogeneous differential equations, the Abel differential equation and higher-order nonlinear differential equations all have accuracy better than $O(h^2)$.

Abel dynamic equations of first and second kind

This paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations. By using appropriate transformations, expressions of Abel dynamic equations of second kind are derived on the general time scale. This also leads to a specific class of Abel dynamic equations of first kind. Finally, the canonical Abel dynamic equation is defined and examined.