3,961 research outputs found
Solving Second-Order Differential Equations by Decomposition
The subject of this article are linear and quasilinear differential equations of second order that may be decomposed into a first-order component with guaranteed solution procedure for obtaining closed-form solutions. These are homogeneous or inhomogeneous linear components, special Riccati components, Bernoulli, Clairaut or d’Alembert components. Procedures are described how they may be determined and how solutions of the originally given second order equation may be obtained from them. This makes it possible to solve new classes of differential equations and opens up a new area of research. Applying decomposition to linear inhomogeneous equations a simple procedure for determining a special solution follows. It is not based on the method of variation of constants of Lagrange, and consequently does not require the knowledge of a fundamental system. Algorithms based on these results are implemented in the computer algebra system ALLTYPES which is available on the website www.alltypes.de
Nonlocal symmetries of Riccati and Abel chains and their similarity reductions
We study nonlocal symmetries and their similarity reductions of Riccati and
Abel chains. Our results show that all the equations in Riccati chain share the
same form of nonlocal symmetry. The similarity reduced order ordinary
differential equation (ODE), , in this chain yields
order ODE in the same chain. All the equations in the Abel chain also share the
same form of nonlocal symmetry (which is different from the one that exist in
Riccati chain) but the similarity reduced order ODE, , in
the Abel chain always ends at the order ODE in the Riccati chain.
We describe the method of finding general solution of all the equations that
appear in these chains from the nonlocal symmetry.Comment: Accepted for publication in J. Math. Phy
Integrability of Lie systems through Riccati equations
Integrability conditions for Lie systems are related to reduction or
transformation processes. We here analyse a geometric method to construct
integrability conditions for Riccati equations following these approaches. This
approach provides us with a unified geometrical viewpoint that allows us to
analyse some previous works on the topic and explain new properties. Moreover,
this new approach can be straightforwardly generalised to describe
integrability conditions for any Lie system. Finally, we show the usefulness of
our treatment in order to study the problem of the linearisability of Riccati
equations.Comment: Corrected typo
The nonlinear superposition principle and the Wei-Norman method
Group theoretical methods are used to study some properties of the Riccati
equation, which is the only differential equation admitting a nonlinear
superposition principle. The Wei-Norman method is applied to obtain the
associated differential equation in the group . The superposition
principle for first order differential equation systems and Lie-Scheffers
theorem are also analysed from this group theoretical perspective. Finally, the
theory is applied in the solution of second order differential equations like
time-independent Schroedinger equatio
Lie systems: theory, generalisations, and applications
Lie systems form a class of systems of first-order ordinary differential
equations whose general solutions can be described in terms of certain finite
families of particular solutions and a set of constants, by means of a
particular type of mapping: the so-called superposition rule. Apart from this
fundamental property, Lie systems enjoy many other geometrical features and
they appear in multiple branches of Mathematics and Physics, which strongly
motivates their study. These facts, together with the authors' recent findings
in the theory of Lie systems, led to the redaction of this essay, which aims to
describe such new achievements within a self-contained guide to the whole
theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure
A geometric approach to integrability conditions for Riccati equations
Several instances of integrable Riccati equations are analyzed from the
geometric perspective of the theory of Lie systems. This provides us a unifying
viewpoint for previous approaches.Comment: 14 page
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