Discretization of second-order ordinary differential equations with symmetries

A number of publications (indicated in the Introduction) are overviewed that address the group properties, first integrals, and integrability of difference equations and meshes approximating second-order ordinary differential equations with symmetries. A new example of such equations is discussed in the overview. Additionally, it is shown that the parametric families of invariant difference schemes include exact schemes, i.e., schemes whose general solution coincides with the corresponding solution set of the differential equations at mesh nodes, which can be of arbitrary density. Thereby, it is shown that there is a kind of mathematical dualism for the problems under study: for a given physical process, there are two mathematical models: continuous and discrete. The former is described by continuous curves, while the latter, by points on these curves.     

Complete separability of time-dependent second-order ordinary differential equations

Extending previous work on the geometric characterization of separability in the autonomous case, necessary and sufficient conditions are established for the complete separability of a system of time-dependent second-order ordinary differential equations. In deriving this result, extensive use is made of the theory of derivations of scalar and vector-valued forms along the projection :J ¹ EE of the first jet bundle of a fibre bundleE . Two illustrative examples are discussed, which fully demonstrate all aspects of the constructive nature of the theory.     

Singular boundary-value problems for ordinary second-order differential equations

This article gives an exposition of the fundamental results of the theory of boundary-value problems for ordinary second-order differential equations having singularities with respect to the independent variable or one of the phase variables. In particular criteria are given for solvability and unique solvability of two-point boundary-value problems and problems concerning bounded and monotonic solutions. Several specific problems are considered which arise in applications (atomic physics, field theory, boundary-layer theory of a viscous incompressible fluid, etc.)Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 105–201, 1987.     

Lagrange formula for ordinary continual second-order differential equations

For ordinary continual second-order differential equations, we derive a Lagrange formula and construct their fundamental solutions. We use the Lagrange formula to determine well-posed forms of initial data for these equations and obtain explicit representations of solutions of these initial value problems.     

Point transformations and linearizability of second-order ordinary differential equations

Translated from Matematicheskie Zametki, Vol. 49, No. 1, pp. 146–148, January, 1991.     

Mixed boundary-value problems for singular second-order ordinary differential equations

It is proved that the boundary-value problem

Complex Lie symmetries for scalar second-order ordinary differential equations

A scalar complex ordinary differential equation can be considered as two coupled real partial differential equations, along with the constraint of the Cauchy–Riemann equations, which constitute a system of four equations for two unknown real functions of two real variables. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of the given complex ordinary differential equation. Further, if we restrict the complex function to be of a single real variable, then the complex ordinary differential equation yields a coupled system of two ordinary differential equations and their invariance can be obtained in a non-trivial way from the invariance of the restricted complex differential equation. Also, the use of a complex Lie symmetry reduces the order of the complex ordinary differential equation (restricted complex ordinary differential equation) by one, which in turn yields a reduction in the order by one of the system of partial differential equations (system of ordinary differential equations). In this paper, for simplicity, we investigate the case of scalar second-order ordinary differential equations. As a consequence, we obtain an extension of the Lie table for second-order equations with two symmetries.     

Group classification of projective type second-order ordinary differential equations

Group classification with respect to admitted point transformation groups is carried out for second-order ordinary differential equations with cubic nonlinearity of the first-order derivative. The result is obtained with use of the invariants of the equivalence transformation group of the family of equations under consideration. The corresponding Riemannian metric is found for the equations that are the projection of the system of geodesics to a two-dimensional surface.     

On two comparison tests for linear second-order ordinary differential equations

We prove two comparison tests for second-order linear differential equations. The well-known Sturm comparison theorem is a straightforward corollary of the first of them. The second (integral) test permits one to use specific integral relations between the coefficients of two equations to prove the oscillation of one of the equations assuming the oscillation of the other.     

On periodic solutions of linear degenerate second-order ordinary differential equations

We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity.     

Nonclassical behavior of solutions of linear second-order ordinary differential equations

We present a number of unstable second-order equations of the form

$$y'' + (1 + g(x))y = 0,$$

where the coefficient g(x) satisfies the conditions g(x) ∈ C(0, ∞) and lim_x→+∞ g(x)=0 but the maximum absolute values of solutions grow unboundedly (as power-law functions or even as exponentials).

     

A procedure for solving some second-order linear ordinary differential equations

The purpose of this work is to introduce the concept of pseudo-exactness for second-order linear ordinary differential equations (ODEs), and then to try to solve some specific ODEs.     

Positive solutions for a multi-parameter system of second-order ordinary differential equations

In this paper, we establish the product formula for the fixed point index on product cone, andthen, as applications, consider the existence, nonexistence and multiplicity of positive solutions for a second-order ordinary differential system with parameters. The discussion is based on the product formula and thefundamental properties of the fixed point index.