Solving a system of linear ordinary differential equations with redundant conditions

A system of linear ordinary differential equations is examined under the assumption that, in addition to the basic conditions, which in general are nonlocal and are specified by a Stieltjes integral, certain redundant (and possibly also nonlocal) conditions are imposed. Generically, such a problem has no solution. A principle for constructing an auxiliary system is proposed. This system replaces the original one and is normally consistent with all the conditions prescribed. A method for solving this auxiliary problem is analyzed. The method is numerically stable if the auxiliary problem is numerically stable.     

Solving boundary value problems of ordinary differential equations with non-separated boundary conditions

This paper focuses on solving the two point boundary value problem, in which boundary conditions are systems of nonlinear equations. The shooting method was used together with a combination of Newton’s method and Broyden’s method, to update the initial values of the differential equations. The experiments showed that the proposed method performed well, in the sense that the overall amount of work was less than that of the Newton Shooting method.     

A nonlinear singular eigenvalue problem for a linear system of ordinary differential equations with redundant conditions

A nonlinear eigenvalue problem for a linear system of ordinary differential equations is examined on a semi-infinite interval. The problem is supplemented by nonlocal conditions specified by a Stieltjes integral. At infinity, the solution must be bounded. In addition to these basic conditions, the solution must satisfy certain redundant conditions, which are also nonlocal. A numerically stable method for solving such a singular overdetermined eigenvalue problem is proposed and analyzed. The essence of the method is that this overdetermined problem is replaced by an auxiliary problem consistent with all the above conditions.     

Boundary-value problems for systems of ordinary differential equations

This article contains an exposition of fundamental results of the theory of boundary-value problems for systems of linear and nonlinear ordinary differential equations. In particular, criteria are given for problems with functional, many-point, and two-point boundary conditions to be solvable and well-posed, as well as methods of finding approximate solutions. We also examine questions of existence, uniqueness, and stability of periodic and bounded solutions of nonautonomous differential systems.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 30, pp. 3–103, 1987.     

Some problems of the linear theory of systems of ordinary differential equations

We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet–Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula to the introduction of amplitude–phase coordinates in a neighborhood of a periodic trajectory of a dynamical system.     

On some two-point boundary value problems for two-dimensional systems of ordinary differential equations

Sufficient conditions for the solvability of two-point boundary value problems for the systemx _i =f_i(t, x₁, x₂) (i=1,2) are given, wheref ₁ andf ₂: [₁, ₁] ×R ₂ R are continuous functions.     

Nonlinear boundary-value problems for systems of ordinary differential equations

We consider nonlinear boundary-value problems (with Noetherian operator in the linear part) for systems of ordinary differential equations in the neighborhood of generating solutions. By using the Lyapunov — Schmidt method, we establish conditions for the existence of solutions of these boundary-value problems and propose iteration algorithms for their construction. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 162–171, February, 1998.     

Solving linear ordinary differential equations by exponentials of iterated commutators

Summary A sequence of transformations of a linear system of ordinary differential equations is investigated. It is shown that these transformations produce new systems which represent progressively smaller perturbations of the original set of equations.The transformations are implemented as a basis of a numerical method. Order, stability and error control of this method are analyzed. Numerical examples demonstrate the potential of this approach.     

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.     

Double resonance with Landesman-Lazer conditions for planar systems of ordinary differential equations

We prove the existence of periodic solutions for first order planar systems at resonance. The nonlinearity is indeed allowed to interact with two positively homogeneous Hamiltonians, both at resonance, and some kind of Landesman-Lazer conditions are assumed at both sides. We are thus able to obtain, as particular cases, the existence results proposed in the pioneering papers by Lazer and Leach (1969) [27], and by Frederickson and Lazer (1969) [18]. Our theorem also applies in the case of asymptotically piecewise linear systems, and in particular generalizes Fabry's results in Fabry (1995) [10], for scalar equations with double resonance with respect to the Dancer-Fu?ik spectrum.