A. A. Abramov

A modification of one method for solving nonlinear self-adjoint eigenvalue problems for hamiltonian systems of ordinary differential equations

A modification of the method proposed earlier by the author for solving nonlinear self-adjoint eigenvalue problems for linear Hamiltonian systems of ordinary differential equations is examined. The basic assumption is that the initial data (that is, the system matrix and the matrices specifying the boundary conditions) are monotone functions of the spectral parameter.

A nonlocal problem for singular linear systems of ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite half-interval. This system is supplemented by the boundedness condition for solutions and a nonlocal linear condition specified by the Stieltjes integral. A method for approximating the resulting problem by a problem posed on a finite interval is proposed, and the properties of the latter are investigated. A numerically stable method for solving this problem is examined. This method uses an auxiliary boundary value problem with separated boundary conditions.

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.

On certain properties of a nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations

Properties of the eigenvalues are examined in a nonlinear self-adjoint eigenvalue problem for linear Hamiltonian systems of ordinary differential equations. In particular, it is proved that, under certain assumptions, every eigenvalue is isolated and there exists an eigenvalue with any prescribed index.

Numerical solution of the Cauchy problem for the Painlevé I and II equations

A numerical method for solving the Cauchy problem for the first and second Painlevé differential equations is proposed. The presence of movable poles of the solution is allowed. The positions of the poles are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to an auxiliary system of differential equations in a neighborhood of a pole. The equations in this system and its solution have no singularities in either the pole or its neighborhood. Numerical results confirming the efficiency of this method are presented.

General Nonlinear Self-Adjoint Eigenvalue Problem for Systems of Ordinary Differential Equations

The general nonlinear self-adjoint eigenvalue problem for systems of ordinary differential equations is considered. A method is proposed for reducing the problem to one for a Hamiltonian system. Results for Hamiltonian systems previously obtained by the authors are extended to this system.

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.

Solving a singular nonlocal problem with redundant conditions for a system of linear ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite or semi-infinite interval. The basic conditions are nonlocal and are specified by a Stieltjes integral; moreover, certain redundant (and also nonlocal) conditions are imposed. At infinity, the solution is required to be bounded. A method for solving such an over-determined problem is proposed and analyzed. The method is numerically stable if an auxiliary problem that replaces the original one is numerically stable.

Erratum to: “Nonlinear Spectral Problem for a Hamiltonian System of Differential Equations with Redundant Conditions”

We consider a nonlinear spectral problem for a self-adjoint Hamiltonian system of differential equations. The boundary conditions correspond to a self-adjoint problem. It is assumed that the input data (the matrix of the system and the matrices of the boundary conditions) satisfy certain conditions of monotonicity with respect to the spectral parameter. In addition to the main boundary conditions, a redundant nonlocal condition given by a Stieltjes integral is imposed on the solution. For the nontrivial solvability of the over-determined problem thus obtained, the original problem is replaced by an auxiliary problem that is consistent with the entire set of conditions. This auxiliary problem is obtained from the original one by allowing a discrepancy of a specific form. We study the resulting problem and give a numerical method for its solution.

Numerical Solution of the Cauchy Problem for Painleve III

We suggest a numerical method for solving the Cauchy problem for the third Painlevé equation. The solution of this problem is complicated by the fact that the unknown function can have movable singular points of the pole type, and in addition, the equation has a singularity at the points where the solution vanishes. The position of poles and zeros of the function is not given and is specified in the course of the solution. The method is based on the passage, in a neighborhood of these points, to an auxiliary system of differential equations for which the equation and the corresponding solution has no singularity in that neighborhood and at the pole or zero itself. We present the results of numerical experiments, which justify the efficiency of the suggested method.