Weakly nonlinear boundary-value problem in a special critical case

We investigate the problem of the determination of conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for finding the generating solution of a weakly nonlinear Noetherian boundary-value problem turns into an identity. We improve the classification of critical cases and construct an iterative algorithm for finding solutions of weakly nonlinear Noetherian boundary-value problems in the special critical case.     

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.     

Degenerate nonlinear boundary-value problems

We establish necessary and sufficient conditions for the existence of solutions of weakly nonlinear degenerate boundary-value problems for systems of ordinary differential equations with a Noetherian operator in the linear part. We propose a convergent iterative procedure for finding solutions and establish the relationship between necessary and sufficient conditions.     

Weakly nonlinear boundary-value problems for operator equations with pulse influence

We consider the problem of finding conditions of solvability and algorithms for construction of solutions of weakly nonlinear boundary-value problems for operator equations (with the Noetherian linear part) with pulse influence at fixed times. The method of investigation is based on passing by methods of the Lyapunov—Schmidt type from a pulse boundary-value problem to an equivalent operator system that can be solved by iteration procedures based on the fixed-point principle. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 272–288, February, 1997.     

Introduction to the theory of functional differential equations and their applications. Group approach

In this work, we give an introduction to the theory of nonlinear functional differential equations of pointwise type on a finite interval, semi-axis, or axis. This approach is based on the formalism using group peculiarities of such differential equations. For the main boundary-value problem and the Euler-Lagrange boundary-value problem, we consider the existence and uniqueness of the solution, the continuous dependence of the solution on boundary-value and initial-value conditions, and the “roughness” of functional differential equations in the considered boundary-value problems. For functional differential equations of pointwise type we also investigate the pointwise completeness of the space of solutions for given boundary-value conditions, give an estimate of the rank for the space of solutions, describe types of degeneration for the space of solutions, and establish conditions for the “smoothness” of the solution. We propose the method of regular extension of the class of ordinary differential equations in the class of functional differential equations of pointwise type. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 8, Functional Differential Equations, 2004.     

Boundary-value problems for parametric ordinary differential equations

By using semiinverse matrices and a generalized Green's matrix, we construct solutions of boundary-value problems for linear and weakly perturbed nonlinear systems of ordinary differential equations with a parameter in boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 372–377, April, 1994.     

Some remarks concerning the convergence of the numerical-analytic method of successive approximations

We establish new improved estimates necessary for the justification of the numerical-analytic method for the investigation of the existence and construction of approximate solutions of nonlinear boundary-value problems for ordinary differential equations.     

Positive solutions of boundary-value problems for disfocal ordinary differential equations

We study the existence of positive solutions of a nonlinear eigenvalue problem for a two-point boundary-value problem for a family of second-order disfocal ordinary differential equations. We apply a cone-theoretic fixed-point theorem and obtain both sufficient conditions for the existence of positive solutions and sufficient conditions for the existence of multiple positive solutions.     

Weakly nonlinear boundary-value problems for differential systems with pulse influence

The coefficient sufficient conditions for the existence of solutions and the iteration algorithm of constructing these solutions are obtained for weakly nonlinear boundary-value problems for systems of ordinary differential equations with pulse influence in the general case in which the number of boundary conditions does not coincide with the order of the differential system. The equation is derived for generating amplitudes of these boundary-value problems. This equation determines the amplitude of a solution, which can be regarded as generating for the required solution, and gives necessary conditions for the existence of this solution.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 221–225, February, 1993.     

Generalized green operator of a boundary-value problem with degenerate pulse influence

We establish necessary and sufficient conditions for the solvability of inhomogeneous linear boundaryvalue problems for systems of ordinary differential equations with pulse influence in the case where the number of boundary conditions is not equal to the order of the differential system (Noetherian problems). We construct a generalized Green operator for boundary-value problems not all solutions of which can be extended from the left endpoint to the right endpoint of the interval where these solutions are constructed.     

Investigation and solution of boundary-value problems with parameters by numerical-analytic method

We suggest a modification of the numerical-analytic iteration method. This method is used for studying the problem of existence of solutions and for constructing approximate solutions of nonlinear two-point boundary-value problems for ordinary differential equations with unknown parameters both in the equation and in boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1031–1042, August, 1994.     

Bifurcation of solutions of a linear Fredholm boundary-value problem

We establish constructive conditions for the appearance of solutions of a linear Fredholm boundary-value problem for a system of ordinary differential equations in the critical case and propose an iterative procedure for finding these solutions. The range of values of a small parameter for which the indicated iterative procedure is convergent is estimated. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1148–1152, August, 2007.     

Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems

Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations (PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems with ℝ ⁿ -valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against those obtained by using shooting techniques.     

Asymptotic analysis of some classes of ordinary differential equations with large high-frequency terms

A complete asymptotic expansion is constructed for solutions of the Cauchy problem for nth order linear ordinary differential equations with rapidly oscillating coefficients, some of which may be proportional to ω ^n/2, where ω is oscillation frequency. A similar problem is solved for a class of systems of n linear first-order ordinary differential equations with coefficients of the same type. Attention is also given to some classes of first-order nonlinear equations with rapidly oscillating terms proportional to powers ω ^d . For such equations with d ∈ (1/2, 1], conditions are found that allow for the construction (and strict justification) of the leading asymptotic term and, in some cases, a complete asymptotic expansion of the solution of the Cauchy problem.     

On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations

We study a non-linear semi-periodic boundary-value problem for a system of hyperbolic equations with mixed derivative. At that, the semi-periodic boundary-value problem for a system of hyperbolic equations is reduced to an equivalent problem, consisting of a family of periodic boundary-value problems for ordinary differential equations and functional relation. When solving a family of periodic boundary-value problems of ordinary differential equations we use the method of parameterization. This approach allowed to establish sufficient conditions for the existence of an isolated solution of non-linear semi-periodic boundary-value problem for a system of hyperbolic equations.     

Effective method for solving singularly perturbed systems of nonlinear differential equations

A boundary-value problem for a class of singularly perturbed systems of nonlinear ordinary differential equations is considered. An analytic-numerical method for solving this problem is proposed. The method combines the operational Newton method with the method of continuation by a parameter and construction of the initial approximation in an explicit form. The method is applied to the particular system arising when simulating the interaction of physical fields in a semiconductor diode. The Frechét derivative and the Green function for the corresponding differential equation are found analytically in this case. Numerical simulations demonstrate a high efficiency and superexponential rate of convergence of the method proposed. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 15, Differential and Functional Differential Equations. Part 1, 2006.     

Numerical experiments using Sukhanov's initial-value method for nonlinear two-point boundary-value problems,II

In a recent paper, Sukhanov derived a new method for transforming a nonlinear two-point boundary-value problem into an initial-value problem. Sukhanov's equations involve only the solution of ordinary differential equations and not partial differential equations. An earlier paper by the authors presented their interpretation of Sukhanov's method. An alternative method is presented in this paper. Numerical results are given.     

On Partially Irregular Almost Periodic Solutions of Weakly Nonlinear Ordinary Differential Systems

For weakly nonlinear almost periodic ordinary differential systems, we obtain conditions for the existence of partially irregular almost periodic solutions and propose algorithms for their construction. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1123 – 1130, August, 2005.     

A study of the bending of a short beam

We consider the two-dimensional problem of static deformation ofa short beam under the action of a self-balanced load. We propose an approximate method of solution based on a variational approach and a special choice of the stress function. We prove that the resulting boundary-value problem for a system of ordinary differential equations is well-posed. For special cases of the boundary conditions we give an analysis of the solutions. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 65–73.     

Multipoint boundary-value problems with pulse effects

By using pseudoinverse matrices, we establish conditions for the existence and uniqueness of solutions of linear and weakly linear boundary-value problems for ordinary differential equations with pulse action. We consider the case where the dimension of a differential system does not coincide with the dimension of the boundary conditions.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 770–774, June, 1995.