Projection method for discretization of boundary-value problems for a fourth-order ordinary differential equation

The projection method is applied to construct discrete models of boundary-value problems for a fourth-order ordinary differential equation in which the only unknowns are the solution values.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 14–22, 1989.     

Periodic boundary-value problem for third-order linear functional differential equations

For a linear functional differential equation of the third order {fx481-01}, we establish the theorems on existence and uniqueness of a solution satisfying the conditions {fx481-02}. Here, ℓ is a linear continuous operator transforming the space C([0, ω]; R) into the space L([0, ω]; R) and q ∈ L([0, ω]; R). The problem of nonnegativity of the solution of the analyzed boundary-value problem is also studied. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 413–425, March, 2008.     

On a nonlocal boundary value problem for a fourth-order ordinary differential equation

We study a boundary value problem for a fourth-order ordinary differential equation with a nonlocal boundary condition. We give a necessary and sufficient condition for a minimizer of a specially constructed functional to be a solution of the problem.     

Solution of a three-dimensional boundary-value problem for a fractional differential heat conduction equation

We consider a three-dimensional axisymmetric problem for a differential heat conduction equation with fractional time derivatives. Using the method of homogeneous solutions and integral transformations, we obtain an asymptotic and a numerical solution of the problem. The results of calculations are presented.     

Uniqueness of the Dirichlet problem for a partial differential equation of the fourth-order

We give some uniqueness theorems for regular solutions to the Dirichlet problem associated with a partial differential equation of the fourth-order in an unbounded domain. Moreover, by means of a counterexample, we show that one of these result is sharp.     

On the solvability of a boundary value problem for a fourth-order ordinary differential equation

We study the existence and multiplicity of nontrivial periodic solutions for a semilinear fourth-order ordinary differential equation arising in the study of spatial patterns for bistable systems. Variational tools such as the Brezis–Nirenberg theorem and Clark theorem are used in the proofs of the main results.     

Solvability of the boundary-value problem for a variable-order differential equation on a geometric graph

The solvability of the boundary-value problem for a string-beam model is studied. The model is described by an equation of orders 2 or 4 on dirrerent edges of an arbitrary graph. Criteria for the problem to be degenerate and nondegenerate are obtained; in particular, it is proved that the nondegeneracy of the problem is equivalent to the maximum principle.     

On the two-point boundary-value problem for the Riccati matrix differential equation

Constructive sufficient conditions for univalent resolvability of a two-point boundary value problem for nonlinear Riccati equation are obtained. An illustrative example is given.     

Generalized green operator of Noetherian boundary-value problem for matrix differential equation

We find necessary and sufficient conditions for solvability and the construction of the generalized Green operator for Noetherian linear boundary-value problem for a linear matrix differential equation. We propose an operator, which leads a linear matrix algebraic equation to the traditional linear algebraic system with a rectangular matrix. We use pseudoinverseMoore–Penrose matrices and orthogonal projections for solving a linear algebraic system.     

Solution of a singularly perturbed nonstationary fourth-order boundary-value problem

A difference scheme is constructed by the method of lines for a nonstationary boundary-value problem for a fourth-order equation in the space coordinate. The uniform convergence with respect to a small parameter, of the solution of the linearization scheme to a solution of the original problem is proven.Translated from Vychislitel'naya i Prikladnaya Matematika, Vol. 70, pp. 3–11, 1990.     

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R ² with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H ^m (Ω), m ≥ 4.     

The first boundary-value problem for a singular nonlinear ordinary differential equation of fourth order

The solvability of the boundary-value problem

An approximation method in a boundary-value problem for a linear differential equation with polynomial coefficients

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 2, pp. 248–253, March–April, 1988.     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

Matrix methods for solving the boundary-value problem for a second-order differential equation with delayed argument

Zero-rank matrix numerical differentiation algorithms are applied to construct efficient numerical-analytical methods (so-called zero-rank matrix methods) to find the eigenvalues and eigenfunctions of boundary-value problems for second-order differential equations with a delayed argument. The proposed methods are analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 19–24, 1986.