Spaces with an abstract convolution of measures

The self-adjoint subspace extensions of a possibly nondensely defined symmetric ordinary differential operator in a Hilbert space are described. The operator part of these extensions involve not only the differential operator but boundary-integral terms, and the side conditions which determine the domains of the extensions also involve boundary-integral terms. Corresponding to each self-adjoint subspace extension in a possibly larger Hilbert space an eigenfunction expansion result is obtained. Analogous results for first-order systems of ordinary differential operators are shown to be valid.     

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

A system of ordinary differential equations of mixed order on an interval (0, r₀) is considered, where some coefficients are singular at 0. Special cases have been dealt with by Kako , where the essential spectrum of an operator associated with a linearized MHD model was calculated, and more recently by Hardt , Mennicken and Naboko . In both papers this operator is a selfadjoint extension of an operator on sufficiently smooth functions. The approach in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly defined as the closure of an operator defined on sufficiently smooth functions. This closed operator can be written as a sum of a selfadjoint operator and a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of ℂ, and formulas for the calculation of the essential spectrum in terms of the coefficients are given.     

Constructively Factoring Linear Partial Differential Operators in Two Variables

We study conditions under which a partial differential operator of arbitrary order n in two variables or an ordinary linear differential operator admits a factorization with a first-order factor on the left.The process of factoring consists of recursively solving systems of linear equations subject to certain differential compatibility conditions.In the general case of partial differential operators, it is not necessary to solve a differential equation. In special degenerate cases, such as an ordinary differential operator, the problem eventually reduces to solving some Riccati equation(s). We give the factorization conditions explicitly for the second and third orders and in outline form for higher orders. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 2, pp. 165–180, November, 2005.     

Spectral analysis of singular ordinary differential operators with indefinite weights

In this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions. We prove a general perturbation result on the local spectral properties of selfadjoint operators in Krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares. This result is applied to singular indefinite Sturm-Liouville operators and higher order singular ordinary differential operators with indefinite weight functions.     

Stability in the almost everywhere sense: A linear transfer operator approach

The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron-Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.     

A spectral problem with integral conditions

In this paper, we consider an ordinary differential operator with integral conditions. We obtain an a priori estimate of the solutions and prove the sector structure and the discreteness of the spectrum of this operator.     

The splitting index for linear differential systems

Consider the linear differential operator associated with an n-dimensional first order linear system of time varying ordinary differential equations. Conditions are given on the system for time near plus and minus infinity which guarantee that the operator is Fredholm. The splitting index is introduced and it is shown to be the negative of the ordinary index of a Fredholm operator. The splitting index is shown to be invariant under appropriate perturbations and is computable in terms of the asymptotic properties of the coefficient matrix for a wide class of systems. The asymptotic conditions on the system are discussed in various function space topologies and a new concept of admissibility of a pair of Banach spaces is introduced whereby a pair is admissible with respect to all operators whose coefficients lie in a given function space.     

Iteration scheme for nondensely defined ordinary differential operator equation

An iteration scheme is given for constructing the unique least-squares solution of minimal norm for an ordinary differential operator equation. This operator is subject to generalized two- point boundary conditions and is nondensely defined. The convergence is established in a uniform topology.     

Convexity with respect to a differential equation

The concepts of convexity of a set, convexity of a function and monotonicity of an operator with respect to a second-order ordinary differential equation are introduced in this paper. Several well-known properties of usual convexity are derived in this context, in particular, a characterization of convexity of function and monotonicity of an operator. A sufficient optimality condition for a optimization problem is obtained as an application. A number of examples of convex sets, convex functions and monotone operators with respect to a differential equation are presented.     

一类非线性方程正解的存在唯一性

本文证明了形如Ａ＝Ｂ＋Ｃ的算子，其中Ｂ为线性算子，Ｃ为ψ－凹算子不动点的存在唯一及迭代收敛性，并将所获结果就用于常微分方程，偏微分方程和各人分方程得到了新的结论。     

Degenerate non-stationary differential equations with delay in Banach spaces

A generalization of the operator method by Grisvard is used to ensure weak and strict solutions to some degenerate differential equations with delay in Banach spaces, whose operator coefficients are time depending. Some applications to ordinary and partial differential equations with delay are described.     

Hecke Operators on Linear Ordinary Differential Equations

We construct Hecke operators acting on the space of certain linear ordinary differential equations, and describe a Hermitian inner product on the space of such differential equations. We also determine the adjoint of the Hecke operator with respect to this inner product, and prove that the space of ordinary differential equations associated to an automorphic form for a certain discrete subgroup of SL(2, R) has a basis consisting of common eigenvectors of a class of Hecke operators.     

Spectral Asymptotics of Self-Adjoint Fourth Order Differential Operators with Eigenvalue Parameter Dependent Boundary Conditions

A fourth-order regular ordinary differential operator with eigenvalue dependent boundary conditions is considered. This problem is realized by a quadratic operator pencil with self-adjoint operators. The location of the eigenvalues is discussed and the first four terms of the eigenvalue asymptotics are evaluated explicitly.     

Hypoelliptic ordinary differential operators

Necessary and sufficient conditions for the hypoellipticity of an ordinary differential operator withC ^∞ coefficients in a neighborhood of a zero of finite order of the leading term are given. A sufficient condition for such an operator to be in a certain Hörmander class is also given.     

Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations

In this paper we design higher-order time integrators for systems of stiff ordinary differential equations. We combine implicit Runge–Kutta and BDF methods with iterative operator-splitting methods to obtain higher-order methods. The idea of decoupling each complicated operator in simpler operators with an adapted time scale allows to solve the problems more efficiently. We compare our new methods with the higher-order fractional-stepping Runge–Kutta methods, developed for stiff ordinary differential equations. The benefit is the individual handling of each operator with adapted standard higher-order time integrators. The methods are applied to equations for convection–diffusion reactions and we obtain higher-order results. Finally we discuss the applications of the iterative operator-splitting methods to multi-dimensional and multi-physical problems.     

线性齐次常微分方程(组)的λ-矩阵求解法

本文在文 [2 ]的基础上 ,应用λ-矩阵及微分算子性质给出了一种变系数齐次常微分线性方程 (组 )的λ-矩阵求解法 ,对文 [2 ]作出了更一般的推广 .     

Operator part of infinite system of differential equations

Let T be a differential operator in a Hilbert space generated by a first-order infinite system of an ordinary linear differential expression, which is subject to infinitely many boundary conditions. We solve completely for y from Ty = g. The main tools involved are operator parts of closed linear manifolds, which are closely related to generalized inverses.     

On a Sturm-Liouville problem

In this paper we give a lower bound for the first eigenvalue of an ordinary differential operator which represents the radial part of the Laplace operator restricted to a spherical cap of a sphere, possibly of fractional dimension. The results are obtained by purely analytical methods.Research supported by CONICIT.     

Hyperbolicity of periodic solutions of functional differential equations with several delays

We study conditions for the hyperbolicity of periodic solutions to nonlinear functional differential equations in terms of the eigenvalues of the monodromy operator. The eigenvalue problem for the monodromy operator is reduced to a boundary value problem for a system of ordinary differential equations with a spectral parameter. This makes it possible to construct a characteristic function. We prove that the zeros of this function coincide with the eigenvalues of the monodromy operator and, under certain additional conditions, the multiplicity of a zero of the characteristic function coincides with the algebraic multiplicity of the corresponding eigenvalue.     

Nonlocal Boundary-Value Problem for a Linear Ordinary Differential Equation with Fractional Discretely Distributed Differentiation Operator

Mathematical Notes - A nonlocal boundary-value problem for a linear ordinary differential equation with fractional discretely distributed differentiation operator is considered. The existence and...