Fundamental System Of Solutions Research Articles

On a class of operators related to second-order differential equations

In the present paper, we find a class of linear homogeneous differential equations of order n + 1 (n > 1) whose fundamental system of solutions is constructed from the fundamental system of solutions of a second-order differential equation. The spectral properties of differential operators generated by these differential expressions are investigated. In particular, sufficient conditions are obtained for the coefficients of a second-order differential equation under which the case of maximal deficiency indices is realized.

Fibonacci-Horner decomposition of the matrix exponential and the fundamental system of solutions

This paper concerns the Fibonacci-Horner decomposition of the matrix powers An and the matrix exponential etA (A ∈ M(r; C), t ∈ R), which is derived fromthe combinatorial properties of the generalized Fibonacci sequences in the algebra of square matrices. More precisely, etA is expressed in a natural way in the so–called Fibonacci-Horner basis with the aid of the dynamical solution of the associated ordinary differential equation. Two simple processes for computing the dynamical solution and the fundamental system of solutions are given. The connection to VerdeStar’s approach is discussed. Moreover, an extension to the computation of f(A), where f is an analytic function is initiated. Finally, some illustrative examples are presented.

CAPUTO LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

This paper is devoted to the study of nonsequential linear fractional differential equations with constant coefficients involving the Caputo fractional derivatives. The Laplace transform is applied to obtain the general explicit solutions for the equations being studied in terms of Mittag-Leffler functions and generalized Wright functions. Conditions are given for obtaining linearly independent solutions which form a fundamental system of solutions. Some examples are presented.

Analysis of vibrations in large flexible hybrid systems

The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions makes it impossible to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameters together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different characteristics take place, are defined. The dependence of the vibration frequencies on the physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allows one to control the frequency spectrum, in which excitation of vibrations is possible.

Solution of a Problem of Free Vibrations of a Nonclosed Shell of Revolution under Conditions of its Singular Perturbation

We develop a variational method for solving the spectral problem of free vibrations of a shell of revolution nonclosed in the meridional direction. This method is equally efficient for both medium and small values of the relative thickness of the shell. Coordinate systems of functions are constructed with regard for the structure of formal asymptotic expansions of a fundamental system of solutions of the initial equations. As an example, we calculate frequencies and forms of vibrations of a circular cylindrical shell and show that the algorithm proposed for solving the considered problem guarantees the uniform convergence of solutions and their first three derivatives in the entire region of integration of the equations.

Spectral analysis and system of fundamental solutions for Timoshenko beams

We have found a unified method to analyse Timoshenko beams under various boundary conditions that occurred in practice. Explicit asymptotic expressions for the spectrum are obtained. Our method is very simple but effective because explicit formulas are obtained for the system of fundamental solutions, which are very useful for other purposes such as stability analysis. The eigenfunctions are also shown to form an orthogonal basis.

Four-Vertex Theorems, Sturm Theory and Lagrangian Singularities

We prove that the vertices of a curve γ⊂R n are critical points of the radius of the osculating hypersphere. Using Sturm theory, we give a new proof of the (2k+2)-vertex theorem for convex curves in the Euclidean space R 2k . We obtain a very practical formula to calculate the vertices of a curve in R n . We apply our formula and Sturm theory to calculate the number of vertices of the generalized ellipses in R 2k . Moreover, we explain the relations between vertices of curves in Euclidean n-space, singularities of caustics and Sturm theory (for the fundamental systems of solutions of disconjugate homogeneous linear differential operators L:C ∞(S 1)→C ∞(S 1)).

On stability of the Hill equation with damping

We consider the Hill equation with damping describing the parametric oscillations of a torsional pendulum excited by varying the moment of inertia of the rotating body. Using the method of a small parameter, we analytically calculate a fundamental system of solutions of this equation in the form of power series in the excitation amplitude e with accuracy O(e2) and verify conditions for its stability. In the first-order approximation in e, we prove that the resonance domain exists only if the excitation frequency Ω is sufficiently close to the double natural frequency of the pendulum; the corresponding equation of the stability boundary is obtained.

Differential Systems for Biorthogonal Polynomials Appearing in 2-Matrix Models and the Associated Riemann?Hilbert Problem

We consider biorthogonal polynomials that arise in the study of a generalization of two–matrix Hermitian models with two polynomial potentials V 1 (x), V 2 (y) of any degree, with arbitrary complex coefficients. Finite consecutive subsequences of biorthogonal polynomials (‘‘windows’’), of lengths equal to the degrees of the potentials V 1 and V 2 , satisfy systems of ODE’s with polynomial coefficients as well as PDE’s (deformation equations) with respect to the coefficients of the potentials and recursion relations connecting consecutive windows. A compatible sequence of fundamental systems of solutions is constructed for these equations. The (Stokes) sectorial asymptotics of these fundamental systems are derived through saddle-point integration and the Riemann-Hilbert problem characterizing the differential equations is deduced.

On Deficiency Indices of Singular Operators of Odd Order

One of the main problems of the spectral theory of linear ordinary differential operators is the study of their deficiency indices depending on the behavior of the coefficients of the corresponding differential expression ly . Such problems were investigated by Naimark [1], Fedoryuk [2], Sultanaev, and others (see the bibliography in [2]). As a rule, symmetric operators with real coefficients were studied. For the case of complex coefficients, Fedoryuk obtained asymptotic formulas for the fundamental system of solutions of the equation ly = 0 under the condition that the coefficients of the differential expression ly are polynomials subject to a rigid constraint on their degree. The paper [3] was devoted to the study of a symmetric differential operator of (2n+1)th order with complex-valued coefficients of the odd powers. Classes of differential operators with deficiency indices (m, m) , where 1 ≤ m ≤ 2n − 1 , were described. The papers [4, 5] were also devoted to the study of symmetric operators in the case of complex coefficients under regularity conditions of Titchmarsh– Levitan type; asymptotic formulas for the fundamental system of solutions of the equation ly = λy were obtained, and in certain special cases the deficiency indices of the corresponding minimum operator were found. It is well known that deficiency indices define the dimension of the solution spaces of the equations lu = iy and lu = −iy . The goal of the present paper is to study the deficiency indices of a class of differential operators generated on the semiaxis by a self-adjoint differential expression of odd order with complex-valued coefficients. We consider the equation

Higher-Order Differential Equations Having a Singularity in an Interior Point

Ordinary differential equations of an arbitrary order having a non-integrable singularity inside the interval are considered under additional matching conditions for solutions at the singular point. We construct special fundamental systems of solutions for this class of differential equations, study their asymptotical, analytical and structural properties and the behavior of the corresponding Stokes multipliers. These fundamental systems of solutions are used in spectral analysis of differential operators with singularities.

Duality, Biorthogonal Polynomials¶and Multi-Matrix Models

The statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized Christoffel--Darboux form constructed from sequences of biorthogonal polynomials. For measures involving exponentials of a pair of polynomials V_1, V_2 in two different variables, these kernels may be expressed in terms of finite dimensional ``windows'' spanned by finite subsequences having length equal to the degree of one or the other of the polynomials V_1, V_2. The vectors formed by such subsequences satisfy "dual pairs" of first order systems of linear differential equations with polynomial coefficients, having rank equal to one of the degrees of V_1 or V_2 and degree equal to the other. They also satisfy recursion relations connecting the consecutive windows, and deformation equations, determining how they change under variations in the coefficients of the polynomials V_1 and V_2. Viewed as overdetermined systems of linear difference-differential-deformation equations, these are shown to be compatible, and hence to admit simultaneous fundamental systems of solutions. The main result is the demonstration of a spectral duality property; namely, that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters.

Orthogonal Polynomials with Respect to a Differential Operator. Existence and Uniqueness

A new type of orthogonal polynomial connected with linear differential operators, intimately related with Sobolev orthogonal polynomials and Hermite-Pade polynomials, is introduced. We study the question of uniqueness of the sequence of orthogonal polynomials arising from this construction. As we show, this problem is related to the analytic properties of the fundamental system of solutions of the operator. The notion of T-system of Tchebyshev plays a key role in the analysis. Some examples of general classes of operators which produce a unique system of polynomials are given.

A problem of hartman and wintner: approximation for discrete perturbations

Consider the second-order self-adjoint difference equation Δ( c n Δ x n ) + ( a n + f n ) x n+1 = 0 as a perturbation of the eventually disconjugate difference equation Δ( c n Δ z n ) + a n z n+1 = 0, where c n ≠ 0. Asymptotic approximation for the fundamental system of solutions of the perturbed equation are expressed explicitly in terms of the coefficients and the principal (or recessive) solution of the unperturbed equation. In particular, the coefficient c n is allowed to be oscillatory, and we do not assume absolute summability conditions on f n . Results obtained are applied to a second-order Poincaré difference equation whose unperturbed equation has a double characteristic root. Nonlinear and nonhomogeneous perturbations are also considered. An example is given for illustration.

The solution of boundary-value problems of the longitudinal-transverse bending of orthotropic circular plates on a linearly elastic base

A technique is developed for solving boundary-value problems of non-axisymmetric longitudinal-transverse bending in thin cylindrical orthotropic linearly thermo-elastic annular and solid circular plates on a linearly elastic base in a classical setting. The given and unknown functions are represented by Fourier expansions. The solving system of fourth-order ordinary differential equations is of Bessel type. The solution of the homogeneous system is obtained by a technique developed previously ‡ ‡ KOROL, Ye. Z., Fundamental system of solutions of Nth order differential equations of Bessel type and their application in the Mechanics of deformable solid media. Part 1: Cylindrical functions of Nth order. Generalization of the Neumann—Weber—Schläfli formulae. Inst. Mekhaniki Mosk. Gos. Univ., Moscow, 1998. Dep. at VINITI 03.04.98, No. 990–98. (which generalises the Neumann—Weber—Schläfli technique [1–5]) for determining fundamental solutions in the form of generalized power series — higher-order cylindrical functions of first, second and higher kinds, based on the property of the continuous dependence of the solutions on the parameters. Particular solutions are determined by Lagrange's method (the variation of arbitrary constants). The results of numerical calculations are presented for plates with a hinge-supported outer contour in which the inner contour is loaded with distributed bending moments.

A Formula for the First Regularized Trace of a Perturbed Laplace–Beltrami Operator

The theory of regularized traces is well developed for problems induced by ordinary differential expressions. It was shown in the fundamental paper [1] that the derivation of trace formulas can be reduced to studying zeros of entire functions possessing a specific asymptotic structure related to the particular form of the fundamental solution system of the differential equation. The case of problems induced by partial differential operators is much harder. This is primarily due to the complicated structure of the spectrum. The asymptotic spectrum distribution was obtained in [2] for the operator −∆+V with smooth odd potential V on the sphere S. However, the results of [2] do not allow finding regularized trace formulas. Further substantial progress in this problem was made in [3], where a formula for the first regularized trace was obtained for the first time for the operator −∆ + V with odd smooth potential V . Later, the method of [4] was used in [5] for finding the first regularized trace for an odd potential with n = 2 and in [6] for deriving a regularized trace formula for the operator −∆+V on compact symmetric spaces of rank 1. However, the condition that the potential V is C∞ was substantially used in all these papers. In the present paper, we derive a formula for the first trace of the operator −∆ +V on S in the case of a finite smoothness of the potential V and without the assumption that V is odd. If the potential is odd, then formula (1) coincides with the corresponding formula in [5]. Let L0 = −∆, and let V be the operator of multiplication by a real smooth function in L2 (S), where S is the two-dimensional sphere. By L we denote the operator L0 +V . Further, let R0(z) = (L0 − zE) and R(z) = (L− zE)−1; let λn = n(n+ 1) and μ n , n = 0, 1, . . . , −n ≤ i ≤ n, be the eigenvalues of the operators L0 and L, respectively, and let f (i) n = Y (i) n (φ, θ) be the orthonormal spherical functions, that is, the eigenfunctions of the operator L0. The following assertion is the main result of the present paper.

Lp -Lq Decay Estimates fo the Solutions of Strictly Hyperbolic Equations of Second Order with Increasing in Time Coefficients

One of the most important questions in the theory of nonlinear wave equations is that for global existence of solutions. An essential tool is the Strichartz inequality for special solutions of the wave equation.In the last time different results were proved generalizing the classical one of Strichartz. In the present paper Lp – Lq estimates are proved for the solutions of strictly hyperbolic equations of second order with time dependent coefficients where these are unbounded at infinity. In the first step the WKB method is applied to the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. In a second step the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter.

Integral representation of the solutions to Heun′s biconfluentequation

First, we trace the genesis of the canonical form of Heun′s biconfluent equation. Second, we present a method which allows us to find an integral expression as a solution to our equation, and finally, using the properties of Meijer G‐functions, we give an integral representation of a fundamental system of solutions to the biconfluent equation.

On oscillatory solutions of fourth order ordinary differential equations

The paper deals with the oscillation of a differential equation L 4 y + P(t)L 2 y + Q(t)y ≡ 0 as well as with the structure of its fundamental system of solutions.

An analytical method for solving the problem of unshocked conical gas compression

The two-dimensional unsteady self-similar problem of unlimited unshocked conical compression of a gas is investigated. A solution is constructed in the form of a characteristic series in the domain bounded by a weak discontinuity and the sonic perturbation front. A recursion system of ordinary differential equations is obtained for the coefficients. A boundary-value problem corresponding to the next approximation is investigated in detail, a fundamental system of solutions is found by analytical methods and its asymptotic behaviour is investigated. Essentially independent solutions are determined and different methods are used to seek a solution of the inhomogeneous equation with the required asymptotic behaviour. An algorithm is constructed to compute gas flows induced by the motion of a piston taking the first terms of the series into consideration. The results are compared with those of computations carried out using the method of characteristics.