Fundamental System Of Solutions Research Articles

Jordan — Gauss successive elimination method in solution of pneumohydraulic floatation water purification problems

The emphasis of the article is a potential usage of Jordan - Gauss successive elimination method for reduction of a differential equation system (DES) to one differential equation and determining its solution providing all other solutions of DES. This approach is different from standard method of determining of eigen vectors and eigen values of matrix of DES, which are necessary to determine fundamental system of solutions of homogeneous differential equations of n-th order with constant coefficients denoted in norm form. In the article analytical solution is searched for in general form and can be used for other similar mathematical problems.

Determination of local inhomogeneity of the medium from the natural frequencies of string oscillations

The paper considers the problem of determining the local inhomogeneity of the medium from the natural frequencies of string oscillation. The inhomogeneity is modeled in three sections: in the first and third sections medium is homogeneous, and on average section the elastic characteristics are modeled by a quadratic function. This model is implemented using the conjugation conditions at boundary between media. It is shown that to identify the center of an inhomogeneity and determine its size, two natural frequencies are enough, and in the case of rigid fixing of both ends of the string, the solution of the problem is dual. The problem is solved by expanding the fundamental system of solutions into a power series in the variables x and λ. The estimates of the error of the method are given.

Construction and Floquet–Bloch analysis of analytically solvable Hill equations with smooth potentials

We construct analytically solvable Hill equations with smooth potentials and smooth solutions using a method of Wu and Shih. We then use Floquet–Bloch theory to analyze the band structure and find the Floquet–Bloch fundamental system of solutions of the resulting differential equation. Three examples of constructed potentials and their corresponding solutions are worked out.

Calculation of plate-beam systems by method of boundary elements

The approach to calculation of plate-beam systems by numerical and analytical method of boundary elements is proposed. The method consists in construction in analytical way a fundamental system of solutions, the Green’s function and external load vector for the problem under consideration. Small system of linear algebraic equations to be solved numerically is constructed to take into account the boundary conditions or contact conditions between separate modules.The series of plate-beam system calculations by methods of boundary elements and finite elements at different geometric parameters of plate was performed to assess the limits of applicability of developed theory. The comparison of 45 variants calculations by both methods is shown.

Some boundary-value problems for anisotropic quarter plane

To solve the mixed boundary-value problems of the anisotropic elasticity for the anisotropic quarter plane, a method based on the use of the space of generalized functions with slow growth properties was developed. The two-dimensional integral Fourier transform was used to construct the system of fundamental solutions for the anisotropic quarter plane in this space and a system of eight boundary integral relations was obtained, which allows one to reduce the mixed boundary-value problems for the anisotropic quarter plane directly to systems of singular integral equations with fixed singularities. The exact solutions of these systems were found by using the integral Mellin transform. The asymptotic behavior of solutions was investigated at the vertex of the quarter plane.

To the solution of the problem of bending of a cylindrical shell by the boundary elements method

The solution of the problem of the long cylindrical shell bending by a numerical and analytical boundary elements method is considered. The method is based on the analytical construction of a fundamental system of solutions and Green’s functions for the differential equation of the problem under consideration. This paper is devoted to the determination of these expressions. The semi-moment theory of the cylindrical shell calculation, proposed by V.Z. Vlasov, which for the problem under consideration leads to one eighth-order partial differential equation is used. The problem of the bending of a cylindrical shell is twodimensional, and in the numerical and analytical boundary elements method, plates and shells are considered as generalized one-dimensional modules, so the variational method of Kantorovich-Vlasov was applied to this equation to obtain an ordinary differential equation of the eighth order. Sixty-four expressions of all the fundamental functions of the problem are constructed, as well as an analytic expression for the Green’s function, which makes it possible to construct a load vector (without any restrictions on the nature of its application), and then proceed to the solution of boundary-value problems for the bending of long cylindrical shells under various boundary conditions.

The Calderon-Zygmund Operator and Its Relation to Asymptotic Estimates for Ordinary Differential Operators

We consider the problem of estimating of expressions of the kind Υ(λ)=supx∈[0,1]∣∣∫x0f(t)eiλtdt∣∣. In particular, for the case f∈Lp[0,1], p∈(1,2], we prove the estimate ∥Υ(λ)∥Lq(R)≤C∥f∥Lp for any q>p′, where 1/p+1/p′=1. The same estimate is proved for the space Lq(dμ), where dμ is an arbitrary Carleson measure in the upper half-plane C+. Also, we estimate more complex expressions of the kind Υ(λ) arising in study of asymptotics of the fundamental system of solutions for systems of the kind y′=By+A(x)y+C(x,λ)y with dimension n as |λ|→∞ in suitable sectors of the complex plane.

Monodromy representations of hypergeometric systems with respect to fundamental series solutions

We study the monodromy representation of the generalized hypergeometric differential equation and that of Lauricella's $F_C$ system of hypergeometric differential equations. We use fundamental systems of solutions expressed by the hypergeometric series. We express non-diagonal circuit matrices as reflections with respect to root vectors with all entries 1. We present a simple way to obtain circuit matrices.

RESEARCH ON THE TYMPANIC MEMBRANE FREE VIBRATION MODEL BASED ON THIN PLATE THEORY

Myringoplasty is one of the routine surgeries in the treatment of tympanic membrane (TM) perforation. Since the anatomic structure of the middle ear cannot be simulated in clinical treatment, the surgery is mainly directed by experiences. Based on the mechanical properties of TM in the anatomy, four hypotheses are presented and TM is simplified as a sectorial annulus plate with fixed boundary condition. This paper proposes a free vibration model of TM. Its natural frequencies of free vibration are obtained by variables separation method and Bessel function. The system of fundamental solutions of fourth-order homogeneous equations can be solved for the analytical expressions of corresponding natural vibration mode. The theoretical model is proved to be valid since the natural frequency of the model is consistent with the experimental data. The effect of geometric parameters and material parameters on TM natural frequency is subsequently discussed in the numerical examples. Especially, the diameter and thickness of TM will cause different natural frequency errors above 40%, while the Young’s modulus and density of TM cause errors below 15% as well.

A system of hypergeometric differential equations in two variables of rank 9

We study a hypergeometric function in two variables and a system of hypergeometric differential equations associated with this function. This is a regular holonomic system of rank [Formula: see text]. We give a fundamental system of solutions to this system in terms of this hypergeometric series. We give circuit matrices along generators of the fundamental group of the complement of its singular locus with respect to our fundamental system.

Inverse spectral problems for differential pencils with the turning point in the finite interval

In this paper, we establish the fundamental system of solutions (FSS) for differential pencils (1.1) by the Birkhoff’s method. We obtain the Weyl function which plays an important role in studying inverse problems. Then we give a formulation of the inverse problem and prove the uniqueness theorem.

Zeta-determinants of Sturm–Liouville operators with quadratic potentials at infinity

We consider Sturm–Liouville operators on a half line [a,∞),a>0, with potentials that are growing at most quadratically at infinity. Such operators arise naturally in the analysis of hyperbolic manifolds, or more generally manifolds with cusps. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solutions adapted to the boundary conditions. Despite being the natural objects in the context of hyperbolic geometry, spectral geometry of such operators has only recently been studied in the context of analytic torsion.

Spectral Analysis for Differential Systems with a Singularity

We consider the differential system \({y'-x^{-1}Ay-q(x)y=\rho By }\) with \({n\times n}\) matrices A,B, q(x), where A,B are constant, B is diagonal, A and q(x) are off-diagonal, \({q(\cdot)\in W^1_1[0,\infty)}\). Some distinguished fundamental system of solutions is constructed. Also, we discuss the inverse scattering problem and obtain the uniqueness result.

COMPORTAMIENTO ASINTÓTICO DE SOLUCIONES NO OSCILATORIAS DE CUARTO ORDEN DE ECUACIONES DIFERENCIALES LINEALES

This article deals with the asymptotic behavior of nonoscillatory solutions of fourth order linear differential equation where the coefficients are perturbations of linear constant coefficient equation. We define a change of variable and deduce that the new variable satisfies a third order nonlinear differential equation. We assume three hypotheses. The first hypothesis is related to the constant coefficients and set up that the characteristic polynomial associated with the fourth order linear equation has simple and real roots. The other two hypotheses are related to the behavior of the perturbation functions and establish asymptotic integral smallness conditions of the perturbations. Under these general hypotheses, we obtain four main results. The first two results are related to the application of a fixed point argument to prove that the nonlinear third order equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the nonlinear third order equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise the formulas for the asymptotic behavior of the linear fourth order differential equation. In addition, we present an example to show that the results introduced in this paper can be applied in situations where the assumptions of some classical theorems are not satisfied.

On n-arithmetical triangles constructed for polynomial coefficients

A fundamental solution system to a finite order linear difference equation with constant coefficients is considerably used during the investigation of the n-arithmetical triangles properties. The constructed fundamental solution system is explicitly expressed via the coefficients of the difference equation. We prove the symmetric and unimodal properties of the polynomial coefficients sequence and obtain the formulas for these coefficient calculations.

Affine Vertex Operator Algebras and Modular Linear Differential Equations

In this paper, we list all affine vertex operator algebras of positive integral levels whose dimensions of spaces of characters are at most 5 and show that a basis of the space of characters of each affine vertex operator algebra in the list gives a fundamental system of solutions of a modular linear differential equation. Further, we determine the dimensions of the spaces of characters of affine vertex operator algebras whose numbers of inequivalent simple modules are not exceeding 20.

Matrix Sturm–Liouville equation with a Bessel-type singularity on a finite interval

The matrix Sturm–Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. Special fundamental systems of solutions for this equation are constructed: analytic Bessel-type solutions with the prescribed behavior at the singular point and Birkhoff-type solutions with the known asymptotics for large values of the spectral parameter. The asymptotic formulas for Stokes multipliers, connecting these two fundamental systems of solutions, are derived. We also set boundary conditions and obtain asymptotic formulas for the spectral data (the eigenvalues and the weight matrices) of the boundary value problem. Our results will be useful in the theory of direct and inverse spectral problems.

An exact reanalysis method for structures with local modifications

An exact reanalysis method named Indirect Factorization Updating (IFU) is proposed for the structure with local modifications, including boundary modifications. The IFU method is developed from the Independent Coefficients (IC) and Sherman-Morrison-Woodbury (SMW) formula. Due to the local modifications, the modified equations are divided into two parts: balanced equations and unbalanced equations. Using the theory of the IC, extra constraints are enforced on the unbalanced Degree of Freedoms (DOFs), so that the fundamental solution system of the balanced equations can be obtained by using the SMW formula. Then, a unique solution is derived from the general solution of the balanced equations to satisfy the unbalanced equations. In order to use the SMW formula directly, the change of the stiffness matrix is converted to a low-rank form by using the Cholesky factorization of the initial stiffness matrix. The Cholesky factorization is indirectly updated according to the stiffness matrix of the balanced equations but not directly according to the modified equations. Three examples are presented to verify the performance of the suggested IFU method. The results show that the IFU can efficiently obtain the exact solution of the structures with boundary modifications.

On the disconjugacy property of an equation on a graph

Under study is the disconjugacy theory of forth order equations on a geometric graph. The definition of disconjugacy is given in terms of a special fundamental system of solutions to a homogeneous equation. We establish some connections between the disconjugacy property and the positivity of the Green’s functions for several classes of boundary value problems for forth order equation on a graph. We also state the maximum principle for a forth order equation on a graph and prove some properties of differential inequalities.

Asymptotics of the General Solution of Linear Singularly Perturbed Systems of Higher-Order Differential Equations with Degeneration in the Case of Multiple Spectrum of the Limiting Matrix Pencil

We construct an asymptotic expansion for the fundamental system of solutions of a linear singularly perturbed system of m-order differential equations with degenerate principal matrix at the higher-order derivatives. The case where the corresponding characteristic polynomial has a multiple spectrum is investigated. It is proved that, in this case, the asymptotic expansions are constructed in the fractional powers of the small parameter. The recurrence formulas for the coefficients of these expansions are obtained.