Fundamental System Of Solutions Research Articles

Other fundamental systems of solutions of the differential equation (D 2 – 2αD + α 2 + β 2) n y = 0, β ≠ 0

Abstract This paper generalizes the results of the first part of the paper [Fecenko, J.—Diekema, E.: On the linear (in)dependence of sequences of derivatives of the functions xn sin x and xn cos x, http://arxiv.org/abs/2305.11184]. The main goal is to prove that the sequence of functions f(x), Df(x), …, D 2n–1 f(x), where f(x) is x n−1e αx sin βx or x n−1 e αx cos βx for β ≠ 0, forms a fundamental system of solutions of the differential equation (D 2 – 2αD + α 2 + β 2) n y = 0. Or more generally, that the sequence of functions Dkf(x), D k+1 f(x), …, D 2n+k–1 f(x), k ∈ ℕ also forms a fundamental system of solutions of the aforementioned differential equation. The problem is solved using a suitable transformation of the Wronskian matrix, by solving a nonlinear differential equation and then calculating the determinant of the modified Wronskian matrix by applying Laplace’s generalized expansion.

On the positive invertibility of a differential operator on a graph

In this article, we study a fourth-order differential operator L λ on a graph, depending on a real parameter λ. The operator L λ is connected with a model of the Euler–Bernoulli beam system. Our consideration is devoted to the study of the set of values of the spectral parameter λ for which the operator L λ is inverse positive. It is shown that the operator L λ is positively invertible if and only if there exists a fundamental system of solutions to the corresponding homogeneous equation, consisting of functions that are positive on the graph. We establish necessary and sufficient conditions for the differential operator L λ to be inverse-positive for all positive values of the spectral parameter less than the lowest eigenvalue of the differential operator L 0 corresponding to λ = 0 . In this way, we establish the positivity of eigenvalues and formulate comparison theorems for eigenvalues of the spectral problem. Next, we formulate maximum principles for a fourth-order differential inequalities on a graph.

Graceful bases in solution spaces of differential and difference equations

We construct fundamental systems of solutions to linear ordinary differential equations, linear difference equations, and systems of partial differential equations whose elements remain linearly independent for all values of algebraically independent symbolic parameters.

Formation of convective roll patterns over a longitudinal throughflow in an active inhomogeneous porous medium with weak clogging

The convective stability of a plane-parallel longitudinal throughflow in a system consisting of two porous sublayers with distinct permeability is studied taking into account the weak pore clogging effect and given a finite difference in concentration between the upper and lower boundaries of the system. A system of differential equations for convection is derived within the framework of the continuum approach and the nonlinear MIM model. A linear stability analysis of the basic longitudinal flow is carried out. The boundary value problem is solved using the numerical technique for constructing a fundamental system of solutions. The neutral stability curves of the basic flow with respect to the disturbances in the form of rolls with different wavelength are obtained. For the absence of pore clogging, the convective stability maps of the critical solutal Rayleigh-Darcy number versus the permeability ratio for the initial uncontaminated sublayers, as well as the plots of the critical disturbance frequency and wave number versus the permeability ratio for some values of the Peclet number, are constructed. It is shown that these results have symmetry with respect to the solution for equal initial permeability of sublayers. The clogging effect associated with the solute adsorption and desorption in a porous medium is studied in the two-layered systems with the values 0.1 and 10 of the ratio of initial permeabilities of the sublayers; convection localization occurs in distinct sublayers. It was found that clogging increases the stability of the longitudinal basic flow relative to solutal inhomogeneity and breaks the symmetry of solutions with respect to the case when the two-layered system is reduced to a single layer with the same filtration properties of the porous medium throughout its volume. The influence of sorption effects on the critical parameters of oscillatory convection is investigated by plotting the critical solutal Rayleigh-Darcy number, disturbance frequency and wave number against the adsorption coefficient at fixed values of the desorption coefficient. The isolines of the stream function are constructed to analyze convective flow patterns. It was established that there is a weak relationship between the typical size of local convective roll patterns and the clogging parameters. It is shown that the sorption processes, which can lead to clogging, have a significant impact on the velocity of convective rolls moving along the two-layered system at a slight variation in their sizes.

On infinite spectra of oscillation exponents of third order linear differential equations

The research topic of this work is at the junction of the theory of Lyapunov exponents and oscillation theory. In this paper, we study the spectra (i.e., the sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential equations with coefficients continuous on the positive semi-axis. In the first part of the paper, we build a third order linear differential equation with the following property: the spectra of all upper and lower strong and weak exponents of oscillation of strict and non-strict signs, zeros, roots and hyper roots contain a countable set of different essential values, both metrically and topologically. Moreover, all these values are implemented on the same sequence of solutions of the constructed equation, that is, for each solution from this sequence, all of the oscillation exponents coincide with each other. In the construction of the indicated equation and in the proof of the required results, we used analytical methods of the qualitative theory of differential equations and methods from the theory of perturbations of solutions of linear differential equations, in particular, the author’s technique for controlling the fundamental system of solutions of such equations in one special case. In the second part of the paper, the existence of a third order linear differential equation with continuum spectra of the oscillation exponents is established, wherein the spectra of all oscillation exponents fill the same segment of the number axis with predetermined arbitrary positive incommensurable ends. It turned out that for each solution of the constructed differential equation, all of the oscillation exponents coincide with each other. The obtained results are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential equations.

On the completeness of root function system of the Dirac operator with two‐point boundary conditions

AbstractThe paper is concerned with the completeness property of root functions of the Dirac operator with summable complex‐valued potential and nonregular boundary conditions. We also obtain an explicit form for the fundamental solution system of the considered operator.

On the Asymptotic of Solutions of Odd-Order Two-Term Differential Equations

This work is devoted to the development of methods for constructing asymptotic formulas as x→∞ of a fundamental system of solutions of linear differential equations generated by a symmetric two-term differential expression of odd order. The coefficients of the differential expression belong to classes of functions that allow oscillation (for example, those that do not satisfy the classical Titchmarsh–Levitan regularity conditions). As a model equation, the fifth-order equation i2p(x)y‴″+p(x)y″‴+q(x)y=λy, along with various behaviors of coefficients p(x),q(x), is investigated. New asymptotic formulas are obtained for the case when the function h(x)=−1+p−1/2(x)∉L1[1,∞) significantly influences the asymptotics of solutions to the equation. The case when the equation contains a nontrivial bifurcation parameter is studied.

ЕКВІВАВАЛЕНТНІСТЬ ФУНДАМЕНТАЛЬНОГО СПЛАЙНУ ТА ФУНКЦІЇ ГРІНА ДЛЯ ПОБУДОВИ ТОЧНОГО СКІНЧЕННОВИМІРНОГО АНАЛОГА КРАЙОВОЇ ЗАДАЧІ ДЛЯ ЗВИЧАЙНОГО ДИФЕРЕНЦІАЛЬНОГО РІВНЯННЯ 4-ГО ПОРЯДКУ

The author considers a problem with the main and natural boundary conditions on an interval. A new method for constructing an exact discrete analog of the problem is proposed. The method deals with the projection of the differential equation on local splines, formed by the fundamental system of solutions of the Cauchy problems for the homogeneous equation. A system of linear algebraic equations with a 5-diagonal matrix is obtained for the values of the exact solutions of the original problem at the points of a uniform grid. To implement an exact analog, we recommend using high-order accuracy schemes which are formed by partial sums of series in even powers of the grid step for solving the Cauchy problems. Keywords: boundary-value problem, ordinary differential equation of the 4th order, Cauchy problem, Wronskian, local spline, superposition of solutions, exact discrete analog, system of linear algebraic equations, 5-diagonal matrix.

Principal fundamental system of solutions, the Hartman-Wintner problem and correct solvability of the general Sturm-Liouville equation

Principal fundamental system of solutions, the Hartman-Wintner problem and correct solvability of the general Sturm-Liouville equation

Embedding of Unimodular Row Vectors

In this paper, we mainly study the embedding problem of unimodular row vectors, focusing on avoiding the identification of polynomial zeros. We investigate the existence of the minimal syzygy module of the ZLP polynomial matrix and demonstrate that the minimal syzygy module has structural properties that are similar to the fundamental solution system of homogeneous linear equations found in linear algebra. Finally, we provide several embedding methods for unimodular vectors in certain cases.

GKZ hypergeometric systems of the three-loop vacuum Feynman integrals

We present the Gel’fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of the Feynman integrals of the three-loop vacuum diagrams with arbitrary masses, basing on Mellin-Barnes representations and Miller’s transformation. The codimension of derived GKZ hypergeometric systems equals the number of independent dimensionless ratios among the virtual masses squared. Through GKZ hypergeometric systems, the analytical hypergeometric series solutions can be obtained in neighborhoods of origin including infinity. The linear independent hypergeometric series solutions whose convergent regions have non-empty intersection can constitute a fundamental solution system in a proper subset of the whole parameter space. The analytical expression of the vacuum integral can be formulated as a linear combination of the corresponding fundamental solution system in certain convergent region.

GKZ-system of the 2-loop self energy with 4 propagators

Applying the system of linear partial differential equations derived from the Mellin–Barnes representation and the Miller transformation, we present the GKZ-system of the Feynman integral of the 2-loop self energy diagram with 4 propagators. The codimension of the derived GKZ-system equals the number of independent dimensionless ratios among the external momentum squared and virtual mass squared. In total 536 hypergeometric functions are obtained in the neighborhoods of the origin and infinity, in which 30 linearly independent hypergeometric functions whose convergent regions have nonempty intersection constitute a fundamental solution system in a proper subset of the whole parameter space.

On essential values of Sergeev's frequencies and exponents of oscillation for solutions of a third-order linear differential periodic equation

In this paper, we study various types of Sergeev's frequencies and exponents of oscillation for solutions of linear homogeneous differential equations with continuous bounded coefficients. For any preassigned natural number $N$, a periodic third-order linear differential equation is constructively built in this paper, which has the property that its upper and lower Sergeev frequency spectra of strict signs, zeros and roots, as well as the spectra of all upper and lower strong and weak oscillation indices of strict and non-strict signs, zeros, roots and hyperroots contain the same set, consisting of $N$ different essential values, both metrically and topologically. Moreover, all these values are implemented on the same set of solutions of the constructed equation, that is, for each solution from this set, all the frequencies listed above and the oscillation exponents coincide with each other. When constructing the indicated equation and proving the required results, analytical methods of the qualitative theory of differential equations were used, in particular, methods of the theory of perturbations of solutions of linear differential equations, as well as the author's technique for controlling the fundamental system of solutions of such equations in one particular case.

A hypersingular integro-differential equation with linear functions in coefficients

In this paper, we investigated a new linear integro-differential equation of arbitrary order given on the closed curve located on a complex plane. The coefficients of the equation are variables and have a special form. The characteristic feature is the presence of linear functions in the coefficients. The equation is reduced to the consecutive solution of a Riemann boundary value problem on an initial curve and two linear differential equations. Differential equations are solved for analytic functions in areas into which the initial curve separates a complex plane. The corresponding fundamental systems of solutions are found, after that the arbitrary-constant variation method is applied. To achieve the analyticity of the obtained solutions the restrictions are imposed. All the arising conditions of resolvability of the input equation are written down explicitly, and if they are carried out then the solution is written in an explicit form. We represent the example demonstrating the existence of the cases when all conditions of resolvability are satisfied.

Feynman integrals of Grassmannians

We embed Feynman integrals in the subvarieties of Grassmannians through homogenization of the integrands in projective space, then obtain Gel'fand-Kapranov-Zelevinsky systems satisfied by those scalar integrals. The Feynman integral can be written as linear combinations of the hypergeometric functions of a fundamental solution system in neighborhoods of regular singularities of the Gel'fand-Kapranov-Zelevinsky system, whose linear combination coefficients are determined by the integral on an ordinary point or some regular singularities. Taking some Feynman diagrams as examples, we elucidate in detail how to obtain the fundamental solution systems of Feynman integrals in neighborhoods of regular singularities. Furthermore we also present the parametric representations of Feynman integrals of the two-loop self-energy diagrams that are convenient to embed in the subvarieties of Grassmannians.

Hypersingular integro-differential equation with recurrent relations in coefficients

A new hypersingular integro-differential equation is considered on a closed curve located on the complex plane. The equation refers to linear equations with variable coefficients of a special kind. A characteristic feature is the presence of constant multipliers in the coefficients, given by some recurrent relations. The equation is first reduced to solving the Riemann boundary value problem on the original curve. A class of functions is established for solving the Riemann problem, after which this problem is solved. Next, it is necessary to solve two linear differential equations of arbitrary order for analytical functions in two different regions of the complex plane. The corresponding fundamental systems of solutions are found, after which the method of variation of arbitrary constants is used for the solution. Restrictions are imposed on the obtained solutions of differential equations in order to achieve their analyticity. As a result, all the resulting solvability conditions of the original equation are written explicitly. The solution of the original equation after solving the differential equations can be written explicitly. Solved the example.

$L^p-L^q$ estimates for wave equations with strong time-dependent oscillations

In this paper we consider the following Cauchy problem for the linear wave equation with time-dependent propagation speed: \begin{align} \label{Abstract.Cauchy} \tag{$\star$} \begin{cases} u_{tt}-\lambda^2(t)\omega^2(t)\Delta u=0, & (t,x)\in[0,\infty)\times \mathbb{R}^n, \\ u(0,x)=u_0(x), ~~ u_t(0,x)=u_1(x), & x\in\mathbb{R}^n, \end{cases} \end{align} where $\lambda=\lambda(t)$ is an increasing shape function and $\omega=\omega(t)$ is a bounded oscillating function which has very fast oscillations. The goal is to prove $L^p-L^q$ estimates on the conjugate line for Sobolev solutions of the Cauchy problem \eqref{Abstract.Cauchy} in the case that $\omega=\omega(t)$ has very fast oscillations. Basically, we apply the WKB analysis for the construction of a fundamental system of solutions for ordinary differential equations depending on a parameter. Then, the method of stationary phase yields the asymptotical behaviour of Fourier multipliers with nonstandard phase functions depending on a parameter. This research continues the research of the paper [17].

On Dirac systems with multi-eigenparameter-dependent transmission conditions

In this work, we investigate a Dirac system which has discontinuities at finite interior points and contains eigenparameter in both boundary and transmission conditions. By defining a suitable Hilbert space H associated with the problem, we generate a self-adjoint operator T such that the eigenvalues of the considered problem coincide with those of T. We construct the fundamental system of solutions of the problem and get the asymptotic formulas for the fundamental solutions, eigenvalues and eigen-vectorfunctions. Also, we examine the asymptotic behaviour for the norm of eigenvectors corresponding to the operator T. We construct Green?s matrix, and derive the resolvent of the operator T in terms of Green?s matrix. Finally, we estimate the norm of resolvent of the operator T. In the special case, when our problem has no eigenparameter in both boundary and transmission conditions, the obtained results coincide with the corresponding results in Tharwat (Boundary Value Problems, DOI:10.1186/s13661-015-0515-1, 2016).

Спектральные свойства дифференциального оператора четного порядка с разрывной весовой функцией

This article proposes a new method for studying differential operators with a discontinuous weight function. It is assumed that the potential of the operator is a piecewise smooth function on the segment of the operator definition. The conditions of «conjugation» at the point of discontinuity of the weight function are required. The spectral properties of a differential operator defined on a finite segment with separated boundary conditions are studied. The asymptotics of the fundamental system of solutions of the corresponding differential equation for large values of the spectral parameter is obtained. With the help of this asymptotics, the «conjugation» conditions of the differential operator in question are studied. The boundary conditions of the operator under study are investigated. As a result, we obtain an equation for the eigenvalues of the operator, which is an entire function. The indicator diagram of the eigenvalue equation, which is a regular polygon, is studied. In various sectors of the indicator diagram, the asymptotics of the eigenvalues of the investigated differential operator is found. The formula for the first regularized trace of this operator by using the found asymptotics of the eigenvalues by the Lidsky–Sadovnichy method is obtained. In the case of the passage to the limit, the resulting formula leads to the trace formula for the classical operator with a smooth potential and constant weight function.

New results for the non-oscillatory asymptotic behavior of high order differential equations of Poincaré type

<abstract><p>This paper discusses the study of asymptotic behavior of non-oscillatory solutions for high order differential equations of Poincaré type. We present two new and weaker hypotheses on the coefficients, which implies a well posedness result and a characterization of asymptotic behavior for the solution of the Poincaré equation. In our discussion, we use the scalar method: we define a change of variable to reduce the order of the Poincaré equation and thus demonstrate that a new variable can satisfies a nonlinear differential equation; we apply the method of variation of parameters and the Banach fixed-point theorem to obtain the well posedness and asymptotic behavior of the non-linear equation; and we establish the existence of a fundamental system of solutions and formulas for the asymptotic behavior of the Poincaré type equation by rewriting the results in terms of the original variable. Moreover we present an example to show that the results introduced in this paper can be used in class of functions where classical theorems fail to be applied.</p></abstract>