Theory Of Linear Differential Equations Research Articles

Solvability of Linear Differential Equations

We propose a new approach to the solvability of ordinary as well as partial differential equations in the theory of linear differential equations and also in the theory of integral equations.

CONSTRUCTION OF THE SOLUTION OF THE SYSTEM OF SINGULARLY PERTURBATED DIFFERENTIAL EQUATIONS WITH A DIFFERENTIAL TURNING POINT

In this work, a system of singularly perturbed differential equations of the 4th order with a small parameter at the highest derivative and a turning point is considered. The turning point x=0 is located in the end of the segment [-l;0] under consideration. The coefficients of a given matrix are infinitely differentiable functions on a given interval. The goal is to construct a uniform solution asymptotics for a system of singularly perturbed differential equations with a differential turning point on the segment [-l;0]. In this case, the spectrum of the limit operator contains multiple and identically zero elements. The uniform asymptotic solution is constructed by the method of essential-singular functions using the Airy functions [0;l] and their derivatives. Studies have shown that the Apparatus of Airy functions and model equations is quite effective for constructing a uniform asymptotic solution of problems with turning points. It was also established that in order to select these functions along with the independent variable it is necessary to introduce a new variable t of the segment [0;l]. As a result, an extended problem will be obtained, which will make it possible to formally find the solution in the form of series with a small parameter. The constructions of the solutions are obtained by sequentially solving the systems of iterative equations, which at a certain step of the iterations make it possible to determine all the components of the sought-after vector functions with an accuracy of two scalar factors that form an arbitrary vector. The regularizing function Bi(t) is chosen in such a way that the function increases indefinitely when t approaches the infinity. The conducted studies showed that for any ratio of the signs of the matrix coefficient near which the inflection point is located, it is not possible to write down a smooth solution in the form of a single analytical expression. The solution of a degenerate vector equation in the general case has a discontinuity of the second kind at the inflection point, so it cannot be used explicitly to construct the asymptotic solution of this problem. For this purpose, a technique from the classical theory of linear differential equations was applied and solutions for a degenerate equation of the 2nd order were obtained.

On integer parts of the reciprocal remainders of some sums

We present generalizations of some results on the integer parts of the reciprocal remainders of the zeta function zeta (s) with s=2 and s=3, and a very short and elegant proof of a recent result on the integer parts of the reciprocal remainders of the series zeta (3). We also give some historical and theoretical remarks to problems of this type, conduct some analyses, and make some connections with the theory of linear difference equations with constant coefficients.

Algebraic groups as difference Galois groups of linear differential equations

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field C ( x ) with derivation d d x and endomorphism f ( x ) ↦ f ( x + 1 ) . Our main result is that every linear algebraic group , considered as a difference algebraic group, occurs as the difference Galois group of some linear differential equation over C ( x ) .

Mean value theorems for polyharmonic functions

The problem of characterization of polyharmonic functions by mean value expressions is analyzed. A sufficient condition for the biharmonic function is formulated and proved in the paper. Mean value theorems and their operators have various applications in function theory (approximation of functions, description of functional spaces) and in the qualitative theory of linear differential equations with partial derivatives (boundary properties, elimination of features, differential properties of solutions, etc.). The main properties of polyharmonic functions, in particular, properties with mean value are investigated in the work. The results of L. Salcman and V. Volchkov, which generalize the classical mean value theorems, are analyzed. These results also reveal deep connections between complex analysis, the theory of linear differential equations with partial derivatives, harmonic analysis, integral geometry and the theory of special functions. The new directions of research in the theory of polyharmonic functions, differential equations, as well as in computational mathematics are established with the mean value theorems (algorithms "walk on spheres"\ , application of the Monte Carlo method). Particular attention is paid to the case of biharmonic functions. It is found that when performing the equality with the mean value on the circle from the set on the complex plane, the function of class $C^2$ on the specified set satisfies the Laplace equation of the second order, i.e. the function is polyharmonic of the second order. By using a smooth function with a compact support, the required function is obtained as a weak solution of the corresponding Laplace equation. In the future, this result can be used to establish a sufficient condition of biharmonicity of a twice continuously differentiated function on a plane that satisfies the above-mentioned relation only for a pair of positive values of $r_1$, $r_2$. Thus, it is possible to formulate a new theorem on two radii.

On existence of quasi-Strebel structures for meromorphic $k$-differentials

In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order $k\\ge 4$ satisfying certain natural conditions at its singular points admits such a structure. The case of differentials of odd order is quite different and our existence result involves some arithmetic conditions. We discuss the set of quasi-Stebel structures associated to a given differential and introduce the subclass of positive $k$-differentials. Finally, we provide a family of examples of positive rational differentials and explain their connection with the classical Heine–Stieltjes theory of linear differential equations with polynomial coefficients.

MODAL ANALYSIS OF REINFORCED CONCRETE AND FIBER CONCRETE BEAMS

Analytical, experimental and numerical results of determination of natural frequencies and forms of oscillations of reinforced concrete and fiber concrete beams are given. Modern analytical, numerical and experimental methods of studying the dynamics of reinforced concrete and fiber concrete beams are analyzed. The problem of determining the natural frequencies and forms of oscillations of reinforced concrete and fiber concrete beams at the initial modulus of elasticity and taking into account the nonlinear diagram of deformation of materials is solved analytically. Computer modeling of the considered constructions in four software complexes is done and the technique of their modal analysis on the basis of the finite element method is developed. Experimental researches of free oscillations of the considered designs and the comparative analysis of all received results are carried out. It is established that all involved complexes determine the imaginary frequency and imaginary form of oscillations. The frequency spectrum calculated by the finite element method is approximately 4% lower than that calculated analytically; the results of the calculation in SOFiSTiK differ by 2% from the results obtained in the PC LIRA; the discrepancy with the experimental data reaches 20%, and all frequencies calculated experimentally, greater than the frequencies calculated analytically or by the finite element method. This rather significant discrepancy is explained, according to the authors, by the incorrectness of the used dynamic model of the reinforced beam. The classical dynamics of structures is known to be based on the theory of linear differential equations, and the oscillations of structures are considered in relation to the unstressed initial state. It is obvious that in the study of free and forced oscillations of reinforced concrete building structures such an approach is unsuitable because they are physically nonlinear systems. The concept of determining the nonlinear terms of these equations is practically not studied. Numerous experimental researches and computer modeling for the purpose of qualitative and quantitative detection of all factors influencing a spectrum of natural frequencies of fluctuations are necessary here.

An Empirical Study on The Relationship Between Social Responsibility and Sustainable Development of Enterprises Based on The Stability Theory of Linear Differential Equations

An Empirical Study on The Relationship Between Social Responsibility and Sustainable Development of Enterprises Based on The Stability Theory of Linear Differential Equations

ON THE CONSTRUCTION OF A FUNDAMENTAL SYSTEM OF SOLUTIONS OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS OF AN ARBITRARY ORDER

The paper proposes a method of presentation topics «On the construction of a fundamental system of solutions of a linear homogeneous differential equation with constant coefficients of an arbitrary order». In the traditional presentation of this topic in the case when the characteristic equation has complex roots, the particular solutions of the equation corresponding to them are constructed by applying the elements of complex analysis. In consequence of that, for students in the field, whose training programs included the theory of linear differential equations with constant coefficients and at the same time does not include the study of the theory of complex analysis, types of private solving the equation in this case is given without substantiation, or as a known fact, only for this case, previously issued elements complex analysis. Offered in the presentation technique differs from the traditional presentation of the topic in that it partial solutions scheme for constructing fundamental system of homogeneous linear equation with constant coefficients of arbitrary order is based only on the basis of the properties of the differential form corresponding to the left side of the equation, without using the elements of the theory of complex analysis.

A differential Galois approach to path integrals

We point out the relevance of the differential Galois theory of linear differential equations for the exact semiclassical computations in path integrals in quantum mechanics. The main tool will be a necessary condition for complete integrability of classical Hamiltonian systems obtained by Ramis and myself [Morales-Ruiz and Ramis, Methods Appl. Anal. 8, 33–96 (2001); see also Morales-Ruiz, in Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Modern Birkhäuser Classics (Springer, Basel, 1999)]; if a finite dimensional complex analytical Hamiltonian system is completely integrable with meromorphic first integrals, then the identity component of the Galois group of the variational equation around any integral curve must be abelian. A corollary of this result is that, for finite dimensional integrable Hamiltonian systems, the semiclassical approach is computable in a closed form in the framework of the differential Galois theory. This explains in a very precise way the success of quantum semiclassical computations for integrable Hamiltonian systems.

Construction of the Fuchs equation with four given finite critical points and a given reducible monodromy group in the resonance case

One inverse problem of the analytic theory of linear differential equations is considered. Namely, the completely integrable Fuchs equation with four given finite critical points and a given reducible monodromy group of rank 2 on the complex projective line is constructed. Reducibility of the monodromy group of rank 2 means that 2×2-monodromy matrices (the generators of the monodromy group) can be simultaneously reduced by a linear nonsingular transformation to an upper triangular form. In so doing we study the case when the eigenvalue ξj of the diagonal matrix of the monodromy formal exponent at a corresponding Fuchs critical point is equal to an integer different from zero (resonance takes place).

On a Problem Arising in Application of the Re-Quantization Method to Construct Asymptotics of Solutions to Linear Differential Equations with Holomorphic Coefficients at Infinity

The re-quantization method—one of the resurgent analysis methods of current importance—is developed in this study. It is widely used in the analytical theory of linear differential equations. With the help of the re-quantization method, the problem of constructing the asymptotics of the inverse Laplace–Borel transform is solved for a particular type of functions with holomorphic coefficients that exponentially grow at zero. Two examples of constructing the uniform asymptotics at infinity for the second- and forth-order differential equations with the help of the re-quantization method and the result obtained in this study are considered.

Almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks of neutral type with time delays in the leakage term

ABSTRACTIn this paper, we propose a class of quaternion-valued shunting inhibitory cellular neural networks of neutral type with time delays in the leakage term. In order to overcome the difficulty of the non-commutativity of quaternion multiplication, we first decompose the system under consideration into four real-valued systems. Then by using the exponential dichotomy theory of linear differential equations and Banach's fixed point theorem, we establish sufficient conditions to ensure the existence and global exponential stability of almost periodic solutions for this class of neural networks. Finally, a numerical example is given to demonstrate the effectiveness of the obtained result.

A necessary and sufficient condition for local asymptotic stability of a class of nonlinear systems in the critical case

By the theory of linear differential equations, a system described by a chain of integrators with a linear feedback is globally asymptotically stable if and only if the characteristic polynomial is Hurwitz, which implies that all the coefficients in the linear feedback equation are negative. However, negative coefficients may not guarantee the local asymptotic stability of the linear system. In this paper, we reveal that, by monotonizing the powers of the integrators, the strict negativity of the feedback coefficients is not only necessary but also sufficient for the local asymptotic stability of the system. A dual result is also obtained for the dual power integrator systems.

Global Convergence on Asymptotically Almost Periodic SICNNs with Nonlinear Decay Functions

In this paper, we propose a class of asymptotically almost periodic shunting inhibitory cellular neural networks with mixed delays and nonlinear decay functions. Without using the exponential dichotomy theory of linear differential equations, a set of easily verifiable sufficient conditions are established to show that every solution of the considered system is asymptotically almost periodic, and converges to a same almost periodic function as $$t\rightarrow +\infty $$ , which improve and supplement some previously known researches. Finally, a numerical example is given to demonstrate the effectiveness of the obtained results.

Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation

We present some interesting facts connected with the following second-order difference equation: xn+2−qnxn=fn,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x_{n+2}-q_{n}x_{n}=f_{n},\\quad n\\in \\mathbb{N}_{0}, $$\\end{document} where (q_{n})_{ninmathbb{N}_{0}} and (f_{n})_{ninmathbb {N}_{0}} are given sequences of numbers. We give some sufficient conditions for the existence of a unique bounded solution to the difference equation and present an elegant proof based on a combination of theory of linear difference equations and the Banach fixed point theorem. We also deal with the equation by using theory of solvability of difference equations. A global convergence result of solutions to a linear first-order difference equation is given. Some comments on an abstract version of the linear first-order difference equation are also given.

Eringen’s small length scale coefficient for vibration of axially loaded nonlocal Euler beams with elastic end restraints

AbstractThis paper presents the calibration of Eringen’s small length scale coefficient

Unbounded rough drivers

We propose a theory of linear differential equations driven by unbounded operator-valued rough signals. As an application we consider rough linear transport equations and more general linear hyperbolic symmetric systems of equations driven by time-dependent vector fields which are only distributions in the time direction.

DIFFERENCE ALGEBRAIC RELATIONS AMONG SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups, and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski, and Y. Peterzil.

Generalized linear cellular automata in groups and difference Galois theory

Generalized non-autonomous linear cellular automata are systems of linear difference equations with many variables. They can be easily transformed into convolution equations in discrete groups. We study these equations from the standpoint of the Galois theory of linear difference equations and discrete Fourier transform.