Analytic Theory Of Differential Equations Research Articles

Analytic in the unit polydisc functions of bounded L-index in direction

The concept of bounded $L$-index in a direction $\mathbf{b}=(b_1,\ldots,b_n)\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ is generalized for a class of analytic functions in the unit polydisc, where $L$ is some continuous function such that for every $z=(z_1,\ldots,z_n)\in\mathbb{D}^n$ one has $L(z)>\beta\max_{1\le j\le n}\frac{|b_j|}{1-|z_j|},$ $\beta=\mathrm{const}>1,$ $\mathbb{D}^n$ is the unit polydisc, i.e. $\mathbb{D}^n=\{z\in\mathbb{C}^n: |z_j|\le 1, j\in\{1,\ldots,n\}\}.$ For functions from this class we obtain sufficient and necessary conditions providing boundedness of $L$-index in the direction. They describe local behavior of maximum modulus of derivatives for the analytic function $F$ on every slice circle $\{z+t\mathbf{b}: |t|=r/L(z)\}$ by their values at the center of the circle, where $t\in\mathbb{C}.$ Other criterion describes similar local behavior of the minimum modulus via the maximum modulus for these functions. We proved an analog of the logarithmic criterion desribing estimate of logarithmic derivative outside some exceptional set by the function $L$. The set is generated by the union of all slice discs $\{z^0+t\mathbf{b}: |t|\le r/L(z^0)\}$, where $z^0$ is a zero point of the function $F$. The analog also indicates the zero distribution of the function $F$ is uniform over all slice discs. In one-dimensional case, the assertion has many applications to analytic theory of differential equations and infinite products, i.e. the Blaschke product, Naftalevich-Tsuji product. Analog of Hayman's Theorem is also deduced for the analytic functions in the unit polydisc. It indicates that in the definition of bounded $L$-index in direction it is possible to remove the factorials in the denominators. This allows to investigate properties of analytic solutions of directional differential equations.

The Jacobi - Poisson Method of Building First Integrals for System of Ordinary Differential Equations

In this paper, the Jacobi-Poisson method of building first integrals by known integral manifolds for Hamiltonian differential systems is developed. The existence theorem of first integrals in the form of Poisson brackets from known integral manifolds for Hamiltonian systems is proved. The statement of finding additional first integrals by known first integrals and integral manifolds for Hamiltonian systems is given. In the case of polynomial Hamiltonian systems, we use the concept of partial integrals and concretize ours results. The Jacobi - Poisson method of construction first integrals in the form of Poisson brackets from known integral characteristics (integral manifolds, partial integrals, and first integrals) for general ordinary differential systems is proposed. Statements of finding first integrals for an ordinary differential system by integral characteristics of the auxiliary Hamiltonian system are obtained. The generalized Poisson theorem of construction first integrals is indicated. And, the question of the existence of partial integrals and first integrals in the form of Poisson brackets for ordinary polynomial differential systems is investigated. The results obtained in this paper can be used in the analytical theory of differential equations and in the analytical mechanics.

INDEX BOUNDEDNESS OF ANALYTIC SOLUTIONS OF A DIFFERENTIAL EQUATION GENERATED BY MATHEMATICAL MODEL OF ASYNCHRONOUS MOTOR

In paper, the authors use the methods of the analytical theory of differential equations to study the equation arising in a mathematical model of an asynchronous short-circuit motor by its angular velocity. Recent results of the theory of functions having bounded index are applied in order to find conditions on the coefficients of this equation, under which each its analytic in a disc solution or an entire solution is a function of the bounded l-index for some positive continuous function l obtained from the conditions on the coefficients of the equation.

LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS

The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $\mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{i\psi_n},$ $-\pi/2+\eta<\psi_n<\pi/2-\eta,$ $r_n\to0+$ as $n\to+\infty,$ of the Blaschke product $B(z);$ $\Gamma_m=\bigcup\limits_{j=1}^{m}\{z:|z|<1,\mathop{\text{arg}}(1-z)=-\theta_j\}=\bigcup\limits_{j=1}^{m}l_{\theta_j},$ $-\pi/2+\eta<\theta_1<\theta_2<\ldots<\theta_m<\pi/2-\eta,$ be a finite system of rays, $0<\eta<1$; $\upsilon(t)$ be continuous on $[0,1)$, $\upsilon(0)=0$, slowly increasing at the point 1 function, that is $\upsilon(t)\sim\upsilon\left({(1+t)}/2\right),$ $t\to1-;$ $n(t,\theta_j;B)$ be a number of zeros $a_n=1-r_ne^{i\theta_j}$ of the product $B(z)$ on the ray $l_{\theta_j}$ such that $1-r_n\leq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-i\varphi}\to1,$ $-\pi/2<\varphi<\pi/2,$ $\varphi\neq\theta_j,$ under the condition that zeros of $B(z)$ lay on $\Gamma_m$ and $n(t,\theta_j;B)\sim \Delta_j\upsilon(t),$ $t\to1-,$ for all $j=\overline{1,m},$ $0\leq\Delta_j<+\infty.$ We also considered the inverse problem for such $B(z).$

О коградиентности и контрградиентности линейных дифференциальных уравнений и систем

The Siegel-Shidlovskii method remains one of the basic methods in the theory of transcendental numbers. By using this method, it is possible to prove the transcendence and algebraic independence of the values of entire functions of a certain class (so-called E-functions). A necessary condition for applying this method is that all of the considered functions must constitute a solution of a system of linear differential equations and were algebraically independent over . The question about algebraic independence of the solutions of linear differential equations and systems of such equations is of great importance in differential algebra, analytical theory of differential equations, theory of special functions, and calculus (in the broad sense of the word). As is shown in papers by E. Kolchin, F. Beukers, W.D. Brownawell, and G. Heckman, this question boils down in many instances to verification of the cogredience and contragredience condition. Two systems of 1st order linear homogeneous differential equations with coefficients from are said to be cogredient (or, respectively, contragredient), if for arbitrary fundamental matrices and Ψ of these systems one of the equations Φ=gBΨC, Φ (ΨC)^*=gB, is fulfilled, where C∈GL(C), B∈GL(C(z)), g=g(z) is a function with the condition g^'/g∈C(z), and A^* is the matrix transposed to . The notions of cogredience and contragredience for linear homogeneous differential equations of arbitrary order are defined similarly. Another, more restricted definitions of cogredience and contragredience were in fact used in some papers of the author, devoted to generalized hypergeometric functions. According to these definitions, the function in the presented equalities is the product of a power function and an exponential function of some kind. The conditions for equivalence of these definitions are found.

Asymptotics of Solutions of Linear Differential Equations with Holomorphic Coefficients in the Neighborhood of an Infinitely Distant Point

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier.

The first integrals and rational solutions of some fourth-order differential equations

The object of this research is fourth-order differential equations. The aim of the research is to study the analytical properties of the solutions of these differential equations. The general form of the considered equations is indicated, and also the choice of the research object is justified. Herein we studied fourth-order differential equations for which sets of resonances with all positive nontrivial resonances are absent. Besides, three of these equations satisfy the conditions of absence in the solutions of moving multivalued singular points. The solutions of the next three equations have movable special points of multivalued character. Moreover, we also investigated the analytical properties of one more fourth-order differential equation of another general form for which it is also possible to construct a two-parameter rational solution as there is a nontrivial negative resonance in the related set of resonances. The first integrals of the equations under study are found and their rational solutions are constructed from negative non-trivial resonances. The resonance method was used in this study. The obtained results can be used in the analytical theory of differential equations.

Boundedness of l-Index and Completely Regular Growth of Entire Functions

We study the relationship between the class of entire functions of completely regular growth of order 𝜌 and the class of entire functions with bounded l-index, where l(z) = |z|𝜌−1 + 1 for |z| ≥ 1. Possible applications of these functions in the analytic theory of differential equations are considered. We formulate three new problems on the existence of functions with given properties that belong to the differences of these classes. For the fourth problem, we obtain an affirmative answer, namely, we present sufficient conditions for an infinite product to be an entire function of completely regular growth of order 𝜌 with unbounded l𝜌-index whose zeros do not satisfy the well-known Levin conditions (C) and (C′). We also construct an entire function of completely regular growth of order 𝜌 with unbounded l𝜌-index whose zeros do not satisfy the Levin conditions (C) and (C′).

Обмеженістьl-індексу та цілком регулярне зростання цілих функцій

УДК 517.547.22 Досліджується зв'язок між класом цілих функцій цілком регулярного зростання порядкуρ і класом цілих функцій обмеженогоl-індексу, деl(z)=|z|ρ-1+1для|z|≥1. Розглядаються можливі застосування цих функцій в аналітичній теорії диференціальних рівнянь. Сформульовано три нові проблеми про існування функцій із заданими властивостями, які належать до різниць цих класів. Отримано ствердну відповідь на проблему 4, а саме наведено достатні умови того, що нескінченний добуток є цілою функцією цілком регулярного зростання порядкуρ,необмеженогоlρ-індексу, а його нулі не задовольняють відомі умови Левіна (C) та (C'). Також побудовано цілу функцію цілком регулярного зростання порядкуρй обмеженого індексуlρ,нулі якої не задовольняють відомі умови Левіна (C) і (C').

Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables

We investigate the slice holomorphic functions of several complex variables that have a bounded L-index in some direction and are entire on every slice {z0 + tb : t ∈ ℂ} for every z0 ∈ and for a given direction b ∈ ℂn {0}. For this class of functions, we prove some criteria of boundedness of the L-index in direction describing a local behavior of the maximum and minimum moduli of a slice holomorphic function and give estimates of the logarithmic derivative and the distribution of zeros. Moreover, we obtain analogs of the known Hayman theorem and logarithmic criteria. They are applicable to the analytic theory of differential equations. We also study the value distribution and prove the existence theorem for those functions. It is shown that the bounded multiplicity of zeros for a slice holomorphic function F : → ℂ is the necessary and sufficient condition for the existence of a positive continuous function L : → ℝ+ such that F has a bounded L-index in direction.

Some weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball

We partially reinforce some criteria of $L$-index boundedness in direction for functions analytic in the unit ball. These results describe local behavior of directional derivatives on the circle, estimates of maximum modulus, minimum modulus of analytic function, distribution of its zeros and modulus of directional logarithmic derivative of analytic function outside some exceptional set. Replacement of universal quantifier on existential quantifier gives new weaker sufficient conditions of $L$-index boundedness in direction for functions analytic in the unit ball. The results are also new for analytic functions in the unit disc. The logarithmic criterion has applications in analytic theory of differential equations. This is convenient to investigate index boundedness for entire solutions of linear differential equations. It is also apllicable to infinite products.Auxiliary class of positive continuous functions in the unit ball (so-denoted $Q_{\mathbf{b}}(\mathbb{B}^n)$) is also considered. There are proved some characterizing properties of these functions. The properties describe local behavior of these functions in the polydisc neighborhood of every point from the unit ball.

Some criteria of boundedness of the L-index in direction for slice holomorphic functions of several complex variables

We investigate the slice holomorphic functions of several complex variables that have a bounded \(L\)-index in some direction and are entire on every slice \(\{z^0+t\mathbf{b}: t\in\mathbb{C}\}\) for every \(z^0\in\mathbb{C}^n\) and for a given direction \(\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}\). For this class of functions, we prove some criteria of boundedness of the \(L\)-index in direction describing a local behavior of the maximum and minimum moduli of a slice holomorphic function and give estimates of the logarithmic derivative and the distribution of zeros. Moreover, we obtain analogs of the known Hayman theorem and logarithmic criteria. They are applicable to the analytic theory of differential equations. We also study the value distribution and prove the existence theorem for those functions. It is shown that the bounded multiplicity of zeros for a slice holomorphic function \(F:\mathbb{C}^n\to\mathbb{C}\) is the necessary and sufficient condition for the existence of a positive continuous function \(L: \mathbb{C}^n\to\mathbb{R}_+\) such that \(F\) has a bounded \(L\)-index in direction.

On rational solutions of two differential equations with а moving singular line

The study is devoted to the analytical theory of ordinary differential equations. In the introduction, it is said that the object of investigation is nonlinear third-order autonomous differential equations with a moving singular line, and whose two-parameter rational solutions are known. The study aims to clarify how to obtain two-parameter rational solutions from the general solutions of these equations. In the analytical theory of differential equations, as a rule, nonlinear differential equations have negative resonances. Among these resonances is the resonance equal to –1 (trivial case). However, it is asserted in some of the researchers’ papers that the nature of these negative resonances has not been found until now. The equations only with non-trivial negative resonance arose the interest of researchers. And it appears that the rational solutions of the equations can be constructed by their negative resonances. In this paper, a necessary and sufficient condition is indicated, at which the two-parameter rational solution of the equation with a moving singular line can be obtained from its general solution.

VII International Conference Problems of Mathematical Physics and Mathematical Modelling

Mathematical models, mathematical methods and program codes for the prediction and analysis of experiments in recent years have become a necessary element not only in research projects, but also an integral part of many technological processes. The basis of many mathematical models are both partial differential equations and ordinary differential equations.The aim of this conference is to bring together researchers with interests in: Analytical Theory of Differential Equations, Optimization Problems, Analytical and Solid Mechanics, Fluid Mechanics, Mathematical Modelling.This annual conference was established at the National Nuclear Research University – Moscow Engineering Institute in the year 2012. The Conference was organized by three leading departments of the National Research Nuclear University MEPhI in the field of mathematical physics.This year, above 120 participants from different countries including Belorussia, Czech Republic, France, Great Britain, Japan, Moldova, Russia, Slovakia, Spain, Sweden and the USA take part in the Conference with oral talks.Valery V. KozlovNikolay A. KudryashovOleg V. Nagornov

Meromorphic solutions in the FitzHugh–Nagumo model

The FitzHugh–Nagumo model is studied in the framework of analytic theory of differential equations. The Nevanlinna theory is used to find all meromorphic solutions of a second-order ordinary differential equation related to the FitzHugh–Nagumo model. As a consequence new exact solutions of the FitzHugh–Nagumo system are obtained in explicit form.

О НЕУСТОЙЧИВОСТИ РЕШЕНИЙ ОДНОРОДНОЙ КРАЕВОЙ ЗАДАЧИ РИМАНА ДЛЯ КВАЗИГАРМОНИЧЕСКИХ ФУНКЦИЙ В КРУГОВЫХ ОБЛАСТЯХ

We consider the boundary value problem of Riemann type (the conjugation problem) in the classes of piecewise quasiharmonic functions. A homogeneous problem of a Riemann type problem in the classes of piecewise quasiharmonic functions of the second kind in circular domains is studied in details. In particular, in this case a clear solution method is developed for a homogeneous problem of Riemann type, the logical essence of which consists in reducing the solution of the homogeneous problem under consideration to a sequential solution of the common homogeneous Riemann problem for analytic functions and two second-order linear differential Euler equations. Moreover, instability of the solutions of the homogeneous problem is determined with respect to the change in the radius value of the considered circular domain, and a complete picture of its solvability for different values of the index of the problem and the radius of the circular domain is constructed. It is proved that the main reason for the instability of the solutions of a homogeneous problem of Riemann type in classes of piecewise quasiharmonic functions of the second kind in circular domains with respect to the change in the radius value of the considered circular domain is the fact that the number of linear independent analytic solutions of homogeneous differential Euler equations, to which the desired Riemann type problem is reduced, depends essentially on the radius of the considered circular region. In this paper we use the methods of the theory of functions of a complex variable, the theory of integral equations, and the analytic theory of differential equations.

On an example of a system of differential equations that are integrated in Abelian functions

The short review of the theory of Abelian functions and its applications in mechanics and analytical theory of differential equations is given. We think that Abelian functions are the natural generalization of commonly used functions because if the general solution of the 2nd order differential equation depends algebraically on the constants of integration, then integrating this equation does not lead out of the realm of commonly used functions complemented by the Abelian functions (Painlevé theorem). We present a relatively simple example of a dynamical system that is integrated in Abelian integrals by “pairing” two copies of a hyperelliptic curve. Unfortunately, initially simple formulas unfold into very long ones. Apparently the theory of Abelian functions hasn’t been finished in the last century because without computer algebra systems it was impossible to complete the calculations to the end. All calculations presented in our report are performed in Sage.

VI International Conference Problems of Mathematical Physics and Mathematical Modelling

PrefaceThis annual conference was established at the National Nuclear Research University – Moscow Engineering Institute in the year 2012. The Conference was organized by three leading departments of the MEPhI in the field of mathematical physics.Each year, more than 100 participants from different countries including Belorussia, Czech Republic, France, Great Britain, Japan, Moldova, Russia, Slovakia, Spain, Sweden and USA take part in the Conference with oral talks.The aim of this conference is to bring together researchers with interests in: Analytical Theory of Differential Equations, Optimization Problems, Analytical and Solid Mechanics, Fluid Mechanics, Mathematical Modelling.Valery V. KozlovNikolay A. KudryashovOleg V. Nagornov

V International Conference on “Problems of Mathematical and Theoretical Physics and Mathematical Modelling”

This annual conference was established at the National Nuclear Research University – Moscow Engineering Institute in the year 2012. The scope was initially on mathematical physics and mathematical modelling, while since this year theoretical physics was also included. The Conference was organized by three leading departments of the MEPhI in the field of theoretical and mathematical physics.Each year, more than 100 participants from different countries including Belorussia, Czech Republic, France, Great Britain, Japan, Moldova, Russia, Slovakia, Spain, Sweden and the USA take part in the Conference with oral talks.The aim of this conference is to bring together researchers with interests in: Analytical Theory of Differential Equations, Interaction of Radiation With Matter and Physics of Extreme Light, Nuclear Physics and Theory of Elementary Particles, Optimization Problems, Solid Mechanics, Soft Matter and Interdisciplinary Physics, Fluid Mechanics, Mathematical Modelling, Time Series.Nikolay A. KudryashovOleg V. NagornovSergey V. Popruzhenko

Divergent CR-equivalences and meromorphic differential equations

Using the analytic theory of differential equations, we construct, in any positive CR-dimension and CR-codimension, examples of formally but not holomorphically equivalent real-analytic CR-submanifolds in complex space.