Solutions Of Linear Differential Equations Research Articles

Growth of meromorphic solution related to linear difference equations and value sharing

In this paper, we consider the growth of meromorphic solutions of complex linear difference equations of certain types and derive some results, which improve the results of Chiang and Feng [On the Nevanlinna characteristic of f ( z + η ) and difference equations in the complex plane. Ramanujan J. 2008;16(1):105–129; On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans Am Math Soc. 2009;361:3767–3791] and Chen [Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl. 2011;373:235–241]. Also in this paper, we consider the sharing value problems of entire functions with their difference operators and obtain a result which improves the result of Li, Yang and Yi [Entire functions sharing an entire function of smaller order with their shifts. Proc Jpn Acad. 89 (A) (2013), 34–39]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.

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.

A quantum algorithm for linear differential equations with layerwise parameterized quantum circuits

Solving linear differential equations is a common problem in almost all fields of science and engineering. Here, we present a variational algorithm with shallow circuits for solving such a problem: given an N×N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N \ imes N$$\\end{document} matrix A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{A}}$$\\end{document}, an N-dimensional vector b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{b}$$\\end{document}, and an initial vector x(0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{x}(0)$$\\end{document}, how to obtain the solution vector x(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{x}(T)$$\\end{document} at time T according to the constraint dx(t)/dt=Ax(t)+b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{d}\\varvec{x}(t)/\ extrm{d} t = {\\varvec{A}}\\varvec{x}(t) + \\varvec{b}$$\\end{document}. The core idea of the algorithm is to encode the equations into a ground state problem of the Hamiltonian, which is solved via hybrid quantum-classical methods with high fidelities. Compared with the previous works, our algorithm requires the least qubit resources and can restore the entire evolutionary process. In particular, we show its application in simulating the evolution of harmonic oscillators and dynamics of non-Hermitian systems with PT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{P}\\mathcal{T}$$\\end{document}-symmetry. Our algorithm framework provides a key technique for solving so many important problems whose essence is the solution of linear differential equations.

A new approximate solution of the fractional trigonometric functions of commensurate order to a regular linear system

This paper introduces a novel approach to the approximate solution of linear differential equations associated with principal fractional trigonometry and the R function. This method proposes a solution that is expressed by adding appropriate fractional linear fundamental functions. Laplace transforms of these functions are irrational. Therefore, we rounded these functions to obtain rational functions in the form of damped cosine, damped sine, cosine, sine and exponential functions. This transformation was achieved by utilizing the concept of fractional commensurate order and, as a result, has direct practical relevance to real-world physics. The precision and effectiveness of the approach are demonstrated through illustrative examples of solving fractional linear systems.

Frobenius method for Mahler equations

Using Hahn series, one can attach to any linear Mahler equation a basis of solutions at 0 reminiscent of the solutions of linear differential equations at a regular singularity. We show that such a basis of solutions can be produced by using a variant of Frobenius method.

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.

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons.

On the Zeros of Exponential Polynomials

We consider the problem of deciding the existence of real roots of real-valued exponential polynomials with algebraic coefficients. Such functions arise as solutions of linear differential equations with real algebraic coefficients. We focus on two problems: theZero Problem, which asks whether an exponential polynomial has a real root, and theInfinite Zeros Problem, which asks whether such a function has infinitely many real roots. Our main result is that for differential equations of order at most 8 the Zero Problem is decidable, subject to Schanuel’s Conjecture, while the Infinite Zeros Problem is decidable unconditionally. We show moreover that a decision procedure for the Infinite Zeros Problem at order 9 would yield an algorithm for computing the Lagrange constant of any given real algebraic number to arbitrary precision, indicating that it will be very difficult to extend our decidability results to higher orders.

Growth of solutions of linear difference equations with meromorphic coefficients in terms of iterated $$p-\phi$$ order

Growth of solutions of linear difference equations with meromorphic coefficients in terms of iterated $$p-\phi$$ order

Growth properties of solutions of linear difference equations with coefficients having $\varphi$-order

"In this paper, we investigate the relations between the growth of entire coefficients and that of solutions of complex homogeneous and non-homogeneous linear difference equations with entire coefficients of $% \varphi $-order by using a slow growth scale, the $\varphi $-order, where $% \varphi $ is a non-decreasing unbounded function. We extend some precedent results due to Zheng and Tu (2011) [15] and others."

An Approximation Solution of Linear Differential Equation using Kantorovich Methods

In our work, we constructed a numerical approximations method to deal with approximations of a linear differential equation. We explained the general framework of the projection method which helps to clarify the basic ideas of the Kantorovich methods. We applied the iterative projection methods and presented a theorem to show the convergence of the constructed solutions to the exact solution. Also, most of the expressions encountered earlier can be used to define functions. Here are some illustrations. A great deal of information can be learned about a functioning relationship by studying its graph. A fundamental objective of section 4, is to acquaint with the graphs of some important functions and develop basic graphing procedures.

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.

Effect of core losses on thermal analysis of 3ɸ- Squirrel Cage Induction Motor using lumped parameter thermal model

The proposed Lumped Parameter Thermal Model (LPTM) is complex enough to predict temperatures in most parts of the 3ɸ- Squirrel Cage Induction Motor (3ɸ-SCIM), including the end winding and rotor surface temperatures. It is easily adaptable to a variety of frame sizes because it is formulated using dimensional information and constant thermal coefficients. The solution of linear differential equations adequately describes the thermal behavior of the 3ɸ-SCIM. The application of the thermal model on 15 kW 3ɸ-SCIM is discussed. The rise in temperature at different parts of motor is estimated considering copper loss, constant and variable core loss. The maximum flux density derived from Finite Element Method (FEM) analysis, used to calculate core loss at various loads of a 3ɸ-SCIM. Variable core loss causes deviation in temperature rise, from the results, it is observed that, the relative percentage error considering variable core loss to constant core loss is about 19 %.

An application of solutions of linear difference equations for obtaining the conditional moments of the trending Ornstein-Uhlenbeck processes

An application of solutions of linear difference equations for obtaining the conditional moments of the trending Ornstein-Uhlenbeck processes

Усреднение по Чернову линейных дифференциальных уравнений

Chernoff averages for random operator-valued functions generated by solutions of linear differential equations with variable coefficients on the real line are considered. Sufficient conditions are obtained for the convergence of the sequence of the Chernoff averages to a semigroup solving the Cauchy problem for the corresponding Fokker-Planck equation. The case of non-stationary random parameters is considered.

Asymptotic representation of some classes of solutions third-order differential equation

The conditions for the existence of one class of solutions of a binomial non-autonomous differential equation of the third order with a nonlinearity close in some sense to a linear one are established. Using the a priori properties of the so-called Pω(λ0)-solutions, asymptotic at t↑ω(ω≤+∞) images were obtained for such solutions connections and their derivatives of the first and second order in the case λ0 = 0. The propositions proved for the nonlinear equation are transferred to linear differential equations of the third order with asymptotically small coefficients. This made it possible, to some extent, to supplement the known results regarding the asymptotic properties of solutions of linear differential equations of the third order.

PROCESSING OF BULK AND ROD MATERIALS IN COMBINED VESSELS

Purpose and task is making research on the dynamics of a vibrating installation to process tubular materials and the kinematics of their movement (circulation) in hemispherical chambers.
 Methodology. To achieve the target goal, an expert assessment of existing approaches to methods of processing bulk and rod materials in air and liquid environments has been carried out.
 Research methods. Using vibrations for heat treatment of bulk and tubular materials makes it possible to intensify the heat exchange process, significantly reduce processing time, save fuel and electricity consumption, improve product quality, reduce costs on maintenance and environmental protection. The materials of theoretical studies on vibrating machine dynamics under different types of working body oscillations and the number of vibration drives are presented. Analytical dependences of vibration parameters have been obtained.
 The main results. Using vibrations to develop highly reliable and efficient technological units incorporating the performance of low technological operations in one device.
 Conclusions and practical significance. Based on the analysis of vibrating machines with a liquid environment, the results of research aimed at improving technologies and appropriate technical means to process bulk, lump and rod materials in air and liquid environments are provided. The solution of linear differential equations resulted in obtaining amplitude parameters of steady oscillations of an installation working body with relevance to the movement of working camera points and the installation angular oscillations as well. It has been proved that the amplitude of the vertical oscillations of the point is equal to the amplitude of the circular oscillations of the installation gravity center. Obtained rational trajectories of installation point oscillations to determine rational parameters are based on the requirements of material processing technology. The developed calculation methods can be used while studying machine dynamics of other types of a similar class containing liquid in the working space. Installation design makes it possible to adjust the oscillation amplitude and the nature of the trajectory of the installation working body points within wide limits.
 Keywords: rod or bulk material, motion kinematics, processing, vibrating machines, vibration trajectories, liquid, or air environment.

Closed‐form solutions of second‐order linear difference equations close to the self‐adjoint Euler type

This paper is dedicated to obtaining closed‐form solutions of linear difference equations which are asymptotically close to the self‐adjoint Euler‐type difference equation. In this sense, the equation is related to the Euler–Cauchy differential equation . Throughout the paper, we consider a system of sequences which behave asymptotically as an iterated logarithm.

ON GENERALIZED SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS OF THE FIRST ORDER

The article deals with the problem of existence of generalized solutions to linear differential equation with a generalized coefficient Q which coincides with a given rational function q on the complement to the set of poles of q. To introduce a generalized solution, we consider approximations of the coefficient Q by a family of smooth functions. Then, the generalized solution is the limit of the solutions to approximating equations. A principal difference from the classical theory here is that even such simplest equations with singular coefficient Q may not have the solution. The main result is the description of those Q from the class considered for whom a generalized solution does exist.

Factors of $E$-operators with an $\eta$-apparent singularity at zero

In 1929, Siegel defined $E$-functions as power series in $\overline{\mathbb{Q}}[[z]]$, with Taylor coefficients satisfying certain growth conditions, and solutions of linear differential equations with coefficients in $\overline{\mathbb{Q}}(z)$. The Siegel–Shidlovskii theorem (1956) generalized to $E$-functions the Diophantine properties of the exponential function. In 2000, André proved that the finite singularities of a differential operator in $\overline{\mathbb{Q}}(z)[d/dz] \setminus \{0\}$ of minimal order for some non-zero $E$-function are apparent, except possibly 0 which is always regular singular. We pursue the classification of such operators and consider those for which 0 is $\eta$-apparent, in the sense that there exists $\eta \in \mathbb C$ such that $L$ has a local basis of solutions at 0 in $z^{\eta} \mathbb C[[z]]$. We prove that they have a $\mathbb{C}$-basis of solutions of the form $Q_{j}(z)z^{\eta} e^{\beta_{j} z}$, where $\eta \in \mathbb{Q}$, the $\beta_{j} \in \overline{\mathbb{Q}}$ are pairwise distinct and the $Q_{j}(z) \in \overline{\mathbb{Q}}[z] \setminus \{0\}$. This generalizes a previous result by Roques and the author concerning $E$-operators with an apparent singularity or no singularity at the origin, of which certain consequences are also given here.