Solutions Of Linear Differential Equations Research Articles

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

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.

Uniform Asymptotics of Solutions of Second-Order Differential Equations with Meromorphic Coefficients in a Neighborhood of Singular Points and Their Applications

In this paper, we consider the problem of obtaining the asymptotics of solutions of differential operators in a neighborhood of an irregular singular point. More precisely, we construct uniform asymptotics for solutions of linear differential equations with second-order meromorphic coefficients in a neighborhood of a singular point and apply the results obtained to the equations of mathematical physics. The main results related to the construction of uniform asymptotics are obtained using resurgent analysis methods applied to differential equations with irregular singularities. These results allow us to construct asymptotics for any second-order equations with meromorphic coefficients—that is, with an arbitrary order of degeneracy. This also allows one to determine the type of a singular point and highlight the cases where the point is non-singular or regular.

IDENTIFICATION OF LINEAR DYNAMIC SYSTEMS IN THE ENVIRONMENT OF POLYNOMYAL SIGNALS

A method for the structural and parametric identification of one-dimensional linear stationary dynamic systems, represented by differential "inputoutput" constraint equations, is proposed. The method is focused on both active and passive experiments. The method is based on a polynomialrepresentation of the input and output signals of the identified dynamic system. A compact vector-matrix representation of polynomials is proposed,which makes it possible to find the forced component of the solution of linear differential equations as a result of performing simple linear algebraicoperations. The vector-matrix representation of polynomials made it possible to quite simply solve the problem of inversion of linear dynamicalsystems and the problem of compensating the measured perturbation. The issues of representing time signals in polynomial form are not considered inthis paper. Based on the obtained linear representation of a one-dimensional dynamic system, which links the parameters of the input and outputsignals with the parameters of the differential equation of the identified dynamic system mathematical model, a linear system of algebraic equations forunknown coefficients of the differential process equation is obtained. In the general case, the resulting system belongs to the class of overdeterminedsystems, and therefore its solution can be obtained by the least-square technique and is reduced to finding a pseudoinverse matrix. A block diagram ofsoftware for solving the problem of structural and parametric identification in the environment of polynomial signals is proposed. The algorithmincludes the procedure of comparing the results of numerical simulation of the identified model with the output experimental signal and correcting thestructure of the model based on the results of the comparison.

Study of Complex Oscillation of Solutions of a Second Order Linear Differential Equation With Entire Coefficients of (α,β, γ )-Order

In this paper, we deal with the complex oscillation of solutions of linear differential equation. We mainly study the interaction between the growth, zeros of solutions with the coefficients of second order linear differential equations in terms of ( α, β, γ )-order and obtain some results in general form which considerably extend some results of [5], [18] and [21].

Forced oscillation and asymptotic behavior of solutions of linear differential equations of second order

Forced oscillation and asymptotic behavior of solutions of linear differential equations of second order

Meromorphic functions of finite φ-order and linear q-difference equations

The φ-order was introduced in 2009 for meromorphic functions in the unit disc, and was used as a growth indicator for solutions of linear differential equations. In this paper, the properties of meromorphic functions in the complex plane are investigated in terms of the φ-order, which measures the growth of functions between the classical order and the logarithmic order. Several results on value distribution of meromorphic functions are discussed by using the φ-order and the φ-exponent of convergence. Instead of linear differential equations, the applications in the complex plane lie in linear q-difference equations.

Oscillation tests for difference equations with non-monotone retarded arguments

This paper presents new sufficient conditions, involving limsup, for the oscillation of all solutions of linear difference equations with general retarded arguments.

Fast growing solutions of linear differential equations with analytic coefficients in the unit disc

In this paper, we investigate the growth of solutions of higher order linear differential equations with analytic coefficients of ϕ-order in the unit disc. We introduce new definitions of the lower order and the type related to the ϕ-order concepts to generalise and extend previous results due to Chyzhykov-Semochko [6], Semochko [14], Belaïdi [1,2,3], Hu-Zheng [12].

Time-normalized conductivity concept for analytical characterization of dynamic-conductivity hydraulic fractures through pressure-transient analysis in tight gas reservoirs

This paper presents a time-normalized conductivity concept to allow an analytical model for characterization of dynamically-closed fractures through pressure-transient analysis in tight gas reservoirs. Specifically, the fracture conductivity in our flow model is considered to be a function of time. With the definition of the time-dependent fracture conductivity in the model, calculation of variable-rate pressure-transient behavior (e.g., pressure buildup tests) can be still based on the solution of linear differential equations governing fluid flow in a gas reservoir. Therefore, the superposition principle can be legitimately utilized in our solution to model pressure buildup tests. We then introduce a modified exponential function to represent the decline behavior of the time-normalized fracture conductivity during the production stage, and the dynamic fracture conductivity is essentially handled in a stair-step manner to calculate bottom-hole pressures. The accuracy of the calculated transient pressures was validated by comparing with a commercial simulator. The pressure drawdowns at early-times, because of dynamically-closed fractures, clearly increase more than in the invariant-fracture-conductivity case. We examine the effects of fracture conductivity behavior and production history on the pressure-transient response extensively, and use them to help identify the presence of dynamically-closed fractures. Finally, we demonstrate the influence of dynamically-closed fractures in synthetic and field cases to show the utility of the methodology. Results indicate that integrating multiple pressure build-up data obtained in different production periods enable us to diagnose and then to understand the changes in fracture conductivity during reservoir depletion. Using the results from the multiple pressure-transient analyses, this method can further be used to forecast the trend of fracture conductivity changes.

Dynamics and exact solutions of non-evolutionary partial differential equations

The article presents a new method for constructing exact solutions of non-evolutionary partial differential equations with two independent variables. The method is applied to the linear classical equations of mathematical physics: the Helmholtz equation and the variable type equation. The constructed method goes back to the theory of finite-dimensional dynamics proposed for evolutionary differential equations by B. Kruglikov, O. Lychagina and V. Lychagin. This theory is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of PDEs. The proposed method is used to construct exact particular solutions of linear differential equations (Helmholtz equations and equations of variable type).

Procedures for Constructing Truncated Solutions of Linear Differential Equations with Infinite and Truncated Power Series in the Role of Coefficients

In this paper, we propose a package for symbolic construction of exponential–logarithmic solutions to linear differential equations whose coefficients are represented in incomplete form, i.e., as power series for which only a finite number of initial terms is known. The series involved in the solutions are also represented in incomplete form. For each such series, the maximum possible number of initial terms is constructed, which are uniquely determined by known terms of coefficients of a given equation. Additionally, the truncation degree of each series in a solution should not exceed a value specified by the user. It ensures the termination of the computation even when any number of the terms of the series involved in the solution can be defined by known terms of the coefficients of a given equation.

Bernstein Operator Approach for Solving Linear Differential Equations

In this study, an alternative numerical method having regard to the Bernstein operator is generated for approximate solutions of linear differential equations in the most general form under the initial and boundary conditions. Some applications are also revealed to show how the procedure can be performed for the problems.

The Hopf algebroid structure of differentially recursive sequences

A differentially recursive sequence over a differential field is a sequence of elements satisfying a homogeneous differential equation with non-constant coefficients (namely, Taylor expansions of elements of the field) in the differential algebra of Hurwitz series. The main aim of this paper is to explore the space of all differentially recursive sequences over a given field with a non-zero differential. We show that these sequences form a two-sided vector space that admits, in a canonical way, a structure of Hopf algebroid over the subfield of constant elements. We prove that it is the direct limit, as a left comodule, of all spaces of formal solutions of linear differential equations and that it satisfies, as Hopf algebroid, an additional universal property. When the differential on the base field is zero, we recover the Hopf algebra structure of linearly recursive sequences.

A novel numerical method for the solution of nonlinear equations with applications to heat transfer

PurposeThe purpose of this study is to extend a novel numerical method proposed by the first author, known as the dual mesh control domain method (DMCDM), for the solution of linear differential equations to the solution of nonlinear heat transfer and like problems in one and two dimensions.Design/methodology/approachIn the DMCDM, a mesh of finite elements is used for the approximation of the variables and another mesh of control domains for the satisfaction of the governing equation. Both meshes fully cover the domain but the nodes of the finite element mesh are inside the mesh of control domains. The salient feature of the DMCDM is that the concept of duality (i.e. cause and effect) is used to impose boundary conditions. The method possesses some desirable attributes of the finite element method (FEM) and the finite volume method (FVM).FindingsNumerical results show that he DMCDM is more accurate than the FVM for the same meshes used. Also, the DMCDM does not require the use of any ad hoc approaches that are routinely used in the FVM.Originality/valueTo the best of the authors’ knowledge, the idea presented in this work is original and novel that exploits the best features of the best competing methods (FEM and FVM). The concept of duality is used to apply gradient and mixed boundary conditions that FVM and its variant do not.