Analytical Solution Of Differential Equation Research Articles

Unsteady flow of droplet liquid in hydraulic systems of aircrafts and helicopters: models and analytical solutions

The subject of this study is the unsteady flow of liquid in pipelines, which are part of the design of airplanes and helicopters. This name means, first of all, the phenomenon of a sharp increase in pressure in the pipeline, which is known as a hydraulic shock. Although we have already learned to deal with this phenomenon in some parts of the systems, in many structural elements (flexible pipelines), inside which the working pressure reaches several hundred atmospheres, this phenomenon is still quite dangerous. As you know, the best way to deal with an unwanted phenomenon is through theoretical study. To date, there has been a huge amount of work in the direction of hydraulic shock research. This article does not fully cover these studies. It is limited to references to reviews and relevant works. Because the phenomenon of hydraulic shock has a significantly nonlinear character, analytical solutions of systems of equations corresponding to the simplest models were unknown until recently. This work presents, as an overview, already known analytical solutions describing the process of shock wave propagation. Most importantly, new achievements are given, both for the inviscid approximation and for considering internal viscous friction. It is shown that the internal friction within the considered model is negligible almost everywhere, except for the thin shock layer. The asymptotic is proportional to the tangent function and inversely proportional to the square root of the product of the Reynolds number and the dimensionless parameter characterizing the convection effect. Convection of the velocity field significantly affects the distribution of characteristics in hydraulic shock. If the self-similar solutions that were obtained earlier have a power-law character for the velocity distribution in the shock wave, then the simultaneous consideration in the model of convection and friction on the pipeline walls (according to the Weisbach-Darcy model) made it possible to obtain a distribution in the form of an exponential function that decays with increasing distance from the shock wave front. In addition, the work includes an original approach to solving a nonlinear system of differential equations that describes the propagation of a shock wave without considering the friction on the walls. Analytical solutions were obtained in the form of a function of pressure versus the velocity of shock wave propagation. Research methods. This work uses purely theoretical approaches based on the use of well-known fluid flow models, methods of analytical solution of differential equations and their systems, asymptotic methods, derivation of self-model equations, and finding their solutions. Conclusions. Analytical solutions of systems of differential equations were obtained, which describe models of hydraulic shock without considering viscous effects. A comparison of the obtained results with the results of other studies is given.

Automation of the technological process of forming details with straight rifts

The subject matter of the article is technological line for the production of thick sheet parts with rifts, made of structural and special steels. The goal of the work is analysis and creation of methods of managing basic technological operations at all stages of the technological process of the automated production of such parts. The following tasks were solved in the article: formation of models of the main stages of forming parts with rectilinear rifts and control of dynamic properties of selected systems of their implementation. The following methods used are – methods of numerical integration and numerical methods of solving differential equations; analytical solution of differential equations based on the Dahlembert principle or by using the Lagrange equation; frequency analysis of transfer functions of individual links, presented in the form of fractional-rational functions. The following results were obtained – constructed in the first approximation the composition and sequence of the main operations of the generalized technological process of the automated production of thick-walled sheet parts with rifts; according to the results of the analysis of the process of elastic-plastic change of the shape of the region with many connections, the input and output parameters necessary for building a control model of the specified process were determined; it is determined that for its implementation it is necessary to ensure free deformation of at least 0.25 % of the rift surface area outside of contact with the forming tool, and the necessary technological methods for this are proposed; proposed variants of the technical implementation of the selected technological methods; the results of modeling schemes and modes of operation of the necessary technical equipment are presented. Conclusions: The application of the proposed structure, composition and sequence of main operations of the generalized technological process of the automated production of thick-walled sheet parts with cracks, based on its implementation by the selected methods on the proposed equipment, will allow to ensure the effective management of the quality of the production of the specified parts at each stage of the technological process.

Realization of the brachistochronic motion of Chaplygin sleigh in a vertical plane with an unilateral nonholonomic constraint

The paper considers the procedure for determining the brachistochronic motion of the Chaplygin sleigh in a vertical plane, where the blade is such that it prevents the motion of the contact point in one direction only. The position of the sleigh mass center and orientation at the final positions is specified, as well as the initial value of mechanical energy. The simplest formulation of a corresponding optimal control problem is given and it is solved by applying Pontryagin?s maximum principle. For some cases, analytical solutions of differential equations of the two-point boundary value problem (TPBVP) of the maximum principle were found. Numerical integration was carried out for other cases using the shooting method, where the assessment of missing terminal conditions was given and it was shown that the solution obtained represents the global minimum time for the brachistochronic motion. The method of the brachistochronic motion by means of a single holonomic and a single unilateral nonholonomic mechanical constraint is presented.

A simple way to solve boundary value problems in technological processes

The paper proposes a simple approximate method for solving differential equations for boundary value problems. A method is proposed for averaging differential equations over a moving volume, which allows obtaining approximate analytical solutions of differential equations. The control volume is the only one in the considered area of the boundary value problem. In this case, the control volume is considered to be moved in the area under consideration. Based on the averaging of boundary value problems over the volume being moved, an algebraic equation is obtained. When averaging over one of the variables (in the case of a two-dimensional problem), ordinary differential equations are obtained. Examples are given.

Semi-analytical implicit direct time integration scheme on example of 1-D wave propagation problem

The most common approach in dynamic analysis of engineering structures and physical phenomenas consists in finite element discretization and mathematical formulation with subsequent application of direct time integration schemes. The space interpolation functions are usually the same as in static analysis. Here on example of 1-D wave propagation problem the original implicit scheme is proposed, which contains the time interval value explicitly in space interpolation function as results of analytical solution of differential equation for considered moment of time. The displacements (solution) at two previous moments of time are approximated as polynomial functions of position and accounted for as particular solutions of the differential equation. The scheme demonstrates the perfect predictable properties as to dispersion and dissipation. The crucial scheme parameter is the time interval – the lesser the interval the more correct results are obtained. Two other parameters of the scheme – space interval and the degree of polynomial approximation have minimal impact on the general behavior of solution and have influence on small zone near the front of the wave.

Electromagnetic Fields around Black Holes in Einstein Æther Gravity

Axial symmetry and stationary properties of spacetime allow to find exact analytical solutions of differential equations describing fields and particles in a gravitational background. The present work is mainly devoted to derivation of exact solutions of Maxwell’s equations for magnetic fields generated by current loops around static black holes (BHs) in Einstein-aether gravity based on the spacetime symmetries in both regions: (i) interior and (ii) exterior to the current loop for a proper observer. The spacetime symmetries are applied in separating variables to solve the second order ordinary differential equation for vector potential of electromagnetic field and the equations of motion of test particles around the aether BH. We also study effects of the aether field on innermost stable circular orbits (ISCOs) of the test particles assuming the current loop position is placed there. It is obtained that the ISCO radius, as well as dipole magnetic moment of the current loop decrease with the increase of the aether parameter c14. Moreover, the performed analysis indicates that the aether field causes a decrease in the magnetic field inside and outside the current loop due to the change of its position.

Analysis of Fractional Order Differential Equation Using Laplace Transform

Exploring the functioning of analytical solutions of differential equations defined by the fractional order operators is one of the hottest topics in the field of research in stability problems. This article analyze the stability of a certain type of fractional order differential equation. Employing Banach contraction mapping principle, existence and uniqueness of solutions are obtained. A sufficient condition to assure the reliability of solving the fractional order differential equation by Laplace Transform method and Generalized Laplace Transform are presented. Exponential stability results for the solution is discussed using Gronwall inequality. Application of the generalized Gronwall inequality to fractional order differential equation under investigation yields new sufficient condition for stability of fractional order differential equation. Applicability of the theoretical results on stability are demonstrated with an example. In the analysis of fractional order differential equation, Laplace transform is proved to be a valid tool under certain conditions.

ON NOTEWORTHY APPLICATIONS OF DIFFERENTIAL EQUATIONS WITH LEGUERRE POLYNOMIAL

The Kamal Transform is a mathematical tool used in solving the differential equations. Kamal Transform makes it easier to solve the problem in engineering application and make differential equations simple to solve. In this paper, we will discuss analytic solutions of differential equations including leguerre polynomial by using kamal transform. KEY WORDS: Kamal Transform, Leguerre Polynomial, Differential Equations.

Non-planar elliptic vertex

We consider the problem of obtaining higher order in regularization parameter ε analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct integration of parametric representations in terms of iterated integrals. Taking non-planar elliptic vertex as an example we show that in addition to two mentioned methods one can use analytical solution of differential equations in terms of power series. Moreover, in the last case it is possible to obtain the exact in ε results.

Analytical solutions of some mechanics problems by Elzaki transform

Objectives: To discuss the applicability of new kind of transform technique named as Elzaki transform in solving the higher order ordinary linear differential equations occurred in various categories of vibrations in the field of engineering mechanics. Methods: We have selected some problems in the field of engineering mechanics to study the solutions with the help of Elzaki transform technique. We have used Elzaki transform method and also MATLAB to draw graphs for the obtained solutions of selected problems. Findings: We have successfully applied this new transform technique in finding the analytical solutions and compared with the graphs drawn with MATLAB. Novelty/Applications: We may apply this new kind of transform technique named as Elzaki transform to get the analytical solutions of differential equations. Keywords: Elzaki transform; differential equations; vibrations

ЧИСЕЛЬНИЙ АНАЛІЗ КРУГЛИХ ПЛАСТИН НА ПРУЖНІЙ ОСНОВІ ЗІ ЗМІННИМ КОЕФІЦІЄНТОМ ПОСТЕЛІ

Abstract. The paper presents the results of a study of the stress-strain state of a circular plate of constant cylindrical stiffness, which lies on a variable elastic foundation and is under the influence of a continuously distributed transverse load. Twelve variants of calculation are considered ‒ six for a steel round plate and six more ‒ for a concrete round plate under two conditions of fixing and three different laws of variation of the bed coefficient. To solve the problem, the finite element method implemented in the LIRA-SAPR software package is used. It is noted that in the case when the bedding coefficient is a variable value depending on the coordinate in which the foundation settlement is determined, the analytical approach leads to the need to solve the corresponding differential equations with variable coefficients. Therefore, calculations of circular and annular plates lying on a variable elastic foundation by means of analytical solutions of differential equations are extremely rare in scientific periodicals and are of a private nature. An effective method for the analytical solution of differential equations with variable coefficients for a number of problems in mechanics was proposed by one of the authors of the article, however, the application of the method to the calculation of a circular plate on an elastic foundation with a variable bed coefficient requires verification, therefore, here we consider the features of the finite element analysis of such a plate under different boundary conditions and different laws of variation of the bed coefficient. In all versions, the results completely coincide with the known results of bending of slabs that do not have an elastic base and in the case when this base exists and its resistance is constant. The discrepancy here is very insignificant ‒ in the third significant digit after the decimal point for deflection when hinged and in the second for moments. In case of rigid clamping, the deflections and moments also differ from the corresponding values of the known solutions in the second significant digit after the decimal point.

Adaptive Algorithm for Controlling the Soft Vertical Landing of an Unmanned Return Spacecraft. I.

A new adaptive algorithm for multistep terminal control (MSTC) of a soft landing of an unmanned return spacecraft (URS) is proposed that is based on more accurate analytical solutions of differential equations with identification of real parameters of the URS and the atmosphere.

О равновесном уклоне станционного профиля

Objective: Defi nition of slope gradient for station track profi le elevation at which self-induced runningoff of loose running stock. Methods: Simulation of rolling stock movement based on analytical solution of differential equation of equilibrium of cut of wagons. Cut of wagons on station track is treated as a solid on a gradient plane with a mass equal to the mass of the cut of wagons. Track profi le elevation is approximated by piecewise-linear right line. Results: The paper shows that hopper wagons have the highest probability of self-induced running-off due to high air resistance coeffi cient. For three-element profi le the hypothetic wind speed at which running-off of wagons is possible remains relatively low, about 14 metres per second, which leaves the problem of effective fastening of wagons at stations still current. Practical importance: The solution obtained permits evaluating the probability of running-off of loose rolling stock under the action of gravity and under the impact of wind loading, as well as refi ne the norms for fastening of wagons at a station. The proposed calculation scheme also permits gauging the strength with which moving wagons would impact the backing thrust in case it is installed at the end of a track section.

ЧИСЛЕННОЕ РЕШЕНИЕ НЕСТАЦИОНАРНОГО ДРОБНОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ В ЗАДАЧАХ МОДЕЛИРОВАНИЯ РАСПРОСТРАНЕНИЯ ТОКСИЧНЫХ ВЕЩЕСТВ В ПОДЗЕМНЫХ ВОДАХ

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

Convergence of numerical schemes for the solution of partial integro-differential equations used in heat transfer

Integro-differential equations play an important role in may physical phenomena. For instance, it appears in fields like fluid dynamics, biological models and chemical kinetics. One of the most important physical applications is the heat transfer in heterogeneous materials, where physician are looking for efficient methods to solve their modeled equations. The difficulty of solving integro-differential equations analytically made mathematician to search about efficient methods to find an approximate solution. The present article is designed to supply numerical solution of a parabolic Volterra integro-differential equation under initial and boundary conditions. We have made an attempt to develop a numerical solution via the use of Sinc-Galerkin method, the convergence analysis via the use of fixed point theory has been discussed, and showed to be of exponential order. For comparison purposes, we approximate the solution of integro-differential equation using Adomian decomposition method. Sometimes, the Adomian decomposition method is a highly efficient technique used to approximate analytical solution of differential equations, applicability of Adomian decomposition method to partial integro-differential equations has not been studied in details previously in the literatures. In addition, we present numerical examples and comparisons to support the validity of these proposed methods.

Dynamic Regulation Responding to an External Stimulus: A Differential Equation Model

This study examines the dynamic regulation process responding to an external stimulus. The damped oscillator model has been used to describe this process. However, the model does not allow a nonzero steady state, even though the oscillations may continue and do not necessarily damp toward zero. This study introduces the driven damped oscillator model which has an additional parameter to identify different patterns of the steady state. Three methods, generalized local linear approximation, continuous time structural equation modeling, and analytic solutions of differential equations are provided to estimate model parameters. A simulation study indicates that parameters in the driven damped oscillator model are well recovered. The model is then illustrated using a data set on the daily reports of sales after a sale promotion. Potential applications and possible expansions of this model are also discussed.

On new scaling laws in a temporally evolving turbulent plane jet using Lie symmetry analysis and direct numerical simulation

A temporally evolving turbulent plane jet is studied both by direct numerical simulation (DNS) and Lie symmetry analysis. The DNS is based on a high-order scheme to solve the Navier–Stokes equations for an incompressible fluid. Computations were conducted at Reynolds number $\mathit{Re}_{0}=8000$, where $\mathit{Re}_{0}$ is defined based on the initial jet thickness, $\unicode[STIX]{x1D6FF}_{0.5}(0)$, and the initial centreline velocity, $\overline{U}_{1}(0)$. A symmetry approach, known as the Lie group, is used to find symmetry transformations, and, in turn, group invariant solutions, which are also denoted as scaling laws in turbulence. This approach, which has been extensively developed to create analytical solutions of differential equations, is presently applied to the mean momentum and two-point correlation equations in a temporally evolving turbulent plane jet. The symmetry analysis of these equations allows us to derive new invariant (self-similar) solutions for the mean flow and higher moments of the velocities in the jet flow. The current DNS validates the consequence of Lie symmetry analysis and therefore confirms the establishment of novel scaling laws in turbulence. It is shown that the classical scaling law for the mean velocity is a specific form of the current scaling (which has a more general form); however, the scaling for the second and higher moments (such as Reynolds stresses) has a completely different structure compared to the classical scaling. While the failure of the classical scaling for the second moments of the fluctuating velocities has been noted from the jet data for many years, the DNS results nicely match with the present self-similar relations derived from Lie symmetry analysis. Key ingredients for the present results, in particular for the scaling laws of the higher moments, are symmetries, which are of a purely statistical nature. i.e. these symmetries are admitted by the moment equations, however, they are not observed by the original Navier–Stokes equations.

A one-step method for modelling longitudinal data with differential equations.

Differential equationmodels are frequently used to describe non-linear trajectories of longitudinal data. This study proposes a new approach to estimate the parameters in differential equationmodels. Instead of estimating derivatives from the observed data first and then fitting a differential equationto the derivatives, our new approach directly fits the analytic solution of a differential equationto the observed data, and therefore simplifies the procedure and avoids bias from derivative estimations. A simulation study indicates that the analytic solutions of differential equations (ASDE) approach obtains unbiased estimates of parameters and their standard errors. Compared with other approaches that estimate derivatives first, ASDE has smaller standard error, larger statistical power and accurate Type I error. Although ASDE obtains biased estimation when the system has sudden phase change, the bias is not serious and a solution is also provided to solve the phase problem. The ASDE method is illustrated and applied to a two-week study on consumers' shopping behaviour after a sale promotion, and to a set of public data tracking participants' grammatical facial expression in sign language. R codes for ASDE, recommendations for sample size and starting values are provided. Limitations and several possible expansions of ASDE are also discussed.

Моделирование экономических колебаний добычи сланцевой нефти

Shale oil production is considered to be one of the important determinants of oil price slump in 2014-2016. According to some experts, shale oil became a new market regulator, ousted OPEC. Shale oil production is characterized by non-trivial dynamics. After an accelerated growth in 2010-2015, a period of fluctuations in production caused by the volatility of oil prices followed. The paper develops approach to the shale production modelling on the base of the analytical solutions of differential equation with retarded argument. The differential equation reflects the fact that production change consists of a sum of new debits and the natural process of base production decline. Besides, possibilities of origination and prerequisites of endogenous economic oscillations of shale oil production are studied. Calculations, implemented for economic and technological characteristics given for period from December 2014 to May 2017, showed a possibility of existence of oil production oscillation with period of 33 month, that does not contradict to observations. In general, the proposed approach can be used as a nonlinear structural method for forecasting oil production and corresponding price component, connected with production change.

IMPROVING SUPERVISORY CONTROL WATER DISTRIBUTION OF IRRIGATION CANALS RECLAMATION SYSTEMS

Background: Examine issues of dispatching management of water distribution systems in the reclamation channels using a systematic approach. Materials and methods: Integrated automated control systems are actively developed implemented to manage water distribution in irrigation canals. It needs to take into account the dynamic processes of water flow while the automation of water distribution in open channel irrigation network system must. Imitating mathematical modeling of water distribution during transient driving mode is the process of studying the dynamic properties of these automated control systems on the basis of analytic solutions of differential equations in partial derivatives. Results: Algorithms and mathematical models in the form of a software package, which describes the behavior of object of control, while it’s depending on its condition, control actions and possible disturbances. The elements functional water distribution mathematical model constructed on the basis of control algorithms taking into account the work of the majority of water consumers “on demand”. Conclusion: Based on the simulation and field research there were presented recommendations on the calculation of the propagation time of the disturbance waves in open channels, regarding the selection and appointment of the optimum parameters of channels and structures on them, the lengths of the calculated areas, slope of the bottom of the distribution channels, pressures and quantities shutter opens on structures, the choice of cross-sections sections of the channels for the installation of control equipment at unsteady flow regime.