Analytical Solution Of Differential Equation Research Articles

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.

An analytical model to predict the impact response of one-dimensional structures

An analytical solution method is presented for the transient axial response of one-dimensional structures subjected to impact loading. The transient structural response is expressed as a series of the impact loading function and its progressive shifting values depending on both material position and time scale. The governing differential equation on the impact force is derived considering the general solution and constitutive equation of the contact interface. The analytical solution of differential equation yields a recursive function describing the impact force and displacement function of the total geometry. Depending on the contact surface roughness and the material properties, an impact function is introduced as a base function for impact response. The procedure is implemented to determine the shock waves generated at the collision of the elastic rod on the rigid surface and two elastic rods. The analytical solution also derives the steady-state response of the structure after the impact loading.

Method for constructing approximate analytic solutions of differential equations with a polynomial right-hand side

A method for constructing approximate analytic solutions of systems of ordinary differential equations with a polynomial right-hand side is proposed. The implementation of the method is based on the Picard method of successive approximations and a procedure of continuation of local solutions. As an application, the problem of constructing the minimal sets of the Lorenz system is considered.

Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach

Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the $C^{\mathbf {k}}$ category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems and give a number of computer assisted proofs in the analytic framework.

Evenand odd-parity kinetic equations of particle transport. 3: Finite analytic scheme on tetrahedra

A finite analytic (not finite-difference) scheme is derived for the numerical solution of evenand odd-parity transport equations of neutral particles. This discrete algebraic equation is constructed by cross-linking analytic solutions of differential equations in tetrahedral cells. Such a scheme makes it possible to simulate the three-dimensional transport of neutrons and photons in heterogeneous absorbing, scattering, and multiplying media (simulate problems of nuclear reactors, radiation shielding, radiative heat exchange, and radiation gas dynamics) without restrictions on the optical depth of the cell (the product of the beam extinction coefficient by the cell chord) and on the value of the jump in the extinction coefficient under the particle transition from one cell to another. A change in the sign of the extinction coefficient is allowed. The scheme is combined with efficient iterative methods for solving deterministic transport problems in which the angular (direction-of-flight) variable is discretized using the discrete-ordinate (Sn) approximation.

The Dynamic Mechanical Analysis of a Clamped-Free Timoshenko Nano-Beam Subjected to the Moving Force the Nonlocal Effects

Based on nonlocal beam theories, the dynamic mechanical behavior of a clamped-free Timoshenko nano-beam subjected to a variable speed moving force are studied in this paper. The analytical solution of differential equation is obtained using state-space method. The effects of the nonlocal stress and the magnitude of the moving force acceleration on the dynamic responses of the nano-beam are discussed in detail. The results indicate that nonlocal effects and moving force acceleration play a significant role on the dynamic mechanical response of nano-beam.

DISTRIBUTION NORMAL CONTACT STRESSES IN THE ROLL GAP AT A CONSTANT SHEAR STRESS

Differential equation describing stress state in the rolling gap was derived first time by Karman. Since the solution of the differential equation is not easy many authors try to simplify of entry conditions. Some authors have replaced the circular arch of the contact zone of rolls by straight line, polygonal curve or parabola for simplifications of solution. These simplifications allow to obtain analytical solution differential equation but with acceptation some inaccuracy of the final results. Another solution of the differential equation was focused on the substitution of the analytical solution by the numerical solution, but should also expect some uncertainty of the final results. A more sophisticated solution was given by Gubkin is based on defining a constant shear stress and the approximation of the circular arch through the straight line. Gubkin for analytic solution of differential equation used one constant that includes the friction coefficient and second constant which is including the geometry of the rolling gap. The contribution of this paper is an original analytical solution of the differential equation based on the description of the contact arc by the equation of a circle. The proposed solution for the calculation of normal stress distribution is described by two constants. The first constant is describing the geometry of the rolling gap and the second describes friction coefficient. The final solution of differential equation is sum of two independent functions involving the shear stress as a variable value. The proposed solution does not consider with material work hardening during processing.

The Drainage Consolidation Modeling of Sand Drain in Red Mud Tailing and Analysis on the Change Law of the Pore Water Pressure

In order to prevent the occurring of dam failure and leakage, sand-well drainages systems were designed and constructed in red mud tailing. It is critical to focus on the change law of the pore water pressure. The calculation model of single well drainage pore water pressure was established. The pore water pressure differential equation was deduced and the analytical solution of differential equation using Bessel function and Laplace transform was given out. The impact of parameters such as diameterd, separation distancel, loading rateq, and coefficient of consolidationCvin the function on the pore water pressure is analyzed by control variable method. This research is significant and has great reference for preventing red mud tailings leakage and the follow-up studies on the tailings stability.

Numerical Simulation of Orientation Distribution on Fibers Suspensions in Planar Extensional Field

A mathematical model of evolution process is adopted to simulate orientation distribution of fibers suspensions in planar extensional flow, i.e., specific form of Fokker-Plank partial differential equation and Jeffery equation. The analytical solution of differential equation on forecast fiber orientation distribution is deduced.

Orientation Distribution of Fiber Suspensions in Extensional Flow

An orientation distribution function is adopted to describe three-dimensional orientation distribution of short fibers suspensions in extensional flow. A mathematical model of evolution process on fiber orientation distribution function is established by analytical method. Numerical simulation is also used to describe two and three dimensional orientation distribution of fibers. Therefore, analytical solution of differential equation on forecast fiber orientation distribution is deduced.

ON A METHOD FOR CONSTRUCTING SOLUTIONS OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER WITH A HADAMARD TYPE OPERATOR

In this paper we investigate a method for constructing solutions of differential equations of fractional order with Hadamard type derivative. The basis of this method is the construction of systems concerning the operator of fractional differentiation with Hadamard type derivative. We define new classes of special functions generalizing the Mittag-Leffler functions, and related to the differential equations of fractional order. Properties of these functions and their applications to solutions of fractional order differential equations are established. Analytical solutions of differential equations of fractional order are found as a series. Moreover, we consider normed systems of non-homogeneous differential equations of fractional order.

The Method of Multiple Scales in Solving Nonlinear Dynamic Differential Equations of Gear Systems

To obtain exact analytical solutions of differential equations of gear system dynamics due to the difficulty of solving complicated differential equations. Only the approximate analytical solutions can be determined. The method of multiple scales is one of the most powerful, popular perturbation methods. The dynamic model which describes the torsional vibration behaviors of gear system has been introduced accurately in this paper. The differential equation of gear system nonlinear dynamics exhibiting combined nonlinearity influence such as time-varying stiffness, tooth backlash and dynamic transmission error (DTE) has been proposed. The theory of multiple scales method has been presented in solving nonlinear differential equations of gear systems and the frequency response equation has been obtained. The fact that the approximate analytical solution by using the method of multiple scales is in good agreement with the exact solutions by numerically integrating differential equations has proved that the method of multiple scales is one of the most frequently used methods in solving differential equations, especially for large and complicated differential equations.

Analytical Solution for Free Vibrations of a Moderately Thick Rectangular Plate

In the present thick plate vibration theory, governing equations of force-displacement relations and equilibrium of forces are reduced to the system of three partial differential equations of motion with total deflection, which consists of bending and shear contribution, and angles of rotation as the basic unknown functions. The system is starting one for the application of any analytical or numerical method. Most of the analytical methods deal with those three equations, some of them with two (total and bending deflection), and recently a solution based on one equation related to total deflection has been proposed. In this paper, a system of three equations is reduced to one equation with bending deflection acting as a potential function. Method of separation of variables is applied and analytical solution of differential equation is obtained in closed form. Any combination of boundary conditions can be considered. However, the exact solution of boundary value problem is achieved for a plate with two opposite simply supported edges, while for mixed boundary conditions, an approximate solution is derived. Numerical results of illustrative examples are compared with those known in the literature, and very good agreement is achieved.

Approximate Analytical Solutions of Fractional Perturbed Diffusion Equation by Reduced Differential Transform Method and the Homotopy Perturbation Method

The approximate analytical solutions of differential equations with fractional time derivative are obtained with the help of a general framework of the reduced differential transform method (RDTM) and the homotopy perturbation method (HPM). RDTM technique does not require any discretization, linearization, or small perturbations and therefore it reduces significantly the numerical computation. Comparing the methodology (RDTM) with some known technique (HPM) shows that the present approach is effective and powerful. The numerical calculations are carried out when the initial conditions in the form of periodic functions and the results are depicted through graphs. The two different cases have studied and proved that the method is extremely effective due to its simplistic approach and performance.

On approximate analytical solutions of differential equations in enzyme kinetics using homotopy perturbation method

Homotopy perturbation method is used to extend the approximate analytical solutions of non-linear reaction equations describing enzyme kinetics for combinations of parameters for which solutions obtained in previous works are not valid. Also, by constructing a new homotopy, alternative approximate analytical expressions for substrate, substrate-enzyme complex and product concentrations are found. These first-order approximate solutions give more accurate results than the second-order approximations derived in previous works.

Аналитическое решение дифференциальных уравнений ориентации круговой орбиты космического аппарата

Аналитическое решение дифференциальных уравнений ориентации круговой орбиты космического аппарата

Effective Parameters Determining the Information Flow in Hierarchical Biological Systems

Signaling networks are abundant in higher organisms. They play pivotal roles, e.g., during embryonic development or within the immune system. In this contribution, we study the combined effect of the various kinetic parameters on the dynamics of signal transduction. To this end, we consider hierarchical complex systems as prototypes of signaling networks. For given topology, the output of these networks is determined by an interplay of the single parameters. For different kinetics, we describe this by algebraic expressions, the so-called effective parameters.When modeling switch-like interactions by Heaviside step functions, we obtain these effective parameters recursively from the interaction graph. They can be visualized as directed trees, which allows us to easily determine the global effect of single kinetic parameters on the system's behavior. We provide evidence that these results generalize to sigmoidal Hill kinetics.In the case of linear activation functions, we again show that the algebraic expressions can be immediately inferred from the topology of the interaction network. This allows us to transform time-consuming analytic solutions of differential equations into a simple graph-theoretic problem. In this context, we also discuss the impact of our work on parameter estimation problems. An issue is that even the fitting of identifiable effective parameters often turns out to be numerically ill-conditioned. We demonstrate that this fitting problem can be reformulated as the problem of fitting exponential sums, for which robust algorithms exist.

Fitting evolutionary process of matrix protein 2 family from influenza A virus using analytical solution of differential equation

The evolution of protein family is a process along the time course, thus any mathematical methods that can describe a process over time could be possible to describe an evolutionary process. In our previously concept-initiated study, we attempted to use the differential equation to describe the evolution of hemagglutinins from influenza A viruses, and to discuss various issues related to the building of differential equation. In this study, we attempted not only to use the differential equation to describe the evolution of matrix protein 2 family from influenza A virus, but also to use the analytical solution to fit its evolutionary process. The results showed that the fitting was possible and workable. The fitted model parameters provided a way to further determine the evolutionary dynamics and kinetics, a way to more precisely predict the time of occurrence of mutation, and a way to figure out the interaction between protein family and its environment.