Математическое моделирование при учёте силы тяжести течений жидкости, возникающих в результате разрушения плотины

A method for reducing systems of partial differential equations to corresponding systems of ordinary differential equations is proposed. A system of equations describing two-dimensional, cylindrical, and spherical flows of a polytropic gas; a system of dimensionless Stokes equations for the dynamics of a viscous incompressible fluid; a system of Maxwell’s equations for vacuum; and a system of gas dynamics equations in cylindrical coordinates are studied. It is shown how this approach can be used for solving certain problems (shockless compression, turbulence, etc.).

This paper presents a semi-analytical method for solving a system of equations of gas dynamics in Eulerian coordinates. Since only spatial derivatives are replaced by finite differences, the system of gas dynamic equations is reduced to a system of ordinary differential equations on a spatial grid. An approximate analytical solution of this system of differential equations for a small time-interval is used to describe the dynamics of a gas in the entire required time interval. Verification was carried out on one-dimensional test problems on the decay of an arbitrary discontinuity and the propagation of stationary shock waves of various intensities. To compare one-dimensional problems, the solution of test problems is given by the simple-to-implement basic particle-in-cell method. It is shown that the semianalytical method has high accuracy of calculations, and is also the most universal method for calculating applied problems.

Using the Lie symmetry approach, we have examined herein the system of partial differential equations corresponding to the Einstein–Maxwell equations for a static axially symmetric spacetime. The method used reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations is all optimal subgroups. The system of ordinary differential equations is further solved in general to obtain exact solutions. Several new physically important families of exact solutions are derived.

Using the Lie symmetry approach, the authors have examined exact and numerical solutions of coupled short pulse equation with time-dependent coefficients. The method reduces the system of partial differential equations to a system of ordinary differential equations according to the Lie symmetry admitted. In particular, we found the relevant system of ordinary differential equations for all optimal subgroups. The system of ordinary differential equations is further studied in general to obtain exact and numerical solutions. Several new physically important families of exact and numerical solutions are derived.

Numerical use of {[formula omitted]}-approximation method thermoelastic anisotropic thin shell theory equations represented in a special form

This paper considers one-dimensional equations of acoustics equations of inhomogeneous media and the system of gas dynamics equations with constant entropy. Using the Riemann approach, the gas dynamics equations are reduced to a second-order linear hyperbolic equation with variable coefficients. Solutions to this equation are constructed using Euler–Darboux transformations. This allows us to find new exact solutions of the equations of acoustics and gas dynamics, depending on two arbitrary functions.

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

ABSTRACTThe Lie group transformation method is applied for solving the problem of mixed convection flow with mass transfer over a permeable stretching surface with Soret and Dufour effects. The application of Lie group method reduces the number of independent variables by one and consequently the system of governing partial differential equations reduces to a system of ordinary differential equations with appropriate boundary conditions. Further, the reduced non-linear ordinary differential equations are solved numerically by using the shooting method. The effects of various parameters governing the flow and heat transfer are shown through graphs and discussed. Our aim is to detect new similarity variables which transform our system of partial differential equations to a system of ordinary differential equations. In this work a special attention is given to investigate the effect of the Soret and Dufour numbers on the velocity, temperature and concentration fields above the sheet.

We consider a two-component reaction-diffusion model that describes the oxygenation of CO molecules on the surface of platinum in the one-dimensional case. The partial differential equations of the model are reduced to a system of ordinary differential equations. We show that the system of partial differential equations with fixed parameter values has a family of autowave solutions running along the spatial axis at various velocities. These solutions are described by some singular attractors and limit cycles of the corresponding period in the system of ordinary differential equations.

<div class="section abstract"><div class="htmlview paragraph">In this paper, a new solution method is presented to study the effect of wave propagation in engine manifolds, which includes solving one-dimensional models for compressible flow of air. Velocity, pressure, and density profiles are found by solving a system of non-linear Partial Differential Equations (PDEs) in space and time derived from Euler’s equations. The 1D model includes frictional losses, area change, and heat transfer. The solution is traditionally found by utilizing the Method of Characteristics and applying finite difference solutions to the resulting system of ordinary differential equations (ODEs) over a discretized grid. In this work, orthogonal collocation is used to solve the system of ODEs that is defined along the characteristic curves. Orthogonal polynomials are utilized to approximate velocity, pressure, sound speed, and the characteristic curves along which the system of PDEs reduce to a system of ODEs. The approximation polynomials are defined over the whole manifold domain, transforming Euler’s equations into a system of ODEs that can be solved using a generic ODE solver. This reduction is done symbolically using a computer algebra system (Maple). The method results in a system of ODEs that has a higher spatial order along the whole space compared to methods based on finite differences, reducing the number of nodes required to find an acceptable solution that captures the state dynamics at different locations inside the manifold in real time. The proposed model is compared against the Method of Characteristics (MOC) that is used as a reference model; this comparison includes the states at the inlet, outlet, and midpoint. In summary, a high order method that can calculate solutions of the 1D manifold model equations is developed by finding the respective polynomial approximations along the 1D space and solving the resultant system using a generic solver in real time.</div></div>

The analysis of dynamic deformation processes is an important task in civil and in mechanical engineering. One main emphasis is set on the derivation of the objects transmission function, represented by systems of ordinary or partial differential equations (“white box”-models). The parametric identification of the initial and boundary conditions and relevant material parameters is one of the new challenges in the application of least-square algorithms. Embedding the investigation of the deformation process in a combination of a theoretical and an experimental system analysis suitable methods are provided with adaptive KALMAN-filtering as central identification tool. In this paper the approach of a full system analysis is shown quantifying a dynamic “white-box”-model for the calculation of thermal deformations of bar-shaped machine elements. The task was motivied from mechanical engineering searching new methods for the precise prediction and computational compensation of thermal influences in the heating and cooling phases of machine tools (i.e. robot arms, etc.). The quantification of thermal deformations under variable dynamic loads requires the modelling of the non-stationary spatial temperature distribution inside the object. Based upon FOURIERS law of heat flow the high-grade non-linear temperature gradient is represented by a system of partial differential equations within the framework of a dynamic Finite Element topology. It is shown that adaptive KALMAN-filtering is suitable to quantify relevant disturbance influences and to identify thermal parameters (i.e. thermal diffusivity) with a deviation of only 0,2%. As result an identified (and verified) parametric model for the realistic prediction respectively simu-lation of dynamic temperature processes is presented. Classifying the thermal bend as the main deformation quantity of bar-shaped machine tools, the temperature model is extended to a temperature deformation model. In lab tests with an aluminium column control measurements show that the identified model can be used to predict the columns bend with a mean deviation smaller than 10 mgon. Consequently the deformation model is a precise predictor and suitable for realistic simulations of thermal deformations under variable dynamic loads. In future our activities will be primarily focussed on applications in industrial manufacturing. 1. Parametric identification of dynamic deformation processes The creation of a realistic parametric dynamic deformation model is directly associated with the task to combine systems of differential equations (theoretical physical model) with empirical measurements. Consequently basic principles of system theory can be adapted to the geodetic deformation analysis. The parametric identification of a dynamic “white box”-model 3rd IAG / 12th FIG Symposium, Baden, May 22-24, 2006

Classical theory for systems of the first order partial differential equations for a scalar function can be rephrased as the submanifold theory of contact manifolds (geometric first order jet spaces). In the same spirit, we will develop the geometric theory of systems of partial differential equations of second order for a scalar function as the Contact Geometry of Second Order, following E. Cartan.We will formulate the submanifold theory of second order jet spaces as the geometry of PD manifolds (R;D 1 ,D 2 ) of second order. Moreover we will establish the First Reduction Theorem for (R;D 1,D 2) admitting non-trivial Cauchy characteristic systems. By utilizing Parabolic Geometry, we will give, directly or combined with reduction theorems, several classes of systems of partial differential equations of second order, for which the model equation of each class admits the Lie algebra of infinitesimal contact transformations, which is finite dimensional and simple.

Construction of similarity transformations and analytic solutions for a liquid film on an unsteady stretching sheet using lie point symmetries

In this work, the solution of the two temperature dual-lag-phase model in half space under the uncertain thermal loading at boundary has been discussed. A system of fuzzy partial differential equations arise due to study. System of fuzzy partial differential equations are solved by assuming all the partial derivatives as one differentiable. Laplace transformation is used to solve the partial differential equations related to the problem and using the eigen value approach, the system of ordinary differential equations in Laplace transformed domain is solved. The inverse of such Laplace transformations are carried out by the help of numerical computation. The observations based upon the numerical results are illustrated graphically for stress, strain, temperatures and displacement distributions.

Exact solutions of some nonlinear systems of partial differential equations by using the first integral method