scholarly journals Forced oscillations of the main gas pipeline open section during the cleaning piston passage

2020 ◽  
pp. 7-15
Author(s):  
V. Ya. Grudz ◽  
T. F. Tutko ◽  
O. Ya. Dubei
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Infinite Series ◽  
Fourier Method ◽  
Gas Pipeline ◽  
Forced Oscillations ◽  
Inertial Load ◽  
Pipeline Section

The problem of forced oscillations of an open section of a gas pipeline during the cleaning piston passage belongs to the type of problems of forced oscillations of one-dimensional elastic objects under the influence of a moving inertial load on them. Currently, there are two ways to solve such problems. The first way is related to the integration of the partial differential equation and the solution of such problems is a superposition of eigen-oscillations and accompanying oscillations. The second way does not involve the integration of the partial dif-ferential equation. Methods of generalized coordinates, generalized displacements and various numerical methods belong to the second type of solving. None of the mentioned methods is simple. Therefore, the authors suggest the method, in which the first mathematical model provides the determination of forced oscillations of the gas pipeline section during the passage of the cleaning piston without taking into account its inertial load on the gas pipeline. In future, on the basis of the first model it is planned to develop the second mathematical model which will provide an approximate determination of the deflections of the pipeline axis, taking into account the inertial load of the piston on the pipeline. The purpose of this article is to obtain a solution to the problem of forced oscillations of the pipeline section during the passage of the cleaning piston without taking into account the inertial forces on the pipeline. The problem is solved by partial differential equation, Fourier method is applied. The right side of the non-homogeneous differential equation is decomposed into an infinite series, which is the sum of the produc-tions of the eigenfunctions of the pipeline section free oscillations and the unknown function of time. After finding out this function, the authors determine the unknown time function in the Fourier method and hence the solution of the problem in the form of an infinite series, the summands of which lessen rapidly. The authors calculate the deflections of the pipeline axis along the entire section of the gas pipeline for different points of time, as well as deflections of individual sections changing in time and moments of deflection.

A Meshless Method for Solving the Mathematical Model Associated the Leakage Problem in Gas Pipeline

2014 ◽  
Vol 986-987 ◽  
pp. 1418-1421
Author(s):  
Jun Shan Li
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Basis Function ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Gas Pipeline ◽  
The Mathematical Model

In this paper, we propose a meshless method for solving the mathematical model concerning the leakage problem when the pressure is tested in the gas pipeline. The method of radial basis function (RBF) can be used for solving partial differential equation by writing the solution in the form of linear combination of radius basis functions, that is, when integrating the definite conditions, one can find the combination coefficients and then the numerical solution. The leak problem is a kind of inverse problem that is focused by many engineers or mathematical researchers. The strength of the leak can find easily by the additional conditions and the numerical solutions.

scholarly journals Determination of a control parameter in a parabolic partial differential equation

1991 ◽  
Vol 33 (2) ◽  
pp. 149-163 ◽  
Author(s):  
J. R. Cannon ◽  
Yanping Lin ◽  
Shingmin Wang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Global Solution ◽  
Existence And Uniqueness ◽  
Control Parameter ◽  
Parabolic Partial Differential Equation ◽  
Solution Pair ◽  
Image Position

AbstractThe authors consider in this paper the inverse problem of finding a pair of functions (u, p) such thatwhere F, f, E, s, αi, βi, γi, gi, i = 1, 2, are given functions.The existence and uniqueness of a smooth global solution pair (u, p) which depends continuously upon the data are demonstrated under certain assumptions on the data.

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

2013 ◽  
Vol 395-396 ◽  
pp. 1174-1178
Author(s):  
Pei Fang Luo ◽  
Zan Huang
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Fiber Orientation ◽  
Extensional Flow ◽  
Orientation Distribution ◽  
Specific Form ◽  
Fiber Orientation Distribution ◽  
Planar Extensional Flow

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.

scholarly journals Generating Functions for Bessel Functions

1959 ◽  
Vol 11 ◽  
pp. 148-155 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Generating Function ◽  
Lie Group ◽  
Generating Functions ◽  
Bessel Functions ◽  
Partial Differential ◽  
Systematic Determination ◽  
Image Position

On replacing the parameter n in Bessel's differential equation1.1by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where1.2This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

scholarly journals The evolution of pebble size and shape in space and time

2012 ◽  
Vol 468 (2146) ◽  
pp. 3059-3079 ◽  
Author(s):  
G. Domokos ◽  
G. W. Gibbons
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Markov Process ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Direct Simulation ◽  
Size And Shape ◽  
Segregation Process ◽  
Strong Segregation

We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size, are interrelated. Our model is based on a system of ordinary differential equations (ODEs) called the box equations that describe the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's partial differential equation (PDE) describing collisional abrasion and verify them by simple experiments and by direct simulation of the PDE. Although representing a radical simplification of the latter, our system admits the inclusion of additional terms related to frictional abrasion. We show that non-trivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only be stabilized by the ongoing segregation process.

Sign in / Sign up

Export Citation Format

Share Document