scholarly journals PROBABILISTIC SPACE FOR CALCULATION OF PROBABILISTIC CHARACTERISTICS OF A THREE-PARAMETER QUEUEING SYSTEM MODEL

2017 ◽  
pp. 100-109
Author(s):  
Valery Pavsky ◽  
Valery Pavsky ◽  
Kirill Pavsky ◽  
Kirill Pavsky ◽  
Svetlana Ivanova ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Probability Distribution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Function ◽  
Queueing Theory ◽  
Probability Space ◽  
Queueing System ◽  
Probabilistic Space

A model of queueing theory is proposed that describes a queueing system with three parameters, which has important practical applications. The model is based on the continuous time Markov process with a discrete number of states. The model is formalized by a probabilistic space in which the space of elementary events is a set of inconsistent states of the queueing system; and the probabilistic measure is a probability distribution corresponding to a set of elementary events, that is, each elementary event is associated with the probability of the system staying in this state, for each fixed time moment. The model is represented by a system of ordinary differential equations, compiled by methods of queueing theory (Kolmogorov equations). To find the solution of the system of equations, the method of generating functions is used. For the generating function, a partial differential equation is obtained. Finding the generating function completes the construction of a probability space. The latter means that for any random variables and functions defined on the resulting probability space, one can find their probabilistic characteristics. In particular, analytical expressions of the moments (mathematical expectations and variances) of random functions that depend on time are obtained. The peculiarity of finding a solution is that it is obtained not from the probability distribution, but directly from the partial differential equation, which represents a system of ordinary differential equations. For the probability distribution, the solution was found by a combinatorial method, which made it possible to significantly reduce the computations. To apply the formulas in engineering calculations, we consider the stationary case, to which a considerable simplification of the calculations corresponds. A relationship between a system of differential equations and a polynomial distribution known in probability theory is shown. The results are used in the analysis of the reliability of the operation of scalable computing systems; graphical implementation is shown

Parametrizations of sub-attractors in hyperbolic balance laws

2012 ◽  
Vol 142 (3) ◽  
pp. 563-583
Author(s):  
Julia Ehrt
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Global Attractor ◽  
Balance Laws ◽  
Number Of Zeros ◽  
Hyperbolic Balance Laws ◽  
All Solutions ◽  
Finite Dimensionality

We investigate the properties of the global attractor of hyperbolic balance laws on the circle, given byut + f(u)x = g(u).The new tool of sub-attractors is introduced. They contain all solutions on the global attractor up to a given number of zeros. The paper proves finite dimensionality of all sub-attractors, provides a full parametrization of all sub-attractors and derives a system of ordinary differential equations for the embedding parameters that describe the full partial differential equation dynamics on the sub-attractor.

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.

Practical Applications of Optimal-Control Theory to History-Matching Multiphase Simulator Models

1975 ◽  
Vol 15 (04) ◽  
pp. 347-355 ◽  
Author(s):  
M.L. Wasserman ◽  
A.S. Emanuel ◽  
J.H. Seinfeld
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
History Matching ◽  
Single Phase ◽  
Continuous Functions ◽  
Partial Differential

Abstract This paper applies material presented by Chen et al. and by Chavent et al to practical reservoir problems. The pressure history-matching algorithm used is initially based on a discretized single-phase reservoir model. Multiphase effects are approximately treated in the single-phase model by multiplying the transmissibility and storage terms by saturation-dependent terms that are obtained from a multiphase simulator run. Thus, all the history matching is performed by a "pseduo" single-phase model. The multiplicative factors for transmissibility and storage are updated when necessary. The matching technique can change any model permeability thickness or porosity thickness value. Three field examples are given. Introduction History matching using optimal-control theory was introduced by two sets of authors. Their contributions were a major breakthrough in attacking the long-standing goal of automatic history matching. This paper extends the work presented by Chen et al. and Chavent et al. Specifically, we focus on three areas.We derive the optimal-control algorithm using a discrete formulation. Our reservoir simulator, which is a set of ordinary differential equations, is adjoined to the function to be minimized. The first variation is taken to yield equations for computing Lagrange multipliers. These Lagrange multipliers are then used for computing a gradient vector. The discrete formulation keeps the adjoint equations consistent with the reservoir simulator.We include the effects of saturation change in history-matching pressures. We do this in a fashion that circumvents the need for developing a full multiphase optimal-control code.We show detailed results of the application of the optimal-control algorithm to three field examples. DERIVATION OF ADJOINT EQUATIONS Most implicit-pressure/explicit-saturation-type, finite-difference reservoir simulators perform two calculation stages for each time step. The first stage involves solving an "expansivity equation" for pressure. The expansivity equation is obtained by summing the material-balance equations for oil, gas, and water flow. Once the pressures are implicitly obtained from the expansivity equation, the phase saturations can be updated using their respective balance equations. A typical expansivity equation is shown in Appendix B, Eq. B-1. When we write the reservoir simulation equations as partial differential equations, we assume that the parameters to be estimated are continuous functions of position. The partial-differential-equation formulation is partial-differential-equation formulation is generally termed a distributed-parameter system. However, upon solving these partial differential equations, the model is discretized so that the partial differential equations are replaced by partial differential equations are replaced by sets of ordinary differential equations, and the parameters that were continuous functions of parameters that were continuous functions of position become specific values. Eq. B-1 is a position become specific values. Eq. B-1 is a set of ordinary differential equations that reflects lumping of parameters. Each cell has three associated parameters: a right-side permeability thickness, a bottom permeability thickness, and a pore volume. pore volume.Once the discretized model is written and we have one or more ordinary differential equations per cell, we can then adjoin these differential equations to the integral to be minimized by using one Lagrange multiplier per differential equation. The ordinary differential equations for the Lagrange multipliers are now derived as part of the necessary conditions for stationariness of the augmented objective function. These ordinary differential equations are termed the adjoint system of equations. A detailed example of the procedure discussed in this paragraph is given in Appendix A. The ordinary-differential-equation formulation of the optimal-control algorithm is more appropriate for use with reservoir simulators than the partial-differential-equation derivation found in partial-differential-equation derivation found in Refs. 1 and 2. SPEJ P. 347

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

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