Completeness Property of One-Dimensional Perturbations of Normal and Spectral Operators Generated by First Order Systems

AbstractThe deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

Download Full-text

Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

Transport in Porous Media ◽

10.1007/s11242-021-01570-w ◽

2021 ◽

Author(s):

Amarjot Singh Bhullar ◽

Gospel Ezekiel Stewart ◽

Robert W. Zimmerman

Keyword(s):

Approximate Solution ◽

Pore Pressure ◽

Constant Pressure ◽

Initial Permeability ◽

Order Perturbation ◽

Zeroth Order ◽

One Dimensional ◽

First Order ◽

Outflow Boundary ◽

Perturbation Solutions

Abstract Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow. Article Highlights Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure. The predicted flowrate is accurate to within 5% for a wide range of permeability variations. If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

Download Full-text

NEW L1-GRADIENT TYPE ESTIMATES OF SOLUTIONS TO ONE-DIMENSIONAL QUASILINEAR PARABOLIC SYSTEMS

Communications in Contemporary Mathematics ◽

10.1142/s0219199710003725 ◽

2010 ◽

Vol 12 (01) ◽

pp. 85-106 ◽

Cited By ~ 1

Author(s):

S. N. ANTONTSEV ◽

J. I. DÍAZ

Keyword(s):

Type Equation ◽

Parabolic Type ◽

Parabolic Systems ◽

Diffusion Term ◽

One Dimensional ◽

First Order ◽

Estimates Of Solutions ◽

Gradient Type ◽

Degenerate Type ◽

Parabolic Type Equation

We consider a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of the degenerate type. We derive some new L1-gradient type estimates for its solutions which are uniform in the sense that they do not depend on the coefficients nor on the size of the spatial domain. We also give some applications of such estimates to gas dynamics, filtration problems, a p-Laplacian parabolic type equation and some first order systems of Hamilton–Jacobi or conservation laws type.

Download Full-text

System Properties of One-Dimensional Distributed Systems

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3427094 ◽

1977 ◽

Vol 99 (2) ◽

pp. 85-90 ◽

Cited By ~ 4

Author(s):

L. S. Bonderson

Keyword(s):

Distributed Systems ◽

Sufficient Conditions ◽

State Variables ◽

One Dimensional ◽

First Order ◽

Lumped Element ◽

Order Form ◽

Power Product ◽

System Properties ◽

Necessary And Sufficient

The system properties of passivity, losslessness, and reciprocity are defined and their necessary and sufficient conditions are derived for a class of linear one-dimensional multipower distributed systems. The utilization of power product pairs as state variables and the representation of the dynamics in first-order form allows results completely analogous to those for lumped-element systems.

Download Full-text

Study of shocks in a nonideal dusty gas using Maslov, Guderley, and CCW methods for shock exponents

Zeitschrift für Naturforschung A ◽

10.1515/zna-2021-0049 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Swati Chauhan ◽

Antim Chauhan ◽

Rajan Arora

Keyword(s):

Shock Waves ◽

Mass Fraction ◽

Strong Shock ◽

Dust Particles ◽

Excluded Volume ◽

One Dimensional ◽

First Order ◽

Nonideal Gas ◽

Spherical Shock ◽

Strong Shock Waves

Abstract In this work, we consider the system of partial differential equations describing one-dimensional (1D) radially symmetric (i.e., cylindrical or spherical) flow of a nonideal gas with small solid dust particles. We analyze the implosion of cylindrical and spherical symmetric strong shock waves in a mixture of a nonideal gas with small solid dust particles. An evolution equation for the strong cylindrical and spherical shock waves is derived by using the Maslov technique based on the kinematics of 1D motion. The approximate value of the similarity exponent describing the behavior of strong shocks is calculated by applying a first-order truncation approximation. The obtained approximate values of similarity exponent are compared with the values of the similarity exponent obtained from Whitham’s rule and Guderley’s method. All the above computations are performed for the different values of mass fraction of dust particles, relative specific heat, and the ratio of the density of dust particle to the density of the mixture and van der Waals excluded volume.

Download Full-text

Higher-order accurate compact difference solutions for vibration problems of one dimensional continuous systems

European Journal of Computational Mechanics ◽

10.13052/ejcm.19.771-794 ◽

2011 ◽

pp. 771-794

Author(s):

Mondher Yahiaoui

Keyword(s):

Natural Frequencies ◽

Continuous Systems ◽

One Dimensional ◽

First Order ◽

Compact Difference ◽

Transverse Vibrations ◽

Seventh Order ◽

One Step ◽

Vibration Problems ◽

Difference Methods

In this paper, we present a fourth-order accurate and a seventh-order accurate, one-step compact difference methods. These methods can be used to solve initial or boundaryvalue problems which can be modeled by a first-order linear system of differential equations. It is then shown in detail how these methods can be used to solve vibration problems of onedimensional continuous systems. Natural frequencies of a cantilever beam in transverse vibrations are computed and the results are compared to analytical ones to prove the high accuracy and efficiency of both methods. A comparison was also made to a finite element solution and the results have shown that both compact-difference methods yield more accurate values even with a reduced number of intervals.

Download Full-text

Optimal systems and their group-invariant solutions to geodesic equations

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819501354 ◽

2019 ◽

Vol 16 (09) ◽

pp. 1950135

Author(s):

Bismah Jamil ◽

Tooba Feroze ◽

Muhammad Safdar

Keyword(s):

Nonlinear Systems ◽

Separation Of Variables ◽

Noether Symmetries ◽

Invariant Solutions ◽

One Dimensional ◽

Group Invariant ◽

First Order ◽

Geodesic Equations ◽

Integrating Factor ◽

Group Invariant Solutions

We find one-dimensional optimal systems of the Lie subalgebras of Noether symmetries associated with systems of geodesic equations. Further, we find invariants corresponding to each element of the derived optimal system. The derived invariants are shown to reduce systems of geodesic equations (nonlinear systems of quadratically semi-linear second-order ordinary differential equations (ODEs)) to nonlinear systems of first-order ODEs. The resulting systems are solved via known methods (e.g. separation of variables, integrating factor, etc.). In some cases, we provide exact solutions of these systems of geodesic equations.

Download Full-text

First-order calculation of the phonon propagator of a quasi-one-dimensional electron-phonon system

Zeitschrift für Physik B Condensed Matter ◽

10.1007/bf02026430 ◽

1982 ◽

Vol 48 (1) ◽

pp. 59-65 ◽

Cited By ~ 5

Author(s):

W. Gasser

Keyword(s):

Dimensional Electron ◽

One Dimensional ◽

First Order ◽

Order Calculation

Download Full-text

First-order transitions in one-dimensional systems with local couplings

Journal of Statistical Physics ◽

10.1007/bf01053741 ◽

1991 ◽

Vol 65 (3-4) ◽

pp. 483-494 ◽

Cited By ~ 8

Author(s):

M. Asorey ◽

J. G. Esteve

Keyword(s):

One Dimensional ◽

First Order

Download Full-text

A STUDY OF THE DEFLECTIONS OF METAL ROAD GUARDRAIL ELEMENTS

Transport ◽

10.3846/1648-4142.2009.24.225-233 ◽

2009 ◽

Vol 24 (3) ◽

pp. 225-233 ◽

Cited By ~ 10

Author(s):

Olegas Prentkovskis ◽

Andrey Beljatynskij ◽

Rasa Prentkovskienė ◽

Ivan Dyakov ◽

Laima Dabulevičienė

Keyword(s):

Mathematical Model ◽

Finite Elements ◽

Traffic Accidents ◽

Statistical Data ◽

Elastic Deformations ◽

One Dimensional ◽

First Order ◽

Horizontal Beam ◽

Deformation Processes ◽

The Impact

Statistical data on traffic accidents in 2008 in Lithuania is presented. Referring to statistical data, grounding on an obstacle’ makes one‐tenth of all registered traffic accidents ‐ 9.4% (an obstacle may be a road guardrail, a lamp post, a tree, a bar, a gate, etc.). Road guardrails of various types are installed on the shoulders and dividing strips of urban and suburban roads. They are as follows: reinforced concrete guardrails, cable guardrails and metal guardrails. Metal guardrails, consisting of S‐shape metal posts and a protective W‐shape horizontal beam, are most popular. The authors of the present paper examine the deformation processes of the elements of the above mentioned guardrail. A mathematical model of metal road guardrail was developed. Metal road guardrail was modelled using one‐dimensional first‐order finite elements, taking into account only elastic deformations, as well as the effect of soil on the buried post section of the guardrail. Based on the developed mathematical model of metal road guardrail, the deflections of its elements caused by the impact of a vehicle moving at varying speed were determined. The obtained values of deflections of guardrail elements (a protective W‐shape horizontal beam and a S‐shape post) presented in paper do not exceed the admissible values (of beam deflections).

Download Full-text