V—The Brownian Movement of a One-Dimensional Non-Harmonic Oscillator

1968 ◽  
Vol 68 (1) ◽  
pp. 70-82
Author(s):  
A. J. Allnutt
Keyword(s):  
Approximate Solution ◽  
Simple Model ◽  
Harmonic Oscillator ◽  
Langevin Equation ◽  
Iterative Procedure ◽  
Anharmonic Oscillator ◽  
Crystalline Solid ◽  
Brownian Movement ◽  
One Dimensional ◽  
Numerical Example

SynopsisThe Langevin equation for the harmonic oscillator is solved by a different method from that normally used. The approximate solution for the case of the slightly anharmonic oscillator is then obtained by an iterative procedure and the results are illustrated by a numerical example based on a simple model of a crystalline solid.

On Improved Approximate Solution of the Fredholm Integral Equation

2006 ◽  
Vol 6 (3) ◽  
pp. 264-268
Author(s):  
G. Berikelashvili ◽  
G. Karkarashvili
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Fredholm Integral Equation ◽  
Algebraic Equations ◽  
Order Convergence ◽  
One Dimensional ◽  
Averaging Operator ◽  
Linear Algebraic Equations ◽  
Convergence Solution

AbstractA method of approximate solution of the linear one-dimensional Fredholm integral equation of the second kind is constructed. With the help of the Steklov averaging operator the integral equation is approximated by a system of linear algebraic equations. On the basis of the approximation used an increased order convergence solution has been obtained.

scholarly journals Approximate solution for fractional attractor one-dimensional Keller-Segel equations using homotopy perturbation sumudu transform method

2020 ◽  
Vol 9 (1) ◽  
pp. 370-381
Author(s):  
Dinkar Sharma ◽  
Gurpinder Singh Samra ◽  
Prince Singh
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Simple Technique ◽  
Nonlinear Partial Differential Equations ◽  
Transform Method ◽  
Present Scheme ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
Sumudu Transform

AbstractIn this paper, homotopy perturbation sumudu transform method (HPSTM) is proposed to solve fractional attractor one-dimensional Keller-Segel equations. The HPSTM is a combined form of homotopy perturbation method (HPM) and sumudu transform using He’s polynomials. The result shows that the HPSTM is very efficient and simple technique for solving nonlinear partial differential equations. Test examples are considered to illustrate the present scheme.

Reliable one-dimensional approximate solution of insulated oval duct

2008 ◽  
Vol 49 (8) ◽  
pp. 2214-2224 ◽  
Author(s):  
Wen-Lih Chen ◽  
King-Leung Wong ◽  
Tsung-Lieh Hsien ◽  
Ching-Te Huang
Keyword(s):  
Approximate Solution ◽  
One Dimensional

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

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.

A simple model of a one-dimensional, randomly rough, non-Gaussian surface

2016 ◽  
Author(s):  
E. R. Méndez ◽  
G. D. Jiménez ◽  
A. A. Maradudin
Keyword(s):  
Simple Model ◽  
One Dimensional ◽  
Non Gaussian ◽  
Gaussian Surface

scholarly journals A simple model to estimate ice ablation under a thick debris layer

2006 ◽  
Vol 52 (179) ◽  
pp. 528-536 ◽  
Author(s):  
Han Haidong ◽  
Ding Yongjing ◽  
Liu Shiyin
Keyword(s):  
Heat Flux ◽  
Simple Model ◽  
Iterative Procedure ◽  
Indirect Comparison ◽  
Tien Shan ◽  
Balance Model ◽  
Complex Phase ◽  
Proposed Model ◽  
Koxkar Glacier

AbstractThis paper presents a simple model to estimate ice ablation under a thick supraglacial debris cover. The key method employed in the model is to establish a link between the debris heat flux and the debris temperature at a certain depth when the heat transfer in the debris is described by a diffusion process. Given surface temperature, debris thermal properties and relevant boundary conditions, the proposed model can estimate mean debris temperature at interfaces of different debris layers using an iterative procedure, and then the heat flux for ice ablation. The advantage of the proposed model is that it only requires a few parameters to conduct the modeling, which is simpler and more applicable than others. The case study on Koxkar glacier, west Tien Shan, China, shows, in general, that the proposed model gives good results for the prediction of debris temperatures, except for an apparent phase shift between modeled and observed values. We suggest that this error is mainly due to complex phase relations between debris temperature and debris heat flux. The modeled ablation rates at three experimental sites also show good results, using a direct comparison with observed data and an indirect comparison with a commonly used energy-balance model.

One-Dimensional Harmonic Oscillator in Quantum Phase Space

2011 ◽  
Vol 110-116 ◽  
pp. 3750-3754
Author(s):  
Jun Lu ◽  
Xue Mei Wang ◽  
Ping Wu
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Quantum Phase ◽  
Space Representation ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Matrix Mechanics ◽  
The Matrix ◽  
Quantum Wave ◽  
The One

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

scholarly journals A detailed study of the "radioactive decay" of, and the penetration of α-particles into, a simplified one-dimensional nucleus

1929 ◽  
Vol 124 (795) ◽  
pp. 493-501 ◽  
Keyword(s):  
Simple Model ◽  
Common Knowledge ◽  
Radioactive Decay ◽  
Particle Emission ◽  
Decay Process ◽  
Characteristic Energy ◽  
One Dimensional ◽  
Reverse Process ◽  
The Law ◽  
Α Particles

1. Introduction .—Gamow's elegant deduction by general arguments of the law of radioactive decay by α-particle emission and his subsequent investigations on artificial disintegration suggested to us the desirability of investigating as closely as possible any simple model of a decaying nucleus as a verification of his general approximations. For the model chosen the exact investigation of the decay process is almost trivial. Since we obtained this, now some time ago, Dr. Gamow informed us that he had also obtained equivalent detailed results. Still more recently such results have been published by Kudar. We shall not therefore dwell upon them here. The application of the same ideals, however, to the reverse process of penetration presents points of very definite interest, which we think are well worth discussion. The main point that arises is that the chance of penetration α-particle is or is not equal to a characteristic energy of the nucleus itself. This is a point which is not dealt with by Gamow in his paper. We have discussed it with him, and now put forward the results we have obtained. Since the solution of the decay problem is required in the main discussion of the penetration of α-particles into the nucleus it is included here in 2 for reference. We must emphasise that we claim no novelty, except of detail, for the work of 2; the general lines by now are a matter of fairly common knowledge.

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