scholarly journals APPROXIMATE SOLUTION OF ONE PROBLEM ON ELECTRICAL OSCILLATIONS IN WIRES WITH THE USE OF POLYLOGARITHMS

2017 ◽  
Vol 60 (4) ◽  
pp. 334-340 ◽  
Author(s):  
P. G. Lasy ◽  
I. N. Meleshko
Keyword(s):  
Approximate Solution ◽  
Initial Conditions ◽  
Fourier Method ◽  
Special Kind ◽  
Elementary Functions ◽  
The Real ◽  
First Order ◽  
Homogeneous Wave ◽  
Transcendental Functions ◽  
Current Flows

The article considers a mixed problem with homogeneous boundary conditions for onedimensional homogeneous wave equation. Such a problem can arise, for example, when studying oscillations of current and voltage in the conductor through which electric current flows, while the line is free from distortion. The solution can be found with the use of the Fourier method in the form of trigonometric series. This representation is of purely theoretical interest, because the real calculation should be, first, to find a large number of coefficients of the integrals, which in itself is not a trivial task and, second, it is almost impossible to assess the error of the calculations. An alternative way of solving this problem based on the use of transcendental functions i. e. polylogarithms that represent complex power series of a special kind. The exact solution of the problem is expressed through the imaginary part of a polylogarithm of the first order on the single circle and the approximate one – via the real part of the dilogarithm. In addition, if the initial conditions in the problem are elementary functions, then the solution is also computed using elementary functions. A simple and effective error estimate of the approximate solution has been found. It does not depend on time and it has the first-order of accuracy regarding the step of a partitioning segment of the numerical axis on which the problem is considered. This valuation is uniform with respect to the variables of the problem – both spatial and temporal. 

scholarly journals Application of Polylogarithms to the Approximate Solution of the Inhomogeneous Telegraph Equation for the Distortionless Line

2019 ◽  
Vol 62 (5) ◽  
pp. 413-421
Author(s):  
P. G. Lasy ◽  
I. N. Meleshko
Keyword(s):  
Wave Equation ◽  
Approximate Solution ◽  
Unit Circle ◽  
Imaginary Part ◽  
Initial Conditions ◽  
Special Functions ◽  
Telegraph Equation ◽  
Time Range ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation

The paper deals with a mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortion. This problem is reduced to the analogous one for the one-dimensional inhomogeneous wave equation. Its solution can be found as the sum of the solution for a mixed homogeneous boundary value problem for the corresponding homogeneous wave equation and for the solution of a non-homogeneous wave equation with homogeneous boundary data and zero initial conditions. Solutions to both problems can be found by separating the variables in the form of a series of trigonometric functions of the line point with time-dependent coefficients. Such solutions are inconvenient for real application because they require calculation of a large number of integrals, and it is difficult to estimate the miscalculation. An alternative method for solving this problem is proposed, based on the use of special functions, viz. polylogarithms, which are complex power-series with power coefficients converging in a unit circle. The exact solution of the problem is expressed in the integral form via the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one is expressed in the form of a finite sum via the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All these parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding powers on the segment of the length equal to the period. This makes it possible to effectively find an approximate solution to the problem. Also, a simple and convenient error estimate of the approximate solution of the problem is found. It is linear with respect to the step of splitting the line and the step of splitting the time range in which the problem is considered. The score is uniform along the length of the line at each fixed point of time. A concrete example of solving the problem according to the proposed mode is presented; graphs of exact and approximate solutions are constructed.

scholarly journals Exact boundaries for the analytical approximate solution of a class of first-order nonlinear differential equations in the real domain

2021 ◽  
Vol 25 (2) ◽  
pp. 381-392
Author(s):  
Victor Nikolaevich Orlov ◽  
Oleg Aleksandrovich Kovalchuk
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Analytical Approximate Solution ◽  
The Real ◽  
First Order ◽  
Real Domain

Дано решение одной из задач аналитического приближенного метода для одного класса нелинейных дифференциальных уравнений первого порядка с подвижными особыми точками в вещественной области. Рассматриваемое уравнение в общем случае не разрешимо в квадратурах и имеет подвижные особые точки алгебраического типа. Это обстоятельство требует решение ряда математических задач. Ранее авторами была решена задача влияния возмущения подвижной особой точки на аналитическое приближенное решение. Это решение основывалось на классическом подходе и, при этом, существенно уменьшилась область применения аналитического приближенного решения, по сравнению с областью, полученной в доказанной теореме существования и единственности решения. Поэтому в статье предлагается новая технология исследования, основанная на элементах дифференциального исчисления. Этот подход позволяет получить точные границы для аналитического приближенного решения в окрестности подвижной особой точки. Получены новые априорные оценки для аналитического приближенного решения рассматриваемого класса уравнений, хорошо согласующиеся с известными для общей области действия. При этом, представленные результаты дополняют ранее полученные, существенно расширена область применения аналитического приближенного решения в окрестности подвижной особой точки. Приведенные расчеты согласуются с теоретическими положениями, о чем свидетельствуют эксперименты, проведенные с нелинейным дифференциальным уравнением, обладающим точным решением. Дана технология оптимизации априорных оценок погрешности с помощью апостериорных оценок. В исследованиях применялись ряды с дробными отрицательными степенями.

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.

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

On the Vibration of a Rotating Disk

1972 ◽  
Vol 39 (4) ◽  
pp. 1143-1144 ◽  
Author(s):  
S. Barasch ◽  
Y. Chen
Keyword(s):  
Approximate Calculation ◽  
Initial Conditions ◽  
Rotating Disk ◽  
Equation Of Motion ◽  
Outer Radius ◽  
Order Equation ◽  
System Of Differential Equations ◽  
Fourth Order Equation ◽  
First Order ◽  
Inner Radius

The equation of motion of a rotating disk, clamped at the inner radius and free at the outer radius, is solved by reducing the fourth-order equation of motion to a set of four first-order equations subject to arbitrary initial conditions. A modified Adams’ method is used to numerically integrate the system of differential equations. Results show that Lamb-Southwell’s approximate calculation of the frequency is justified.

scholarly journals Establishment and application of heat flow coupling model

2019 ◽  
Vol 23 (1) ◽  
pp. 207-218
Author(s):  
Jun He ◽  
Gao-Liang Peng ◽  
Ling-Tao Yu ◽  
Chen-Zheng Li ◽  
Chuan-Hao Li ◽  
...  
Keyword(s):  
Heat Flow ◽  
Real Time ◽  
Initial Conditions ◽  
Coupling Model ◽  
Productive Efficiency ◽  
Maximum Deviation ◽  
Inspection Method ◽  
The Real ◽  
Flow Coupling ◽  
Washing Process

Wax deposition on walls of oil pipes is a common occurrence in crude oil extraction and is one of the major impediments to oilfield production. The most common method of paraffin removal is superconducting car thermal washing. This study proposes a heat flow coupling model that can analyze the temperature of the tubing-casing annular space to solve the low efficiency problem caused by adjusting initial parameters empirically. Using the superconducting car thermal washing process at the test oil well in city of Daqing, Chine as research object, the real-time temperature of annulus under various initial conditions is acquired by the fully-distributed Raman optical fiber temperature monitoring system. Compared with the real time data, theoretical data has a maximum deviation of 5?C, this result verifies the accuracy of the model. Based on the model, the study investigates the optimal initial parameters of superconducting car thermal washing by taking effective depth as an optimization goal. The optimal parameters for oil wells with different working conditions are obtained to improve the effectiveness of paraffin removal and increase thermal efficiency. This study provides theoretical support and an inspection method to promote superconducting car thermal washing and paraffin removal as well as to improve productive efficiency.

scholarly journals Direct solution of uncertain bratu initial value problem

2019 ◽  
Vol 9 (6) ◽  
pp. 5075
Author(s):  
N. R. Anakira ◽  
A. H. Shather ◽  
A. F. Jameel ◽  
A. K. Alomari ◽  
A. Saaban
Keyword(s):  
Approximate Solution ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Approximate Analytical Solution ◽  
Order System ◽  
Direct Solution ◽  
Initial Value ◽  
First Order ◽  
Bratu Equation ◽  
Variation Iteration Method

<span>In this paper, an approximate analytical solution for solving the fuzzy Bratu equation based on variation iteration method (VIM) is analyzed and modified without needed of any discretization by taking the benefits of fuzzy set theory. VIM is applied directly, without being reduced to a first order system, to obtain an approximate solution of the uncertain Bratu equation. An example in this regard have been solved to show the capacity and convenience of VIM.</span>

scholarly journals The nonlocal solvability conditions for a system of two quasilinear equations of the first order with absolute terms

2019 ◽  
Vol 21 (3) ◽  
pp. 317-328
Author(s):  
Marina V. Dontsova
Keyword(s):  
Cauchy Problem ◽  
Finite Number ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Local Solution ◽  
Quasilinear Equations ◽  
Solvability Conditions ◽  
First Order ◽  
The Cauchy Problem ◽  
Global Estimates

The Cauchy problem for a system of two first-order quasilinear equations with absolute terms is considered. The study of this problem’s solvability in original coordinates is based on the method of an additional argument. The existence of the local solution of the problem with smoothness which is not lower than the smoothness of the initial conditions, is proved. Sufficient conditions of existence are determined for the nonlocal solution that is continued by a finite number of steps from the local solution. The proof of the nonlocal resolvability of the Cauchy problem relies on original global estimates.

ON THE JOINT APPLICATION OF THE COLLOCATION BOUNDARY ELEMENT METHOD AND THE FOURIER METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTION IN FINITE CYLINDERS WITH SMOOTH DIRECTRICES

2021 ◽  
pp. 15-38
Author(s):  
D.Y. Ivanov ◽  
Keyword(s):  
Fourier Series ◽  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Initial Conditions ◽  
Fourier Method ◽  
Time Interval ◽  
Cylindrical Domain ◽  
Two Dimensional ◽  
Element Method

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .

Sign in / Sign up

Export Citation Format

Share Document