Один подход к численному решению основного уравнения магнитостатики для конечного цилиндра в произвольном внешнем поле

2021 ◽  
pp. 22-34
Author(s):  
В.В. Дякин ◽  
О.В. Кудряшова ◽  
В.Я. Раевский
Keyword(s):  
Magnetic Field ◽  
Hypergeometric Functions ◽  
Three Dimensional ◽  
Legendre Functions ◽  
Arbitrary Configuration ◽  
Special Cases ◽  
Linear Integral ◽  
Magnetostatic Equation ◽  
Homogeneous Cylinder ◽  
Linear Integral Equations

The magnetostatics direct problem of calculating the resulting magnetic field strength from a homogeneous cylinder of finite dimensions placed in an external magnetic field of arbitrary configuration is considered. With the help of sufficiently voluminous analytical transformations using the basic properties of hypergeometric functions and Legendre functions, the solution of the basic three-dimensional magnetostatic equation for this configuration is reduced to solving of a certain number of systems of three one-dimensional linear integral equations. A simplified form of these systems for special cases of a constant external field and the resulting field on the cylinder axis is obtained.

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

A path integral approach to disordered systems

1969 ◽  
Vol 309 (1499) ◽  
pp. 457-472 ◽  
Keyword(s):  
Disordered Systems ◽  
Three Dimensional ◽  
Delta Function ◽  
Exact Expression ◽  
Integral Approach ◽  
Disordered System ◽  
Feynman Measure ◽  
Screened Coulomb Potential ◽  
Linear Integral ◽  
Linear Integral Equations

The exact expression for the average propagator of a completely disordered system is evaluated by using a method of expansion in terms of cumulants defined over the Feynman measure. The expansions are evaluated for a one-dimensional model of delta function potentials and for a three-dimensional screened Coulomb potential to give the asymptotic form of the density of states for | E | →∞. Using a functional Taylor expansion it is shown that the average propagator may be expanded to give approximate non-linear integral equations for either the average propagator or average Green function. Using the cumulant formulation it is shown that an effective quantum potential may be defined in terms of which the propagator may be calculated.

scholarly journals Solving Linear Volterra – Fredholm Integral Equation of the Second Type Using Linear Programming Method

2020 ◽  
Vol 17 (1(Suppl.)) ◽  
pp. 0342
Author(s):  
Muna Mansoor Mustafaf
Keyword(s):  
Linear Programming ◽  
Integral Equations ◽  
New Technique ◽  
Fredholm Integral Equations ◽  
Quadrature Methods ◽  
Special Cases ◽  
A New Technique ◽  
Linear Integral ◽  
Romberg Integration ◽  
Linear Integral Equations

In this paper, a new technique is offered for solving three types of linear integral equations of the 2nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree  and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those methods is produced. Finally, for more explanation, an algorithm is proposed and applied for testing examples to illustrate the effectiveness of the new technique.

On water waves produced by ground motions

1983 ◽  
Vol 126 ◽  
pp. 27-58 ◽  
Author(s):  
Pierre C. Sabatier
Keyword(s):  
Ground Motion ◽  
Water Waves ◽  
Ground Motions ◽  
Three Dimensional ◽  
Surface Displacement ◽  
Physical Parameters ◽  
Response Pair ◽  
Small Times ◽  
Linear Integral ◽  
Linear Integral Equations

A linear and irrotational model is constructed to represent the formation of water waves by ground motions of a sloping bed. A survey of the constant depth case, given first, helps in understanding the mechanism of formation, and, in this oversimplified case, wave propagation away from a source, which is usually very asymmetric. The importance of asymmetry, which may produce trapped waves, is illustrated by an estimate of the propagation in a three-dimensional case. The formation of waves by a ground motion on a slope is then studied in detail. The problem is reduced to linear integral equations of the first kind. Using an inversion technique one constructs a source–response pair in which the source is ‘δ-like’ and the response is close to that which would be found if the depth was constant around the source. A general approximate solution is then derived, in both the two-dimensional and three-dimensional cases. Results for the sloping-bottom case are given for small times. They give initial values of surface displacement. They also enable one to determine the important physical parameters in the ground motion and to evaluate the efficiency of wave production.

Multiple characteristics of three‐dimensional radiative Cross fluid with velocity slip and inclined magnetic field over a stretching sheet

Heat Transfer ◽  
2021 ◽  
Author(s):  
Hafiz Abdul Wahab ◽  
Syed Zahir Hussain Shah ◽  
Assad Ayub ◽  
Zulqurnain Sabir ◽  
Muhammad Bilal ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Stretching Sheet ◽  
Three Dimensional ◽  
Velocity Slip ◽  
Inclined Magnetic Field ◽  
Multiple Characteristics

Magneto convective flow of casson nanofluid due to Stefan blowing in the presence of bio-active mixers

2021 ◽  
pp. 239779142110166
Author(s):  
Venkatesh Puneeth ◽  
Sarpabhushana Manjunatha ◽  
Bijjanal Jayanna Gireesha ◽  
Rama Subba Reddy Gorla
Keyword(s):  
Magnetic Field ◽  
Brownian Motion ◽  
Convective Flow ◽  
Three Dimensional ◽  
Induced Magnetic Field ◽  
Density Profiles ◽  
Casson Nanofluid ◽  
And Brownian Motion ◽  
Similarity Transformations ◽  
The Mathematical Model

The induced magnetic field for three-dimensional bio-convective flow of Casson nanofluid containing gyrotactic microorganisms along a vertical stretching sheet is investigated. The movement of these microorganisms cause bioconvection and they act as bio-active mixers that help in stabilising the nanoparticles in the suspension. The two forces, Thermophoresis and Brownian motion are incorporated in the Mathematical model along with Stefan blowing. The resulting model is transformed to ordinary differential equations using similarity transformations and are solved using [Formula: see text] method. The Velocity, Induced Magnetic field, Temperature, Concentration of Nanoparticles, and Motile density profiles are interpreted graphically. It is observed that the Casson parameter decreases the flow velocity and enhances the temperature, concentration, and motile density profiles and also it is noticed that the blowing enhances the nanofluid profiles whereas, suction diminishes the nanofluid profiles. On the other hand, it is perceived that the rate of heat conduction is enhanced with Thermophoresis and Brownian motion.

scholarly journals The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
P. Kim ◽  
R. Jorge ◽  
W. Dorland
Keyword(s):  
Magnetic Field ◽  
Original Work ◽  
Three Dimensional ◽  
Radial Distance ◽  
Analytical Form ◽  
One Dimensional ◽  
Modern Notation ◽  
Magnetic Well ◽  
First Time ◽  
Dimensional Integral

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

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