scholarly journals Mathematical modeling of a field emitter with a hyperbolic shape

2020 ◽  
Vol 16 (3) ◽  
pp. 238-248
Author(s):  
Nickolay V. Egorov ◽  
◽  
Ekaterina M. Vinogradova ◽  
Keyword(s):  
Linear Equations ◽  
Anode Surface ◽  
Legendre Functions ◽  
Electrostatic Potential Distribution ◽  
Constant Coefficients ◽  
Expansion Coefficients ◽  
Diode System ◽  
Prolate Spheroidal Coordinates ◽  
Variable Separation Method ◽  
Spheroidal Coordinates

This article is devoted to modeling a field emission diode system. The emitter surface is a hyperboloid of rotation. The anode surface is a part of the hyperboloid of rotation, in a particular case, a circular diaphragm. A boundary value problem is formulated for the Laplace equation with non-axisymmetric boundary conditions of the first kind. A 3D solution was found by the variable separation method in the prolate spheroidal coordinates. The electrostatic potential distribution is presented in the form of the Legendre functions expansions. The calculation of the expansion coefficients is reduced to solving a system of linear equations with constant coefficients. All geometric dimensions of the system are the parameters of the problem.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

Convective and Microwave Drying of Prolate Spheroidal Solids: Modeling and Simulation

2019 ◽  
Vol 391 ◽  
pp. 233-238
Author(s):  
E. Gomes da Silva ◽  
E. Santana de Lima ◽  
W.M. Paiva Barbosa de Lima ◽  
A.G. Barbosa de Lima ◽  
J.J. Silva Nascimento ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Heat Balance ◽  
Diffusion Theory ◽  
Microwave Drying ◽  
Drying Process ◽  
Balance Equations ◽  
Prolate Spheroidal ◽  
Moisture Migration ◽  
Prolate Spheroidal Coordinates ◽  
Spheroidal Coordinates

This paper focuses some fundamental aspects of combined convective and microwave drying of prolate spheroidal solids. A transient mathematical modeling based on the diffusion theory (mass and heat balance equations) written in prolate spheroidal coordinates was derived and the importance of this procedure on the analysis of the drying process of wet porous solid, is also presented. Results pointed to the behavior of the moisture migration and heating of the solid with different aspect ratio. Solids with higher area/volume relationships dry and heat faster.

The Hydrogen Molecular Ion With Time-Dependent Magnetic Field Strength as Control

2014 ◽  
Author(s):  
Katherine A. Kime
Keyword(s):  
Magnetic Field ◽  
Field Strength ◽  
Magnetic Field Strength ◽  
Terminal State ◽  
Time Dependent ◽  
Initial State ◽  
Hydrogen Molecular Ion ◽  
Molecular Ion ◽  
Prolate Spheroidal Coordinates ◽  
Spheroidal Coordinates

We consider the hydrogen molecular ion with time-dependent magnetic field strength. We discretize the corresponding Schroedinger equation with the Hamiltonian written in prolate spheroidal coordinates. We formulate a control problem and give an example of steering a restricted initial state to a restricted terminal state.

On a Numerical Method for Solution of the Mathieu-Hill Type Equations

1980 ◽  
Vol 102 (3) ◽  
pp. 619-626 ◽  
Author(s):  
A. Midha ◽  
M. L. Badlani
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Initial Conditions ◽  
Linear Equations ◽  
Solution Process ◽  
Constraint Equations ◽  
Step Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Simultaneous Linear Equations

This paper presents a computer-programmable numerical method for the solution of a class of linear, second order differential equations with periodic coefficients of the Mathieu-Hill type. The method is applicable only when the initial conditions are prescribed and the solution is not requiried to be periodic. The solution is facilitated by representing the coefficient functions as a sum of step functions over corresponding sub-intervals of the fundamental interval. During each sub-interval, the solution form is assumed to be that of the differential equations with “constant” coefficients. Constraint equations are derived from imposing the conditions of “compatibility” of response at the end nodes of the intermediate sub-intervals. This set of simultaneous linear equations is expressed in matrix form. The matrix of coefficients may be represented as a triangular one. This form greatly simplifies the solution process for simultaneous equations. The method is illustrated by its application to some specific problems.

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