Linear Equations with Constant Coefficients

2018 ◽  
pp. 88-104
Keyword(s):  
Linear Equations ◽  
Constant Coefficients

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.

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.

HIGHER-ORDER GEODESIC DEVIATIONS AND THE CALCULUS OF RELATIVISTIC ORBITS

2002 ◽  
Vol 17 (20) ◽  
pp. 2756-2756 ◽  
Author(s):  
R. COLISTETE
Keyword(s):  
Harmonic Oscillator ◽  
Taylor Series ◽  
Gravitational Radiation ◽  
Circular Orbit ◽  
Linear Equations ◽  
Higher Order ◽  
Test Particles ◽  
Oscillator Type ◽  
Constant Coefficients ◽  
Systems Of Linear Equations

Starting with an exact and simple geodesic, we generate approximate geodesics1,2 by summing up higher-order geodesic deviations in a fully relativistic scheme. We apply this method to the problem of orbit motion of test particles in Schwarzschild3 and Kerr metrics; from a simple circular orbit as the initial geodesic we obtain finite eccentricity orbits as a Taylor series with respect to the eccentricity. The explicit expressions of these higher-order geodesic deviations are derived using successive systems of linear equations with constant coefficients, whose solutions are of harmonic oscillator type. This scheme is best adapted for small eccentricities, but arbitrary values of M/R. We also analyse the possible application to the calculation of the emission of gravitational radiation from non-circular orbits around a very massive body3.

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