The Numerical Solution of a Challenging Class of Turning Point Problems

2003 ◽  
Vol 25 (3) ◽  
pp. 927-941 ◽  
Author(s):  
Ping Lin ◽  
R. E. O'Malley
Keyword(s):  
Numerical Solution ◽  
Turning Point

The numerical solution for problems of turning point without resonance

1985 ◽  
Vol 6 (5) ◽  
pp. 439-446 ◽  
Author(s):  
Lin Peng-cheng ◽  
Yan Peng-xiang
Keyword(s):  
Numerical Solution ◽  
Turning Point

Pendulum Motion Damped by Speed-Independent Friction

2021 ◽  
Vol 03 (03) ◽  
pp. 2150008
Author(s):  
Carl E. Mungan
Keyword(s):  
Numerical Solution ◽  
Turning Point ◽  
Analytic Solution ◽  
Angular Position ◽  
Rolling Friction ◽  
Pendulum Motion ◽  
Circular Trajectory ◽  
Air Drag ◽  
Small Block

A pendulum without a supporting string or rod is obtained if a small block or marble is released at the rim of a spherical bowl or cylindrical half-pipe. This setup also applies to the familiar loop-the-loop demonstration. However, the bob will then experience sliding or rolling friction, which is speed independent in contrast to the linear or quadratic air drag which is more commonly used to model damping of oscillators. An analytic solution can be found for the speed of the bob as a function of its angular position around the vertical circular trajectory. A numerical solution for the time that the object takes to move from one turning point to the next shows that it is smaller than it would be in the absence of friction.

Numerical solution of a turning point problem in elasticity

CALCOLO ◽  
1989 ◽  
Vol 26 (1) ◽  
pp. 93-102
Author(s):  
A. Schiaffino ◽  
V. Valente
Keyword(s):  
Numerical Solution ◽  
Turning Point ◽  
Point Problem

Correction for the Uniform WKB Wave Functions and the WKB Phase Shifts

1974 ◽  
Vol 52 (18) ◽  
pp. 1805-1815 ◽  
Author(s):  
Byung Chan Eu ◽  
Hervé G. Guerin
Keyword(s):  
Numerical Solution ◽  
Present Method ◽  
Turning Point ◽  
Numerical Test ◽  
Equation Of Motion ◽  
Phase Shifts ◽  
Wave Functions ◽  
Wkb Approximation ◽  
Wkb Method ◽  
Connection Formulas

A method of improving the uniform WKB solution for single turning point problems is discussed and the correction formulas for the WKB phase shifts are obtained. The method involves the equation of motion which is in a form similar to that used by Fröman and Fröman to obtain the connection formulas in the WKB approximation. A sequence of solutions to the equation of motion is obtained in a remarkably compact form—and in a novel way—by utilizing the involutionality of certain 2 × 2 matrices appearing in the equations. A numerical test of the result thus obtained is presented in comparison with the corresponding numerical solution result. Unlike the ordinary WKB method, the present method is completely free of the connection formulas of the WKB approximation.

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