Linear systems of first order ordinary differential equations: fuzzy initial conditions

A set of first-order ordinary differential equations with initial conditions is derived for the exact, nonlinear, inextensional deformation of a loaded plate bounded by two straight edges and two curved ones. The analysis extends earlier approximate work of Mansfield and Kleeman, Ashwell, and Lin, Lin, and Mazelsky. For a plate clamped along one straight edge and subject to a force and couple along the other, there are 13 differential equations, but an independent set of 9 may be split off. In a subsequent paper, we consider alternate forms of these 9 equations for plates that twist as they deform. Their structure and solutions are compared to Mansfield’s approximate equations and particular attention is given to tip-loaded triangular plates.

Download Full-text

Dynamics of a Basketball Rolling Around the Rim

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2194073 ◽

2005 ◽

Vol 128 (2) ◽

pp. 359-364

Author(s):

C. Q. Liu ◽

Fang Li ◽

R. L. Huston

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Initial Conditions ◽

Equations Of Motion ◽

Dynamical Equations ◽

First Order ◽

Coaching Strategies ◽

Ball Motion

Governing dynamical equations of motion for a basketball rolling on the rim of a basket are developed and presented. These equations form a system of five first-order, ordinary differential equations. Given suitable initial conditions, these equations are readily integrated numerically. The results of these integrations predict the success (into the basket) or failure (off the outside of the rim) of the basketball shot. A series of examples are presented. The examples show that minor changes in the initial conditions can produce major changes in the subsequent ball motion. Shooting and coaching strategies are recommended.

Download Full-text

The numerical solution of implicit first order ordinary differential equations with initial conditions

The Computer Journal ◽

10.1093/comjnl/14.2.173 ◽

1971 ◽

Vol 14 (2) ◽

pp. 173-178 ◽

Cited By ~ 3

Author(s):

M. A. Wolfe

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Ordinary Differential Equations ◽

Initial Conditions ◽

First Order

Download Full-text

The numerical solution of systems of stiff ordinary differential equations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100049136 ◽

1974 ◽

Vol 76 (2) ◽

pp. 443-456

Author(s):

J. R. Cash

Keyword(s):

Differential Equations ◽

Numerical Solution ◽

Ordinary Differential Equations ◽

Initial Conditions ◽

Complete Solution ◽

Step Length ◽

Rate Of Change ◽

First Order ◽

Stiff Ordinary Differential Equations ◽

Iterative Procedures

AbstractAlgorithms are developed for the numerical solution of systems of first-order ordinary differential equations, the solutions of which have widely different rates of variation. The iterative procedures described use a step length of integration proportional to the rate of change of the required slowly varying solution in a region of integration, where either the transient components of the complete solution have become negligible compared with the chosen working accuracy or in a region where rapidly increasing components of the solution are theoretically possible but are made absent by the initial conditions. Several numerical examples are given to demonstrate the algorithms.

Download Full-text

L-Stable Block Hybrid Numerical Algorithm for First-Order Ordinary Differential Equations

Journal of the Nigerian Society of Physical Sciences ◽

10.46481/jnsps.2020.108 ◽

2020 ◽

pp. 160-165

Author(s):

B. I. Akinnukawe ◽

K. O. Muka

Keyword(s):

Differential Equations ◽

Ordinary Differential Equations ◽

Hybrid Methods ◽

Initial Conditions ◽

Multistep Method ◽

Stability And Convergence ◽

First Order ◽

One Step ◽

Grid Points ◽

The Stability

In this work, a one-step L-stable Block Hybrid Multistep Method (BHMM) of order five was developed. The method is constructed for solving first order Ordinary Differential Equations with given initial conditions. Interpolation and collocation techniques, with power series as a basis function, are employed for the derivation of the continuous form of the hybrid methods. The discrete scheme and its second derivative are derived by evaluating at the specific grid and off-grid points to form the main and additional methods respectively. Both hybrid methods generated are composed in matrix form and implemented as a block method. The stability and convergence properties of BHMM are discussed and presented. The numerical results of BHMM have proven its efficiency when compared to some existing methods.

Download Full-text