Ordinary differential equations

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Software Design ◽  
Multiple Scales ◽  
Numerical Algorithms ◽  
Object Oriented ◽  
Step Size ◽  
Derivative System ◽  
Adaptive Step ◽  
Classical And Quantum Mechanics

This chapter surveys the ordinary differential equations (ODEs) that occur in classical and quantum mechanics, and describes both numerical algorithms and appropriate software design for solving them. Systems of ordinary differential equations, together with a few constants of integration, can in most cases be regarded as a means of defining a function (the “solution”). In this chapter, we develop an object-oriented architecture that applies integrators of the Runge-Kutta family to create these functions. Together with an automatic derivative system for generating partial derivatives from functions of one or more variables, the differential equation solver becomes a powerful tool for solving a variety of few-body problems in classical Hamiltonian systems. This chapter presents a blend of numerical algorithms, physics, and computing techniques. The phenomenon of energy drift is discussed and used to motivate symplectic solvers. Techniques such as adaptive step size and possible problems with stability and multiple scales are also discussed.

Nonlinear Vibrations of a Thin Plate Under Simultaneous Internal and External Resonances

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Usama H. Hegazy
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Thin Plate ◽  
Nonlinear Response ◽  
Phase Plane ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Response ◽  
The Stability ◽  
Parametric And External Excitations

The dynamic behavior of a rectangular thin plate under parametric and external excitations is investigated. The motion of the thin plate is modeled by coupled second-order nonlinear ordinary differential equations. Their approximate solutions are sought by applying the method of multiple scales. A reduced system of four first-order ordinary differential equations is determined to describe the time variation of the amplitudes and phases of the vibration in the horizontal and vertical directions. The steady-state response and the stability of the solutions for various parameters are studied numerically, using the frequency-response function and the phase-plane methods. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. Moreover, the chaotic motion of the thin plate is found by numerical simulation.

scholarly journals A Quantitative Comparison of Numerical Method for Solving Stiff Ordinary Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
S. A. M. Yatim ◽  
Z. B. Ibrahim ◽  
K. I. Othman ◽  
M. B. Suleiman
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Computation Time ◽  
Quantitative Comparison ◽  
Initial Value ◽  
Step Size ◽  
Stiff Ordinary Differential Equations ◽  
Block Methods ◽  
Variable Step

We derive a variable step of the implicit block methods based on the backward differentiation formulae (BDF) for solving stiff initial value problems (IVPs). A simplified strategy in controlling the step size is proposed with the aim of optimizing the performance in terms of precision and computation time. The numerical results obtained support the enhancement of the method proposed as compared to MATLAB's suite of ordinary differential equations (ODEs) solvers, namely, ode15s and ode23s.

scholarly journals Step Change Strategies for Multistep Methods

1985 ◽  
Vol 14 (196) ◽  
Author(s):  
Ole Østerby
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multistep Methods ◽  
Step Change ◽  
Change Strategies ◽  
Step Size ◽  
Step Method ◽  
Adams Methods

When a system of ordinary differential equations is solved using a step-by-step method it is often desirable to change the step size during the course of the integration. We show that the commonly used formulas for calculating the new step sizes are not correct for multistep methods and we derive correct formulas for Adams methods.

One-leg Integration of Ordinary Differential Equations with Global Error Control

2005 ◽  
Vol 5 (1) ◽  
pp. 86-96 ◽  
Author(s):  
Gennady Yu. Kulikov ◽  
Sergey K. Shindin
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Error Control ◽  
Size Control ◽  
Adaptive Algorithms ◽  
Reasonable Accuracy ◽  
Step Size ◽  
Step Size Control ◽  
The Family

AbstractIn this paper we study the family of one-leg two-step second-order methods developed by Dahlquist et al., which possess the A-stability and G-stability properties on any grid. These methods are implemented with the local-global step size control derived by Kulikov and Shindin with the aim to obtain automatically the numerical solution with any reasonable accuracy set by the user. We show that the error control is more complicated in one-leg methods, especially when applied to stiffproblems. Thus, we adapt our local-global step size control for the methods indicated above and test these adaptive algorithms in practice.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

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