Question

What is the basis of energy balance method in numerical analysis? How does it
differ from the formal finite difference method using Taylor series
approximation? For a specified nodal network, will these two methods result in
the same or a different set of equations?

Answer 1

Answer:

This section briefly introduces energy balance method proposed by He [5]. In this method, a variational principle for the oscillation is established, then the corresponding Hamiltonian is considered from which the angular frequency can be easily obtained by Galerkin method.

Let us consider the motion of a general oscillator with the initial conditions in the form



+()=0,(0)=,′(0)=0,(2.1)

where

is the initial amplitude.

Its variational can be written as

()=

0/4

−1

2

2

+().(2.2)

Here

=2/

is the period of the nonlinear oscillation and

∫()=()

.

The Hamiltonian of (2.1) can be written in the form:

1()=

2

2

+().(2.3.

A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.